The measurement of
Aristoxenus's Divisions of the Tetrachord
© 1999 by Joe Monzo

PREFACE

I have been pondering the mathematics of Aristoxenus's tetrachord intervals for a long time. This is the most rigorous mathematical analysis I have been able to make of them.

Aristoxenus described the various different sizes of intervals in his classification of shades and genera of tetrachord scales. But he never used any mathematics in his theory beyond simple geometric subdivisions of the tone. This was, in fact, the aspect of his music-theory that was distinctively new, a result of the formulation of classic geometry around this time by Euclid. He adamantly maintained that all larger musical intervals could and should be determined soley by ear. Both of these basic tenets were in sharp contrast to Pythagorean theory and other rational harmonic theories.

Aristoxenus's theory was forgotten in Western Europe after the Germanic invasions (400s AD), then rediscovered during the Renaissance. The earliest surviving manuscript of his treatise was written in Constantinople [Istanbul] around 1150, so his theory seems to have remained known the whole time in the Greek-speaking areas.

When he was rediscovered by the Italians around 1450, the idea of 12-tone equal-temperament (the oridinary scale in everyday use today) was a topic of much debate, and many theorists who advocated that tuning interpreted Aristoxenus's approximations of small intervals as a justification for it. They were wrong, but the idea has persisted for over 500 years.

The key thing I discovered about Aristoxenus's theory is that he was dividing *two* different sizes of semitone: the limma (about 90 cents) and the apotome (about 114 cents). Therefore, his smaller divisions (1/4-tones, etc.) also come in two sizes. No one else ever realized this; they all assumed that his theory used one size of semitone and therefore had only one size of each of the small intervals. But I'm certain that his 'basic scale' used the usual (in his time) Pythagorean tuning, and thus had two different semitones.

There are features to Aristoxenus's theories that go far beyond mere description of intervals, and indeed features that go beyond music alone and into philosophy; some of this has been treated in Barker 1978 and Litchfield 1988. The 'General Introduction' to Barker 1989 (p 1-27) provides a good overview of all the musical aspects of Aristoxenus's theory. I will focus here only on his intervallic divisions of the tetrachord.

Thanks to Paul Erlich for asking a few simple questions which inspired me to finally put all of this together.

NOTE ON MATHEMATICAL NOTATIONAL CONVENTIONS The caret sign (^) indicates exponentiation: thus 2^2 means 'two squared' = 4, 3^3 = 'three cubed' = 9, etc. Whenever Aristoxenus uses terminology which implies an equal division of an interval, that will be expressed mathematically by raising the interval to the power of that division: thus, 'half a tone' will be expressed by raising the 9/8 'tone' to the power 1/2: (9/8)^(1/2), etc. In the tabulations of intervals, the exact mathematical calculation will be on the left, and the geometric expression of it (where '^' is generally rendered as 'times', 'times' rendered as 'plus', 'divided' rendered as 'minus', and so on) on the right. NOTE ON CITATIONS OF ARISTOXENUS'S BOOK As with most other recent authors, my citations follow Meibom's page and line numbers, which run consecutively from Book 1 thru Book 2; thus 1.29 (the last page of Book 1) is followed by 2.30 (the first page of Book 2). They simply appear in square brackets at the appropriate point in my text, in the format: book.page.line. My transliterations of the Greek are based on the text given in Macran 1902 (and I assume full responsibility for their shortcomings); the text has been arranged according to Meibom's page and line breaks, and the punctuation is from Meibom. Text given by Meibom but omitted by Macran is indicated in {}. The English translations are from Barker 1989 unless otherwise indicated; there are bound to be discrepancies with Macran's text, because for his text Barker used da Rios 1954, a source unavailable to me.

ABOUT ARISTOXENUS'S TREATISE

Aristoxenus's treatise _Harmonika stoicheia_ (usually known by its Latin name _Elementa harmonica_, 'the Elements of Harmonics') appears to have been assembled from two different treatises written at different times (see the comments in the introduction on Aristoxenus in Barker 1989, p 119-125), and it seems to me to have been drawn from notes that were taken down by students as he lectured.

The treatise is divided into three books, of which both 1 and 2 cover more-or-less the same material presented in the same order, with some differences in both content and presentation. Accordingly, I will frequently cite parallel passages from both books of the treatise simultaneously; I have not had the need to quote from Book 3.

[Barker 1989, p 122]
Book II ... uses as a key concept the notion of melodic 'function' (_dynamis_), implying in particular that notes are to be understood as _dynameis_, not simply as pitches, and that the character of melodic relations in general is to be understood 'dynamically', by reference to the melodic role or character in which they are perceptually grasped, not just through representations of the sizes of intervals they involve. Book I says nothing of _dynamis_, and pursues its quantitative analyses without any hint that they will need reinterpretation. It explicitly defines a note as a pitch. The books also differ markedly in style, the second being much more leisurely, reflective and methodologically self-conscious. It seems to me that there are strong grounds for supposing that Books I and II are from different treatises, the first having belonged, perhaps, to an earlier and rather less sophisticated attempt at the project. Either book could have had a continuation of roughly the sort we find in Book III, but our Book III must in fact go with Book II, if only in view of its liberal use of the concept of _dynamis_.

Aristoxenus was alive around the mid-300s BC in Athens, but the earliest surviving manuscript of his treatise, known as 'M', was written in Constantinople [now Istanbul] around 1150, in Greek.

A synopsis of Aristoxenus's theory was given in the _Eisagoge_ of Cleonides around 100 AD. Note, however, that Cleonides interpreted Aristoxenus's intervals as specific numerical quantities that I think are incorrect. Cleonides's interpretation has been widely misconstrued as the actual theory of Aristoxenus.

The classic early printing of Aristoxenus is Meibom 1652, which presents the Greek text and parallel Latin translation, and includes several of the other famous Greek music-theory treatises.

Barker 1989 [p 125] calls Laloy 1904 'the classic work on' Aristoxenus. The standard English translation of Aristoxenus has long been Macran 1902, and the most recent one is in Barker 1989, which is comprehensively annotated and was my main source.

CONCEPTS BASIC TO ARISTOXENUS'S THEORY

Aristoxenus sought to separate the study of music from that of both physics and mathematics. He refused to make use of any ratios in his treatise, distinctly contra the Pythagoreans, and he also spoke of the raising and lowering of pitch in terms of tension and relaxation, which in his day would admit of no exact numerical measurement, rather than in terms of string-length, from which ratios could be calculated.

[Landels 1999, p 131]
... ancient acousticians never solved the riddle of the relationship between tension and pitch. ... they seem not to have attempted to assign a numerical value to the tension (in terms of units of weight), or to relate such a numerical value to the pitch. In fairness it should be said that the true formula, by which the pitch varies according to the square of the tension, is not one that would easily be arrived at by chance.

Aristoxenus therefore chose to use the newly formulated methods of geometry to pinpoint examples of his divisions of the tetrachord.

Thus, his music-theory was radically different from anything that had come before in ancient Greece.

As Andrew Barker says [Barker 1989, p 119-120]:

In harmonics his work was truly revolutionary, as he would have been the first to agree. It sets aside the researches of the Pythagoreans as irrelevant and misdirected, and seeks in effect to establish a wholly new science that will study music on the basis of principles intrinsic to itself, not borrowed from physics or mathematics.

The lack of rational accuracy is an intentional and fundamental aspect of Aristoxenus's theory. Rapid developments had been taking place in Greek musical practice in his day, and he wished to describe the wide variety of intervals, which could not be done numerically with the mathematics then available. Scientific music-theory had up to that time been based primarily on arithmetic manipulation of integer ratios as described by the followers of Pythagoras [see Burkert 1972].

Another school of music-theorists called the 'Harmonists' apparently had as their goal the illustration of all the different scales thru the use of a small 'basic unit' of intervallic measurement, with which they diagrammed the scales next to one another within the same overall pitch-space, presenting a geometric view based on a common metric, in a diagram which Aristoxenus calls _katapyknosis_. But Aristoxenus criticizes their methods as not portraying the pitches according to their musical *function*.

His main goal is to assemble all the different tetrachordal divisions into a comprehensive system which allows for modulation between various different scales, which was something that had become an important element in Greek music of his time.

Geometry was a new concept in Greek mathematics and Aristoxenus made use of it in his descriptions because it enabled him to illustrate irrational divisions of musical pitch-space. Thanks to more recent developments in mathematics in modern times, we are now able, thru the manipulation of roots and exponents, to probe further into the calculation of his intervals.

It seems clear to me that Aristoxenus still intended for his audience to understand certain of the harmonic relationships as ratios, even if the others must be rendered in another way.

[2.55.2-12]
Epei de ton diastematikon megethon ta men ton symphonon etoi holos ouk | echein dokei topon all heni megethei horisthai, e pantelos akariaion tina, ta de ton diaphonon pollo hetton touto peponthe kai dia tautas tas aitias poly mallon tois ton symphonon megethesi pi|steiei he aisthesis e tois ton diaphonon; akribestate d' an eie diaphonou diastematos lepsis he dia symphonias.

Of the magnitudes of intervals, those of the concords appear to have either no range of variation at all, being determined to a single magnitude, or else a range which is quite indiscernible, whereas those of the discords possess this quality to a much smaller degree. Hence perception relies much more confidently on the magnitudes of concords than on those of discords. The most accurate way of constructing a discordant interval will therefore be by means of concords.

(We will examine his method of 'Tuning by Concords' shortly.)

This certainly means that the 'octave', '4th', and '5th' are to be determined by ear, which means that they will be measured as their usual Pythagorean ratios 2:1, 4:3, and 3:2, and the intervals smaller than the '4th' (the 'discords') calculated in relation to them, by means of a series of alternating '4ths' and '5ths'.

Because of his adamant stance against using ratios (or string-lengths, from which ratios can be easily calculated), it is quite difficult to ascertain Aristoxenus's interval measurements in a numerically exact way. Clues must be found by paying close attention to his terminology and his described methods of tuning. Our findings will be applied to his ideas about the ranges of the _loci_ for the two moveable notes.

THE BASIC TETRACHORD, AND THE NOTE NAMES

Aristoxenus explains [1.22.5-21 and 2.46.20-22] that he will select one tetrachord taken from the whole system of notes, with which he will illustrate the intervals of various tetrachordal divisions. He describes his intervals in terms of the tetrachord _meson_, spanning a 'perfect 4th' below _mese_, which was the reference pitch in Greek theory, and explains that for any given genus or shade, the same intervals should appear in all other tetrachords of the complete system.

The _tetrachord meson_ has these note-names, from highest to lowest:

1) mese
2) lichanos
3) parhypate
4) hypate

The interval between _mese_ and _hypate_ is, as always in Greek theory, a 'perfect 4th'. These two notes are both fixed in pitch, whereas Aristoxenus states that the _lichanos_ and _parhypate_ are both moveable to various postions in the tetrachord, within their respective ranges [1.22-23, 2.46-47].

The various intervals resulting from these moveable notes determined the different _genera_ of ancient Greek theory, of which three had been acknowledged and described by Archytas [Barker 1989, p 46-52]: the _diatonic_ ('thru the tones'), the _chromatic_ ('thru the colors' or 'shades'), and the _enharmonic_ ('thru proper attunement').

Aristoxenus, no doubt trying to accomodate the proliferation of widely divergent varieties of genera used in practice in his day, divided the 'diatonic' and 'chromatic' genera further into 'shades' or 'colors':

[2.50.25...] treis de chroma-[26]tikai, he te tou malakou chromatos, [27] kai he tou hemioliou, kai he tou toniaiou.

Three divisions [of the tetrachord] are chromatic, those of the soft, the hemiolic and the tonic chromatic.

. . . . . . . .

[2.51.22...] Eisi de duo diatonou di-[23]aireseis, he te tou malakou, kai he tou synto-[24]nou.

There are two divisions of the diatonic, those of the soft diatonic and the tense.

(Barker and almost everyone else translate _malakon_ as 'soft', but I prefer 'relaxed', as it indicates the opposite of _syntonon_ 'tense', with the pair thus illustrating an important concept in Aristoxenus's theory.)

He stated further that these were only examples, and that an infinite variety of other shades was possible, including those for the 'enharmonic'.

Greek theorists reckoned their scales downward (the opposite of the way we do today), and the largest intervals were always at the top of the tetrachord, with the smallest at the bottom. The 'characteristic interval' of a tetrachord is the largest one (or the 'tone' in the case of the 'tense diatonic' genus).

LOCATION OF THE FIXED NOTES

We will give a ratio of 1:1 to _mese_. As we go down in pitch the ratios will fall between the octave bounded by 1 at the top and 1/2 at the bottom. The cents value of _mese_ will always be 0 cents, with cents values of intervals below _mese_ given as negative values. Rather than relating Aristoxenus's tetrachord pitches to our modern ideas of a scale, this will give us a clear impression of the ancient Greek conception.

_Hypate_ is by definition a 'perfect 4th' below _mese_:

1/(4/3)  =  3/4  =  0.75  =  ~-498.045 cents.

Aristoxenus states at several places that 'the fourth is two and a half tones' [1.24.4-10, 2.46.2], and gives an elaborate 'proof' that this is so [2.56.14-58.4]. Litchfield addresses this claim:

[Litchfield 1988, p 64-65]
The crux of this proof is whether the fifth is a full perfect fifth. In concept, there is no reason to admit this interval as anything but a fifth; in practice, however, it does not sound like a fifth. Had Aristoxenus performed this proof, he would surely have heard the discrepancy, as would any other musicician used to manipulationg the monochord. Nevertheless, the discrepancy between theory and practice appears not to have bothered Aristoxenus. He changed nothing in his treatise because of it. ... Thus, the concept alone must have been Aristoxenus's concern, and he appears not as an empiricist but as a conceptualist or conceptual idealist.

I believe that this is an extremely important aspect of Aristoxenus's theory, and that it in fact implies that Aristoxenus tempered all of the 'concords' so that the final one would audibly be indistinguishable from the rest. I feel that it is presumptuous to assume that Aristoxenus would have been incapable of conceiving a tempered system, because the mathematics to approximate it had already been recorded 1000 years earlier by the Babylonians. If Aristoxenus did in fact conceive of his system as being a closed, tempered one, then 144-tET is indeed exactly what he was describing.

But let us proceed on the assumption that his 'concords' were the Pythagorean ratios. So without getting entangled into this argument, and considering Aristoxenus's claim that the 'discords' and their related 'remainders' could be found by means of concords (to be discussed shortly), we can here simply assume that he means that a '4th' may be divided into 2 Pythagorean 'tones' and 1 Pythagorean 'lesser semitone' or 'limma'.

Calculating these intervals downward from _mese_:

     ratio              decimal ratio    cents

1/(((9/8)^2)*(256/243))      0.75      ~-498.045

These notes may be rendered in letter-names which make them easier to understand to modern readers. We shall call _mese_ by the first letter 'A'; the _hypate_ will thus be the 'E' a 'perfect 4th' below 'A'.

THE _PYKNON_

The _pyknon_ indicates 'compression', and refers to the grouping together of small intervals at the bottom of the tetrachord.

[1.24.11-14]
Pyknon de legestho to ek duo diastematon synestekos ha syntethenta elatton diastema periexei tou deipomenou diastematos en toi dia tessaron.

Let us call _'pyknon'_ that which is composed of two intervals which, when put together, cover an interval smaller than that which makes up the remainder of the fourth [i.e., 'perfect 4th'].

. . . . . . .

[2.50.15...] Pyknon de legestho mechri toutou, [16] heos an en tetrachordo dia tessaron [17] symphonounton ton akron, ta duo dia-[18]stemata syntethenta, tou henos elatto [19] topon kateche.

Let us use the term '_pyknon_' for every case where, in a tetrachord whose extremes form the concord of a fourth, the two intervals put together occupy a smaller range than the one.

Or, in mathematical terms, if the _pyknon_ is composed of intervals x and y: x * y < (4/3)^(1/2).

[1.24.15-17] Touton houtos horismenon pros toi baryteroi ton menonton phthongon eilephtho to elachiston pyknon;

let us take the smallest _pyknon_, placed next to the lower of the fixed notes [i.e., _hypate_].

This indicates that the _pyknon_ occurs at the bottom of the tetrachord, with the larger interval at the top.

ARISTOXENUS'S BASIC UNITS OF MEASUREMENT

The 'tone' and its subdivisions

Aristoxenus defines his basic units of measurement:

[1.21.22-23] Esti de tonos he ton proton symphonon kata megethos diaphora.

The tone is the difference in magnitude between the first two concords.

. . . . . .

[2.45.35] Tonos d' estin ho to dia pente [2.46.1] tou dia tessaron meixon;

The tone is that by which the fifth is greater than the fourth.

That is, (3/2)/(4/3) = 9/8 = ~203.910 cents. This is the standard Pythagorean definition of the 'tone'. Aristoxenus continues:

[1.21.24-30] Diaireistho d' eis treis diaireseis; meloideistho gar | autou to te hemisu kai to triton meros kai tetarton; ta de touton elattona diastemata panta esto ameloideta. Kaleistho de to men elachiston diesis enarmonios elachiste, to d' echomenon | diesis chromatike elachiste, to de megiston hemitonion.

It [the tone] is to be divided in three ways, since the half, the third, and the quarter of it should be considered melodic. All intervals smaller than these are to be treated as unmelodic. Let the smallest of them be called the _least enharmonic diesis_, the next the _least chromatic diesis_, and the greatest the _semitone_.

. . . . . . .

[2.46.3-8] Ton de tou tonou meron melodeitai to hemisi, ho kaleitai hemitonion, kai to triton meros, | ho kaleitai diesis chromatike elachiste, kai to tetarton, ho kaleitai diesis enarmonios elachiste; toutou d' elatton ouden melodeitai diastema.

Of the parts of the tone the following are melodic: the half, which is called the semitone, the third part, which is called the least chromatic diesis, and the quarter, which is called the least enharmonic diesis. No interval smaller than that is melodic.

Aristoxenus thus calls every interval smaller than a quarter-tone 'unmelodic', meaning that they were unsuitable for use in melody (i.e., actual music), but were useful in analyzation of the various different shades and genera. He also takes pains to state that more than two small intervals cannot be sung in succession in proper melody, altho conceptually a tone may be divided into 3, 4, or more parts, and 'that from a purely abstract point of view there is no least interval' [2.46.19].

Dividing the 9/8 literally into 2, 3, and 4 equal parts, we get:

  ratio      decimal ratio   cents

(9/8)^(1/2)   ~1.060660172   ~101.955   'half-tone'      semitone
(9/8)^(1/3)   ~1.040041912    ~67.970   'third-tone'     least chromatic diesis
(9/8)^(1/4)   ~1.029883572    ~50.978   'quarter-tone'   least enharmonic diesis

There are other statements in Aristoxenus's treatise, however, which indicate that this general and simple description may not always be all there is to it.

'Tuning by Concords'

Without using ratios, Aristoxenus illustrates the method of determining 'discords' (i.e., any interval smaller than a 'perfect 4th') by ear, by means of successive '4ths' and '5ths':

[2.55.13-23] Ean men oun prostachthe pros to dothenti phthongo labein epi to bary to | diaphonon oion ditonon ... epi to oxy apo tou dothentos phthongou lepteon to dia tessaron, eit epi to bary to dia pente, eita palin epi to | oxy to dia tessaron, eit epi to bary to dia pente. kai houtos estai to ditonon apo tou lephthentos phthongou eilemmenon to epi to bary.

if we have the task of constructing from a given note a discord such as a ditone downwards ... we should construct from the given note a fourth upwards, from there a fifth downwards, then another fourth upwards, and then another fifth downwards. In this way the ditone downwards from the given note will have been constructed.

This may be represented in the following diagram (with ratios, letter-names, and cents for each note):

approx.
cents

 500      4/3 D ~498
 400        /  \
 300    (4:3)   \     32/27 C ~294
 200      /    (3:2)     /  \
 100     /        \   (4:3)  \
   0  1/1 A 0      \   /     (3:2)
-100                \ /        \
-200            8/9 G ~-204     \
-300                             \
-400                         64/81 F ~-408
-500
       n^0   3^-1  3^-2   3^-3   3^-4

This results in standard Pythagorean diatonic tuning, with the 'target' pitches giving, in addition to the 'limma', the following 'discord' intervals reckoned from _mese_ (1/1):

 ratio  decimal ratio    cents     interval from _mese_ (1/1)

 32/27     1.185185   ~ 294.135    'semiditone' ('minor 3rd') up
  8/9     ~0.888889   ~-203.910    'tone' down
 64/81     0.790123   ~-407.820    'ditone' down

[2.55.24-26] ean d' epi tounantion prostachthe labein to diapho|non, enantios poieteon ten ton symphonon lepsin.

And if the task is to construct the discord in the opposite direction, the concords should be constructed the opposite way round.

approx.
cents

 500  
 400                        81/64 C# ~408
 300                              /
 200             9/8 B ~204      / 
 100                /  \      (3:2)
   0  1/1 A 0      /  (4:3)    /
-100     \      (3:2)    \    /
-200    (4:3)    /        \  /
-300       \    /     27/32 F# ~-294
-400        \  /
-500     3/4 E ~-498    

       n^0   3^1    3^2   3^3    3^4

It should be quite clear that tuning by ear in this way will result in a chromatic scale (in the modern sense of the word 'chromatic') comprised of the usual Pythagorean intervals.

The 'semitone'

Aristoxenus goes on:

[2.55.27-31] Gignetai de kai ean apo symphonou diastematos to diaphonon aphairethe dia symphonias kai to loipon dia symphonias eilemmenon; aphaireistho | gar to ditonon apo tou dia tessaron symphonias;

Further, if a discord is subtracted from a concordant interval by means of concords, the remainder will also have been found by means of concords. For instance, let the ditone be subtracted by means of concords from the fourth.

He then describes the procedure diagrammed above, finding a ditone above the starting note, all over again, this time not from a single given note, but from a given '4th', and describes the difference between the last note found and the bottom note of the '4th' as a remainder found by means of concords:

[2.55.31-56.12] delon de hoti hoi ten hyperochen periechontes he to dia tessaron hyperechei tou ditonou dia symphonias esontai pros allelous eilemmenou; hypar||chousi men gar hoi tou dia tessaron horoi symphonoi; apo de tou oxyterou auton lambanetai phthongos symphonos epi to oxy dia tessaron, apo de tou le|phthentos heteros epi to bary dia pente, (eita palin epi to oxy dia tessaron,) eit' apo toutou heteros epi to bary dia pente. kai peproke to teleutaion symphonon epi ton exyterou ton hyperochen horizonton, host' einai pha|neron, hoti, ean apo symphonou diaphonon aphairethe dia symphonias, estai kai to loipon dia symphonias eilemmenon.

It is clear that the notes bounding the remainder by which the fourth exceeds the ditone have also been constructed, by means of concords, in their relation to one another. The notes bounding the fourth are themselves concordant. From the higher of them we find a note concordant at a fourth above, from that note another a fifth below, then again one a fourth above, and then from that note another a fifth below. The last of these concordant intervals falls on the higher of the notes bounding the remainder in question: and it is thus clear that if a discord is subtracted from a concord by means of concords, the remainder will also have been found by means of concords.

approx.
cents

 500        4/3 D ~498
 400          /  \
 300      (4:3)   \     32/27 C ~294
 200        /    (3:2)     /  \
 100       /        \   (4:3)  \
   0    1/1 A 0      \   /     (3:2)
-100       |          \ /        \
-200     (4:3)    8/9 G ~-204     \
-300       |                       \
-400       |                   64/81 F ~-408  \
-500  3/4 E ~-498               3/4 E ~-498   / 'remainder'

This remainder is none other than

(64/81)/(3/4)   1.053498   ~90.225	 cents

or the usual Pythagorean 'lesser semitone' or 'limma', which is ~11.730 cents smaller than (9/8)^(1/2). This difference, between the semitone derived by Aristoxenus's tuning methods and the one implied by his terminology of 'semitone' or 'half a tone', is a noticeable one to most ears, and should probably be resolved.

[2.50.22...] Mia men [23]oun ton diaipeseon estin enarmonios, [24]en he to men pyknon, hemitonion esti; to [25]de loipon, ditonon.

One of these divisions [of the tetrachord] is enharmonic, in which the pyknon is a semitone and the remainder a ditone.

. . . . . .

[1.23.12-22] Hoi men gar tei nun katechousei melopoiiai synetheis monon ontes eikotos ten ditonon lichanon exorizousi; | syntonoterais gar chrontai schedon hoi pleistoi ton nun. toutou d' aition to boulesthai glukainein aei, semeion d' hoti toutou stochazontai, malista men gar kai pleiston chronon en toi chromati dia|tribousin, hotan d' aphikontai pote eis ten harmonian, engus tou chromatos prosagousi synepispomenou tou ethous.

It is to be expected that those who are used only to the style of composition at present in vogue rule out the ditone _lichanos_, since most people nowadays use higher ones. The reason is their endless pursuit of sweetness: that this is their objective is shown by the fact that they spend most time and effort on the chromatic, whereas when they do occasionally come to the enharmonic they force it close to the chromatic, and the melody is correspondingly pulled out of shape.

This almost certainly means narrowing the 'characteristic interval' of the enharmonic tetrachord from a Pythagorean 81/64 ditone [= ~408 cents] to a 5/4 'major 3rd' [= ~386 cents], as had already been documented by Archytas. But it is obvious that in Aristoxenus's view, the 'proper' enharmonic tetrachord contains the ditone.

The 'limma' 256/243 would be the _pyknon_ remaining in the tetrachord after subtracting the 'ditone' characteristic interval. Aristoxenus's terminology concerning the enharmonic genus argues in favor of accepting the 'limma' as his definition of 'semitone'.

There is another interesting point in connection with the 'sweetening' of the enharmonic _lichanos_ to or towards the 5/4, and the 'semitonal' relationship of that note to others in the basic Pythagorean diatonic and chromatic genera.

      ratio        decimal ratio   cents     intervals from _mese_

(32/27)/(9/8)         1.0534979   ~ 90.225   semi-ditone - tone
(5/4)/(32/27)         1.0546875   ~ 92.179   'sweet' ditone - semi-ditone
(81/64)/(32/27)       1.0678711   ~113.685   apotome = true ditone - semi-ditone
(5/4)/(9/8)           1.1111111   ~182.404   smaller tone = 'sweet' ditone - tone
((5/4)/(9/8))^(1/2)   1.0540926   ~ 91.202   1/2 of smaller tone

The 32/27 'semi-ditone' (what we call the 'minor 3rd') is a limma [256/243 = ~90.225 cents] larger than the 9/8 tone. The 5/4 interval that Aristoxenus considered a 'sweetened' distortion of the enharmonic 'ditone' is a 5-limit 'lesser semitone' [135/128 = ~92.179 cents] larger than 32/27, whereas the true 81/64 'ditone' is an apotome [2187/2048 = ~113.685 cents] larger. 32/27 thus almost perfectly bisects the 5-limit 'smaller tone' [10/9 = ~182.404 cents] between 5/4 and 9/8; the exact midpoint is (10/9)^(1/2) = ~91.202 cents.

Even tho Aristoxenus described this 5/4 as being 'forced close to the chromatic', it still functioned as the *enharmonic* _lichanos_. Since the 9/8 and 32/27 measure the _lichanoi_ in the standard Pythagorean diatonic and chromatic genera, respectively, the similarity of 'step size' between these three notes would encourage Aristoxenus to think of the progression 9/8 : 32/27 : 5/4 as a series of nearly even 'semitones'.

Note, however, that the difference between the 'sweet' ditone [5/4] and the true ditone [81/64] is the syntonic comma [= ~21.506 cents].

There are ~9.4814124 [= log(9/8)/log((81/64)/(5/4))] of these commas in a tone, making the comma therefore between a 1/9- and a 1/10-tone. Aristoxenus explicity says in his descriptions that he expects us to perceive a 1/12-tone difference between _lichanoi_ of two of his genera, and implies that we should perceive a 1/24-tone difference between two others [see below].

We have already seen how he stipulated and preferred the 81/64 'ditone'. This blurring of distinctions between functionally-identical pitches was perhaps a further reason why Aristoxenus did not use rational values to locate his pitches. He said nothing else about the measurement of the enharmonic _lichanos_, beyond the fact that the upper boundary of its range is conjunct with the lowest chromatic _lichanos_, while he meticulously located the various chromatic _parhypatai_ at smaller distances. This seems to indicate that he needed to blur the distinctions between the two sizes of semitone in order for his calculations to come out right. We will explore Aristoxenus's 'semitone' further, below.

The 'quarter-tone' and 'enharmonic diesis'

Thus, the enharmonic _pyknon_ will be a 256/243 'semitone', and dividing this exactly in half gives (256/243)^(1/2) for the 'enharmonic diesis':

    ratio        decimal ratio   cents

(256/243)^(1/2)   ~1.026400479   ~45.112   enharmonic diesis.

This interval is ~5.865 cents smaller than (9/8)^(1/4), the ratio implied by use of the term 'quarter-tone'. This may or may not be a noticeable difference, depending on various factors.

Based on his meticulous description of how to tune the 'discords', and on the way he describes the 'proper' enharmonic genus, it makes the most sense to assume that Aristoxenus intends to imply the ratio 256/243 for 'semitone', and its geometrically-equal division, the irrational interval (256/243)^(1/2), for 'enharmonic diesis' or 'quarter-tone'.

The 'third-tone' and 'chromatic diesis'

One of Aristoxenus's most brilliant discoveries is that 1.5 * enharmonic diesis [= limma^(3/4)] is almost exactly the same as '1/3-tone' [= tone^(1/3)], with less than 1/3-cent difference between the two:

    ratio              decimal ratio   cents     cents difference

(9/8)^(1/3)              ~1.040041912   ~67.970
((256/243)^(1/2))^1.5    ~1.039860949   ~67.669    ~0.301

For now we may assume the irrational interval (9/8)^(1/3) [= ~67.97 cents] as the definition of both 'third-tone' and 'least chromatic diesis', altho this will cause problems upon closer analysis, which will be encountered later in this paper.

THE RANGES OF THE MOVEABLE NOTES

Aristoxenus describes the ranges for the two moveable notes in the tetrachord as follows:

The range of the _lichanoi_

[1.24.22-25] katholou gar barytatai men hai enarmonioi lichanoi esan, echomenai d' hai chromatikai, syn|tonotatai d' hai diatonoi.	taken overall, the enharmonic _lichanoi_ are the lowest, the chromatic next, and the diatonic the highest.
[1.22.27-30] Lichanou men oun esti toniaios ho sympas topos en hoi kineitai, oute gar elatton aphistatai meses toniaiou diaste|matos oute meizon ditonou.	The total range in which _lichanos_ moves is a tone, since it does not stand at less than the interval of a tone from _mese_, nor at a greater interval than a ditone.
[2.46.28-33] phainetai de syntonotate men einai lichanos he tonon apo meses apechousa, | poiei d' haute diatonon genos, barytate d' he ditonon, gignetai d' haute enarmonios; host' einai phaneron ek touton, hoti toniaios estin ho tes lichanou topos.	It appears that the highest _lichanos_ is that which lies at a tone from _mese_, and creates the diatonic genus; and the lowest is that at a ditone from _mese_, which belongs to the enharmonic. Hence it is clear that the range of _lichanos_ is a tone.

Since these passages make use only of the terms 'tone' and 'ditone' to describe the intervals, we may assume a fairly straightforward Pythagorean tuning:

total range of _lichanos_ = 9/8 tone highest lichanos = 9/8 below _mese_ lowest lichanos = 81/64 ditone below _mese_

    ratio       decimal ratio  cents

   1/(9/8)        ~0.888889   ~203.91   highest lichanos
   1/((9/8)^2)    ~0.790123   ~407.82   lowest lichanos


   The difference between these two is ~203.91 cents, or a
   Pythagorean 9/8 'tone'.

On the infinite variety of possible _lichanoi_:

[1.26.13-27] Noeteon gar apeirous ton arithmon tas lichanous; ohu gar | an steseis ten phonen tou apodedeigeenou lichanoi topou lichanos estai, diakenon d' ouden esti tou lichanoeidous topou oude toiouton hoion me dechesthai lichanon. Host' einai me peri mikrou ton | amphisbetesin; hoi men gar alloi diapherontai peri tou diastematos monon, hoion poteron ditonos estin he lichanos e syntonotera hos mias ouses enarmoniou; hemeis d' ou monon pleious en | hekastoi genei phamen einai lichanous mias alla kai prostithemen hoti apeiroi eisi ton arithmon.	For it must be understood that the lichanoi are unlimited in number. Wherever you arrest the voice in the range that accommodates the _lichanos_ will be a _lichanos_, and no place in the lichanos range is empty or incapable of receiving a lichanos. Hence the present controversy is of no little importance. Other people argue only about the interval in question, for instance whether the lichanos stands at a ditone or is higher, as if there were only one enharmonic lichanos. But we not only say that there is more than one _lichanos_ in each genus, but also add that they are unlimited in number.
[1.26.24-27] hemeis d' ou monon pleious en | [25] hekastoi genei phamen einai lichanous mias alla kai prostithemen hoti apeiroi eisi ton arithmon.	we not only say that there is more than one _lichanos_ in each genus, but also add that they are unlimited in number.
[1.26.14-18] ohu gar | an steseis ten phonen tou apodedeigeenou lichanoi topou lichanos estai, diakenon d' ouden esti tou lichanoeidous topou oude toiouton hoion me dechesthai lichanon.	Wherever you arrest the voice in the range that accommodates the _lichanos_ will be a _lichanos_, and no place in the lichanos range is empty or incapable of receiving a lichanos.

The range of the _parhypatai_

[1.23.27-29] let the range of ... _parhypate_ [be agreed to be] the smallest diesis, since it never comes closer to _hypate_ than a diesis and is never more than half a tone away from it.

[1.23.27-29] let the range of ... _parhypate_ [be agreed to be] the smallest diesis [i.e., '1/4-tone'], since it never comes closer to _hypate_ than a diesis [i.e., (256/243)^(1/2) above _hypate_] and is never more than half a tone away from it [i.e., either (9/8)^(1/2) or 256/243 above _hypate_].

Aristoxenus's loose terminology here, using 'half a tone' instead of 'semitone', admits of the possibility of interpreting 'half a tone' to be (9/8)^(1/2), rather than the 256/243 'limma' which is more likely. Let us examine both:

Based on '1/2-tone' and '1/4-tone':

     ratio                decimal ratio   cents


   (3/4)*((9/8)^(1/2))       0.795495   ~-396.090   highest parhypate
   (3/4)*((9/8)^(1/4))       0.772413   ~-447.067   lowest parhypate

   The difference between these two, which is the 'enharmonic diesis',
   is of course (9/8)^(1/4), or the literal '1/4-tone' of ~50.978 cents.

Based on a 'limma' semitone, and an 'enharmonic diesis' as half of that:

     ratio                decimal ratio   cents

   (3/4)*(256/243)           0.790123   ~-407.820   highest parhypate
   (3/4)*((256/243)^(1/2))   0.7698     ~-452.933   lowest parhypate

   The difference between these two is the diesis we calculated above
   of ~45.112 cents, which is exactly half of the Pythagorean 'lesser
   semitone' or 'limma'.

We thus have two mutually incompatible explanations of the range of _parhypate_, and must search further for clues to solving this dichotomy.

[1.23.30-24.1] the ranges do not overlap, but their point of conjunction is their limit, since when _parhypate_ and _lichanos_ reach the same pitch, as the one is tensed and the other relaxed, their ranges have their limit, the range below it being that of _parhypate_, the range above it that of _lichanos_.

[2.46.34-47.8] to de parypates diastema elatton men hoti ouk an genoito dieseos || enarmoniou phaneron, epeide panton ton melodoumenon elachiston esti diesis enarmonios; hoti de kai touto eis to diplasion auxetai, katanoeteon. hotan | gar epi ten auten tasin aphikontai he te lichnos aniemene kai he parypate epiteinomene, horizesthai dokei ekateras ho topos. host' einai phaneron,

That the interval between _parhypate_ and _hypate_ cannot be less than an enharmonic diesis is obvious, since the enharmonic diesis is the smallest of all melodic intervals: that it too will increase to double the size remains to be shown. When _lichanos_ in its descent and _parhypate_ in its ascent reach the same pitch, the range of each evidently arrives at its limit, so that it is plain that the range of _parhypate_ is not greater than the smallest diesis.

In view of these statements, as well as what he says about the proper enharmonic containing a 'ditone', with most contemporary musicians explaining it as being higher ('closer to the chromatic'), Aristoxenus must have intended the latter division presented in our table for the range of _parhypate_, as only this _parhypate_ reaches the pitch of the lowest _lichanos_ determined above and gives the true Pythagorean 'ditone' with the ratio 64/81.

Another important point which Aristoxenus stresses frequently is that the genera exhibit characteristic sounds which are capable of accomodating an *infinite* variety of intervals within their defined ranges. Thus:

- all _lichanoi_ from the highest diatonic to the lowest diatonic are to be considered 'diatonic',

- all _lichanoi_ from the lowest diatonic to the lowest chromatic are to be considered 'chromatic', and

- all _lichanoi_ below the lowest chromatic are to be considered 'enharmonic';

- all _parhypatai_ below the lowest chromatic are to be considered 'enharmonic', and

- all other _parhypatai_ are to be considered common to diatonic and chromatic.

He further explains on this last point that mixed genera sometimes occur, which make use of a diatonic _lichanos_ with a chromatic _parhypate_.

[2.52.5 ...] tettaron d' ouson parypaton, he [6] men enarmonios idia esti tes harmo-[7]nias; hai de treis koinai tou te diatonou, [8]kai tou chromatos.

Of the four _parhypatai_, the enharmonic one is peculiar to the enharmonic genus, while the other three are common to the diatonic and chromatic.

Mathiesen illustrates distinctions in Aristoxenus's terminology at this point, which have been missed by most editors and translators:

[1.26.30-33, text from Mathiesen 1976, eliminating emendations] parypates de duo eisi topoi - ho men koinos tou te diatonou kai tou chromatos, ho d' heteros idios tes harmonias - ; koinonei gar duo gene ton parypaton. enarmonios men oun esti parypate pasa he barytera tes barytates chromatikos, ...

[Mathiesen 1976, p 14, translating Aristoxenus 1.26.30-33] There are two positions for the parhypate, the one common to the diatonic and _color_, the other unique to the _Harmonia_, for the two genera share the parhypate. For every parhypate is _enharmonic_ which is lower than the lowest _chromatic_ [parhypate], ...

According to the emphasis Mathiesen gives these two pairs of words, Aristoxenus makes the important point [1.26] that there are two different *types* of _parhypate_: one that can be moved within the locus for all the shades of diatonic and chromatic (regardless of which of these two genera is in use), and the other one unique to the _harmonia_, 'the divine ordering of the Universe', which would be used only in the enharmonic genus.

We will thus assume that the largest interval which may appear at the bottom of the tetrachord, in either the diatonic or chromatic genera, will be the 'limma' 256/243.

[2.51.32] Lichanoi men oun eisin hex. mia enarmo-[33]nios, treis chromatikai, kai duo diato-[34]noi. {parypatai de tettares,} hosai per ai [2.52.1] ton tetrachordon diaipeseis. parypatai [2] de duo{in} elattous. te gar hemitoniaia [3] chrometha pros te tas diatonous, kai [4] pros ten tou toniaiou chromatos diaipe- [5]sin.

There are thus as many _lichanoi_ as there are divisions of the tetrachord [i.e., six], and two fewer _parhypatai_, since we use that which stands at a semitone both for the diatonic divisions and for that of the tonic chromatic.

In order to ascertain as precisely as possible the boundaries of the generic ranges, it is necessary to determine first the exact locations of the 'highest' and 'lowest' _lichanoi_ and _parhypatai_ acceptable into each of Aristoxenus's genera.

LOCATING THE MOVEABLE NOTES

Aristoxenus presents [1.24.16-32] six different _systemata_, each one bounded by an interval which is successively larger than the last. The four smallest are referred to as _pykna_, but the last two are not; as Barker says, they are:

[Barker 1989, p 143, note 100] Called by the general name _systema_ because [they exceed] the magnitude properly called _pyknon_.

===========





[1.24.15-25.11]


[1.24.15-25.11]
> [Meibom:]
15>  Touton houtos horismenon, pros toi
16>  baryteroi ton menonton phthongon eile-
17>  phtho to elachiston. pyknon d' estai to
18>  ek duo dieseon enarmonion kai chro-
19>  matikon elachiston. esontai duo de li-
20>  chanoi eilemmenai duo genon bary-
21>  tatai. he men harmonias; he de, chro-
22>  matos.  katholou gar barytatai
23>  men hai enarmonioi lichanoi esan.
24>  echomenai de, hai chromatikai. syn-
25>  tonotatai de, hai diatonoi.  Meta tau-
26>  ta triton eilephtho pyknon pros toi
27>  autoi. tetarton eilephtho pyknon to-
28>  niaion. pempton de pros toi autoi,
29>  to ex hemitoniou kai hemioliou diastem-
30>  matos synestekos systema eilephtho.
31>  hekton de, ex hemitoniou kai tonou.  Hai
32>  men oun ta duo ta prota lephthen-
33>  ta pykna horizousai lichanoi eiren-
34>  tai; he de to triton pyknon horizousa
[25]
 1>  lichanos, chromatike men estin; kalei-
 2>  tai de to chroma, en hoi estin, hemiolion.
 3>  he de to tetarton pyknon horizousa li-
 4>  chanos, chromatike men estin; kaleitai
 5>  de to chroma, en hoi esti, toniaion.  he de
 6>  to pempton lephthen systema horizousa
 7>  lichanos, ho meizon ede pyknou en.
 8>  epeideper isa esti ta duo toi heni, ba-
 9>  rytate diatonos estin.  he de to hekton
10>  lephthen systema horizousa lichanos,
11>  syntonotate diatonos estin.

> [Macran:]
> Touton  houtos horismenon pros toi baryteroi ton menonton
> phthongon eilephtho to elachiston pyknon; touto d' estai to ek
> duo dieseon  autoi; touto de estai to ek duo dieseon> chromatikon elachiston.
> esontai de  duo li|chanoi eilemmenai duo genon barytatai,
> he men harmonias he de chromatos.  katholou gar barytatai men
> hai enarmonioi lichanoi esan, echomenai d' hai chromatikai,
> syn|tonotatai d' hai diatonoi.  Meta tauta triton eilephtho
> pyknon pros toi autoi; tetarton  eilephtho pyknon toniaion;
> pempton de pros toi autoi, to ex hemitoniou kai hemioliou
> diastem|matos synestekos systema eilephtho; hekton de to ex
> hemitoniou kai tonou.  Hai men oun ta duo [ta] prota lephthenta
> pykna horizousai lichanoi eirentai; he de to triton pyknon
> horizousa || lichanos chromatike men estin, kaleitai de to
> chroma en hoi estin hemiolion. He de to tetarton pyknon horizousa
> lichanos chromatike men estin, kaleitai | de to chroma en hoi
> esti toniaion.  he de to pempton lephthen systema horizousa
> lichanos, ho meizon ede pyknou en, epeideper isa esti ta duo
> toi heni, barytate diatonos estin.  he de to hekton lephthen |
> systema horizousa lichanos syntonotate diatonos estin.
>
>> Given these definitions, let us take the smallest pyknon, placed
>> next to the lower of the fixed notes.  This will be the one
>> composed of two enharmonic or two of the smallest chromatic
>> diesis.  The two lichanoi thus specified will be the lowest in
>> each of the two genera, one in the enharmonic, the other in the
>> chromatic: for we have explained that, taken overall, the
>> enharmonic lichanoi are the lowest, the chromatic next, and the
>> diatonic the highest.  After these, consider a third pyknon
>> placed next to the same note, and then a fourth one, which
>> is a tone: fifthly, from the same note take the systema composed
>> of a semitone and an interval one and a half times as great, and
>> sixthly, that composed of a semitone and a tone.
>>    We have already spoken of the lichanoi bounding the first two
>> pykna listed.  The one that bounds the third is chromatic, and
>> the chroma in which it is is called hemiolic.  That bounding the
>> fourth pyknon is chromatic, and the chroma in which it is is
>> called tonic.  The lichanos bounding the fifth systema mentioned,
>> which was specified as greater than a pyknon, since the two
>> intervals are equal to the one, is the lowest diatonic: that
>> which bounds the sixth systema mentioned is the highest diatonic.



[1.25.11-26.7]
[Meibom:]
[25]
11>  .................. He men oun
12>  barytate chromatike lichanos tes
13>  enarmoniou barytates hektoi merei to-
14>  nou oxytera estin. epeideper he chro-
15>  matike diesis tes enarmoniou dieseos
16>  dodekatemorioi tonou meizon esti.  Dei
17>  gar to tou autou tritemorion tou tetartou
18>  merous dodekatemorioi hyperechein.
19>  hai de duo chromatikai ton duo
20>  enarmonion delon hos toi diplasioi.
21>  touto de estin hektemorion elatton dia-
22>  stema tou elachistou ton meloidoume-
23>  non. Ta de toiauta ameloideta
24>  estin. ameloideton gar legomen, ho me
25>  tattetai kath' heauto en systemati.  He de
26>  barytate diatonos tes barytates
27>  chromatikos hemitonioi kai dodekate-
28>  morioi tonou oxytera estin.  epi men gar
29>  ten tou hemioliou chromatos lichanon
30>  hemitonion en ep' autes. apo de tes hemio-
31>  liou epi ten enarmonion, diesis. apo
32>  de tes enarmoniou epi ten baryta-
33>  ten chromatiken, hektemorion. apo de
34>  tes barytates chromatikes epi ten hemiolion, dodekatemorion tonou.  to
[26]
 1>  de tetartemorion ek trion dodeka-
 2>  temorion synkeitai. host' einai phane-
 3>  ron, hoti to eiremenon diastema estin
 4>  apo tes barytates diatonou, epi ten
 5>  barytaten chromatiken.  he de syn-
 6>  tonotate diatonOS tes barytates
 7> diatonou, diesei esti syntonotera.  

[Macran:]
> He men oun barytate chromatike lichanos tes
> enarmoniou barytates hektoi merei tonou oxytera estin, epeideper
> he chro|matike diesis tes enarmoniou dieseos dodekatemorioi
> tonou meizon esti.  Dei gar to tou autou tritemorion tou tetartou
> merous dodekatemorioi hyperechein, hai de duo chromatikai ton
> duo | enarmonion delon hos toi diplasioi.  touto de estin
> hektemorion, elatton diastema tou elachistou ton meloidoumenon.
> Ta de toiauta ameloideta estin, ameloideton gar legomen ho me |
> tattetai kath' heauto en systemati.  He de barytate diatonos
> tes barytates chromatikos hemitonioi kai dodekatemorioi tonou
> oxytera estin.  epi men gar ten tou hemioliou chromatos lichanon |
> hemitonion en ap' autes, apo de tes hemioliou epi ten
> enarmonion diesis, apo de tes enarmoniou epi ten barytaten
> chromatiken hektemorion, apo de tes barytates chromatikes epi
> ten hemiolion dodekatemorion tonou.  to || de tetartemorion
> ek trion dodekatemorion synkeitai, host' einai phaneron, hoti to
> eiremenon diastema estin apo tes barytates diatonou epi ten |
> barytaten chromatiken.  He de syntonotate diatonos tes barytates
> diatonou diesei esti syntonotera.  

>
>> Thus the lowest chromatic lichanos is higher than the lowest
>> enharmonic by a sixth part of a tone, since the chromatic diesis
>> is greater by a twelfth part of a tone than the enharmonic diesis.
>> A third part of anything must exceed a quarter of the same thing
>> by a twelfth part, and the two chromatic dieses must evidently
>> exceed the two enharmonic ones by twice that amount.  This is
>> a sixth part, an interval smaller than the least of the melodic
>> intervals.  Such intervals are unmelodic, since we call 'unmelodic'
>> any interval that is not placed in a systema in its own right.
>> The lowest diatonic lichanos is higher than the lowest chromatic
>> by a semitone and a twelfth part of a tone.  We said that it is
>> a semitone from the lichanos of the hemiolic chromatic, and from
>> there to the enharmonic lichanos is a diesis: from the enharmonic
>> to the lowest chromatic lichanos is a sixth part of a tone, and
>> from the lowest chromatic to the hemiolic lichanos is a twelfth
>> part of a tone.  Now a quarter is composed of three twelfth parts,
>> so that it is clear that from the lowest diatonic to the lowest
>> chromatic lichanos is the interval stated.  The highest diatonic
>> lichanos is higher than the lowest diatonic by a diesis.

[Monzo edit - it's a poem!]
>>
>> Thus the lowest chromatic lichanos
>> is higher than the lowest enharmonic
>> by a sixth part of a tone,
>>
>> since the chromatic diesis
>> is greater by a twelfth part of a tone
>> than the enharmonic diesis.
>>
>>
>> A third part of anything
>> must exceed a quarter of the same thing
>> by a twelfth part,
>>
>> and the two chromatic dieses
>> must evidently exceed the two enharmonic ones
>> by twice that amount.
>>
>> This is a sixth part,
>> an interval smaller
>> than the least of the melodic intervals.
>>
>> Such intervals are unmelodic,
>> since we call 'unmelodic'
>> any interval that is not placed in a systema in its own right.
>>
>> The lowest diatonic lichanos
>> is higher than the lowest chromatic
>> by a semitone and a twelfth part of a tone.
>>
>> We said that it is a semitone
>> from the lichanos of the hemiolic chromatic,
>>
>> and from there
>> to the enharmonic lichanos
>> is a diesis:
>>
>> from the enharmonic
>> to the lowest chromatic lichanos
>> is a sixth part of a tone,
>>
>> and from the lowest chromatic
>> to the hemiolic lichanos
>> is a twelfth part of a tone.
>>
>> Now a quarter
>> is composed of three twelfth parts,
>>
>> so that it is clear that
>>
>> from the lowest diatonic
>> to the lowest chromatic lichanos
>> is the interval stated.
>>
>> The highest diatonic lichanos
>> is higher than the lowest diatonic
>> by a diesis.


******************

Put them all together into a song that would help Aristoxenus's students remember this important part of his lecture:

> Ek touton de phaneroi gignontai hoi topoi ton lichanon hekastes;

==========

The enharmonic genus --------------------

^[1.24.15-21] > Touton houtos horismenon pros toi baryteroi ton > menonton phthongon eilephtho to elachiston pyknon; touto > d' estai to ek duo dieseon (enarmonion elachiston; epeita > deuteron pros toi autoi; touto de estai to ek duo dieseon) > chromatikon elachiston. esontai de (hai) duo li|chanoi > eilemmenai duo genon barytatai, he men harmonias he de chromatos. > >> let us take the smallest pyknon, placed next to the lower >> of the fixed notes. This will be the one composed of two >> enharmonic or two of the smallest chromatic diesis. The two >> lichanoi thus specified will be the lowest in each of the >> two genera, one in the enharmonic, the other in the chromatic:

Extracting what is relevant to this genus:

^[1.24.15-21] >> let us take the ... _pyknon_ ... composed of two enharmonic >> ... dieses. The ... [_lichanos_] thus specified will be the >> lowest in ... the enharmonic [genus]. THIS ALREADY APPEARS ABOVE vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

[2.50.23-24] > Mia men oun ton diaipeseon estin enarmonios en he > to men pyknon hemitonion esti to | de loipon ditonon. > >> One of these divisions is enharmonic, in which the _pyknon_ >> is a semitone and the remainder a ditone.

> [2.50.22 ...] Mia men [23] oun ton diaipeseon estin enarmonios, > [24] en he to men pyknon, hemitonion esti; > to [25] de loipon, ditonon. > >> One of these divisions [of the tetrachord] is enharmonic, in >> which the pyknon is a semitone and the remainder a ditone. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Assuming the Pythagorean 'limma' 256/243 to be the 'semitone', and the 'enharmonic diesis' to be exactly half of this, we get:

1                         1                0.000   mese
1/((9/8)^2)               0.790123457   -407.820   lowest enharmonic lichanos
(3/4)*((256/243)^(1/2))   0.769800359   -452.933   lowest enharmonic parhypate
3/4                       0.75          -498.045   hypate



Other than the statements already presented, which describe the enharmonic
_lichanoi_ and _parhypatai_ as all those lower than the lowest chromatic,
and most musicians tuning the _lichanos_ higher, near the chromatic, 
Aristoxenus makes no other description of the intervals in the enharmonic
genus.  This table may thus be assumed to designate the enharmonic which
he considers 'that far from being the most contemptible ... is perhaps the
finest', and the one which is 'plain enough to those who are familiar with
the first and second groups of ancient styles'. [1.23.5-10]



The chromatic genera
--------------------


The determination of the two lowest shades of Aristoxenus's chromatic
genera will prove to be the most confusing and difficult of all his
divisions.



The 'relaxed' shade of the chromatic genus - part 1
---------------------------------------------------


Again extracting what is relevant here:


[1.24.15-21]
> let us take the ... _pyknon_ ... composed of ... two of
> the smallest chromatic dieses.  The ... [_lichanos_] thus
> specified will be the lowest in ... the chromatic [genus].



Aristoxenus has already defined the 'least chromatic diesis'
as 1/3-tone.  Assuming an exact 3-part division of the 9/8 'tone'
for this diesis (~67.970 cents), and measuring two of them upward
from _hypate_, we get:


        ratio              decimal ratio   cents

(3/4)*((9/8)^(2/3))           0.811265   ~-362.105   lowest chromatic lichanos
(3/4)*((9/8)^(1/3))           0.780031   ~-430.075   lowest chromatic/diatonic parhypate
((3/4)*((9/8)^(2/3)))/(3/4)   1.081687   ~ 135.940   size of pyknon
(9/8)^(1/3)                  ~1.040042   ~  67.970   least chromatic diesis = 1/3-tone

('Chromatic/diatonic' specifies the lowest _parhypate_ which the
chromatic and diatonic genera both have in common.)



On the next page, however, Aristoxenus presents another description
which contradicts this one.




[1.25.11-14]
> He men oun barytate chromatike lichanos tes enarmoniou
> barytates hektoi merei tonou oxytera estin,
>
>> the lowest chromatic _lichanos_ is higher than the lowest
>> enharmonic by a sixth part of a tone, ...



[1.25.32-33]
> apo de tes enarmoniou epi ten barytaten chromatiken hektemorion,
>
>> from the enharmonic to the lowest chromatic _lichanos_ is a
>> sixth part of a tone, ...



[1.25.14-16]
> epeideper he chro|matike diesis tes enarmoniou dieseos
> dodekatemorioi tonou meizon esti. 
>
>> ... since the chromatic diesis is greater by a twelfth part of
>> a tone than the enharmonic diesis.




Aristoxenus here invokes the concepts of a '1/6-tone' and '1/12-tone'
to use in interval measurement.  Assuming equal divisions of the
'tone' gives us:

    ratio     decimal ratio   cents

(9/8)^(1/6)    1.019824451   ~33.985   'sixth part of a tone'
(9/8)^(1/12)   1.009863581   ~16.993   'twelfth part of a tone'


and it is clear that he expects the ~17-cent size of the latter
to be perceptible.


He explains by simple mathematics that since the 'chromatic diesis'
is 1/3-tone and the 'enharmonic diesis' is 1/4-tone, the difference
between them must be 1/3 [= 4/12] - 1/4 [= 3/12] = 1/12-tone.
But upon closer examination, it appears that he must be using
this as an approximation, as the numbers do not agree with all
of his descriptions.



Let us first use his 1/6- and 1/12-tones in our calculations.


Assuming the 'lowest enharmonic _lichanos_' to be a Pythagorean
'ditone' below _mese_ as described above, the 'enharmonic diesis'
to be exactly half of that, and the '1/6- and 1/12- parts of a
tone' to be as above, this description gives us:


        ratio              decimal ratio   cents

(1/((9/8)^2))*((9/8)^(1/6))                0.805787   ~-373.835   lowest chromatic _lichanos_
(3/4)*(((256/243)^(1/2))*((9/8)^(1/12)))   0.777393   ~-435.940   lowest chromatic/diatonic parhypate
((1/((9/8)^2))*((9/8)^(1/6)))/(3/4)        1.074383   ~ 124.210   size of pyknon
((256/243)^(1/2))*((9/8)^(1/12))           1.036524   ~  62.105   chromatic diesis = 1/2-limma


Two of these dieses do indeed comprise a _pyknon_ which is exactly
1/6-tone higher than the 'ditone' enharmonic _lichanos_, i.e.,
exactly 1/6-tone larger than a 'limma' semitone.



Comparing this with his earlier definition of the 'chromatic diesis' as
'1/3-tone', we find it to be ~5.865 cents smaller, the same amount by
which we found our 'enharmonic diesis' to be less than an exact '1/4-tone'.
It seems thus far that Aristoxenus's characterizations of the smaller
dieses as 1/4- and 1/3-tones is simply an approximation.





[2.51.4-8]
> [4...] Hoti d' esti [5] meizon to hemiolion pyknon
> tou malakou, [6] rhadion synidein.
> to men gar enarmo-[7]niou dieseos leipei tonos einai;
> to de, [8] chromatikos.
>
>> It is easy to see that the hemiolic pyknon
>> is greater than that of the soft chromatic,
>> for the former falls short of being a tone by an enharmonic diesis,
>> the latter by a chromatic diesis.


Extracting what is relevant to the genus under discussion:


>> ... the ... _pyknon_ ... of the soft chromatic ... falls
>> short of being a tone ... by a chromatic diesis.



If we assume the 'chromatic diesis' to equal 1/2-limma plus 1/12-tone,
and thus the _pyknon_ to equal the 'limma' semitone plus 1/6-tone,
subtracting that _pyknon_ from the 9/8 'tone' leaves a small interval
that is close to, but does not equal (and in fact is roughly 1/12-tone
larger than), this chromatic diesis:


        ratio              decimal ratio   cents

(9/8)/((256/243)*((9/8)^(1/6)))   1.047113   ~79.700   tone - (limma + 1/6-tone)



If we measure from the opposite direction, subtracting the same
'chromatic diesis' from the 9/8 'tone', we arrive at a value for the
_pyknon_ which is larger than either of the two we have calculated
thus far:


        ratio              decimal ratio   cents

(9/8)/(((256/243)^(1/2))*((9/8)^(1/12)))   1.085358   ~141.805   pyknon = tone - (1/2-limma + 1/12-tone)	



These values therefore do not work out.



Let us now re-examine Aristoxenus's definitions of the 1/6- and
1/12-tone based on our earlier value for the 'chromatic diesis'
as 1/3-tone.


If we subtract the 'enharmonic diesis' (1/2-limma) from the
'chromatic diesis' (1/3-tone), we get an interval that is actually
much closer to 1/9-tone:


        ratio              decimal ratio   cents

((9/8)^(1/3))/((256/243)^(1/2))   1.013291   ~22.858   '1/12-tone' = 1/3-tone - 1/2-limma	



And doubling this interval (i.e., subtracting the enharmonic _pyknon_
composed of two 1/2-limmas [= one 'limma' semitone] from the relaxed
chromatic _pyknon_ spanning an interval of 2/3-tone) gives a result
that is nothing like 1/6-tone, but rather is almost exactly the same
as our 'enharmonic diesis' of 1/2-limma:


        ratio              decimal ratio   cents

((9/8)^(2/3))/(256/243)   1.026758    ~45.715   2/3-tone - 'limma' semitone
(256/243)^(1/2)           1.0264      ~45.112   1/2-limma


If our value of 1/2-limma for the 'enharmonic diesis' is correct, then
an exact 1/3-tone is certainly the wrong value for the 'chromatic diesis'.



> [2.50.28] malakou men oun chromatos esti diai-[29]pesis,
> en he to men pyknon ek duo chro-[30]matikon dieseon elachiston synkei-[31]tai;
> to de loipon duo metrois metrei-[32]tai;
> hemitonioi men tris, chromatikei [33]de diesei hapax.
> hoste metreisthai trisin hemitoniois, kai tonou tritei merei hapax.
> [34]esti de ton chromatikon pyknon elachiston,
> kai lichanos haute barytate tou [2.51.1] genous toutou.
>
>> [2.50.28] The division of the soft chromatic is that
>> [29] in which the _pyknon_ consists of two of the [30] smallest chromatic dieses,
>> [31] and the remainder is measured by two units of measurement,
>> [32] by the semitone three times, and by the chromatic [33] diesis once [,
>> so that the sum of it amounts to three semitones and the third of a tone].
>> [34] It is the smallest of the chromatic _pykna_,
>> and this _lichanos_ is the lowest in [2.51.1] this genus.

([] Indicates translation omitted by Barker 1989.)



We have a problem here because 'semitone three times' probably does
not refer to an equal multiple of a single size of semitone, but
rather probably means some combination of both sizes of Pythagorean
semitone, since a 9/8 'tone' is comprised of one each of the
256/243 'limma' and 2187/2048 'apotome'.



It seems that we should re-define the 'least chromatic diesis',
but the measurement for the relaxed chromatic genus that best fits
Aristoxenus's descriptions is the definition of 'least chromatic
diesis' as 1/3-tone.


        ratio              decimal ratio   cents

(9/8)/((9/8)^(1/3)) ~1.081687178   ~135.940   pyknon = '1/3-tone' less than a tone
((9/8)^(1/3))^2      1.081687      ~135.940   pyknon = two '1/3-tones'
			




1                                       1                0.000   mese

lichanos: 'pyknon = tone - diesis';  diesis = tone/3
1/((9/8)*(256/243)*((9/8)^(1/3)))       0.811265383   -362.105

parhypate: 'diesis = 1.5 * enh. d.' = tone^(1/3)
NOTE HOW CLOSE TO limma^(3/4)
1/(((9/8)^2)*(256/243))*((9/8)^(1/3))   0.780031434   -430.075

1/(4/3)                                 0.75          -498.045   hypate


xxxxxxxxxxxxxxxxxxxxxx

(256/243)*((9/8)^(1/6))                    1.074383   124.210   pyknon = limma + 1/6-tone	
(9/8)/((256/243)*((9/8)^(1/6)))            1.047113    79.700   tone - pyknon; pyknon = (limma + 1/6-tone) --> DOES NOT EQUAL CHR DIESIS	
				
(9/8)/(((256/243)^(1/2))*((9/8)^(1/12)))   1.085358   141.805   pyknon = tone - (1/2-limma + 1/12-tone)	
-------------------------------------------------------------------------------				
(9/8)/((9/8)^(1/3))                        1.081687   135.940   pyknon = tone - 1/3-tone	
((9/8)^(1/3))^2                            1.081687   135.940   pyknon = two 1/3-tones	
- this is the only soft chr that works				
				
((9/8)^(1/3))/((256/243)^(1/2))            1.013291    22.858   '1/12-tone' = 1/3-tone - 1/2-limma	
9/8                                        1.125      203.910   8.920922   =this means that the above value is really close to a 1/9-tone	

				
((9/8)^(2/3))/(256/243)                    1.026758    45.715   4.460461   =2/3-tone - semitone DOES NOT EQUAL 1/6-tone	

(256/243)^(1/2)                            1.0264      45.112	              1/2-limma, for comparison	



	
	





xxxxxxxxxxxxxxxxxxxxxx




The 'hemiolic' shade of the chromatic genus
-------------------------------------------


The calculation of the hemiolic chromatic _lichanos_ presents an
intriguing problem.


[1.24.26]
> Meta tauta triton eilephtho pyknon pros toi autoi;
>
>> consider a third _pyknon_ placed next to the same note [_hypate_]...



[1.25.30-31]
> apo de tes hemioliou epi ten enarmonion diesis,
>
>> from the _lichanos_ of the hemiolic chromatic ... to the enharmonic
>> _lichanos_ is a diesis ...




To which Barker supplies the note:


[Barker 1989, p 143, note 107]
> The word diesis, unqualified, refers to the enharmonic diesis, 1/4 tone.



But it may not be quite that simple.




> [2.51.1 ...] hemioliou de chromatos di-[2]aipesis estin,
> en he to te pyknon hemiolion [3] esti, tou [t'] enarmoniou,
> kai ton dieseon heka-[4]teras ton enarmonion.
>
>> [2.51.1 ...] The division of the hemiolic chromatic [2] is that
>> in which the _pyknon_ is one and a half times [3] that of the enharmonic,
>> and each of its dieses is [4] one and a half times the corresponding enharmonic diesis.


And again, extracting from the more elaborate phrase previously
quoted, that which is relevant to this genus:


[2.51.4-7]
> the hemiolic _pyknon_ ... falls short of being a tone by
> an enharmonic diesis



Assuming that the 'enharmonic diesis' is always 1/2-limma does
not agree with both of these propositions simultaneously:

       ratio           decimal ratio   cents

(256/243)^1.5             1.081311   ~135.337   1/2-limma * 1.5
(9/8)/((256/243)^(1/2))   1.096063   ~158.798   tone - 1/2-limma



If, however, we assume that the bottom 'tone' of the tetrachord
must be composed of not two 256/243 'limmas' but rather of one
'limma' and one 2187/2048 'apotome', as was normally the case
in Pythagorean tetrachord theory, then Aristoxenus's 'enharmonic
diesis' in this genus would actually indicate two different sizes
of interval: in some places equal to 1/2-limma and in others
equal to 1/2-apotome.


If we make the 'enharmonic diesis' above the _pyknon_ equal to 1/2-apotome,
and make the size of the _pyknon_ itself, spanning an interval 'one and
a half times the enharmonic _pyknon_', equal to the limma plus 1/2-apotome
(that is, three 'quarter-tones' distributed as two 1/2-limmas and one
1/2-apotome), we arrive at a value for _lichanos_ which agrees with both
of Aristoxenus's descriptions:


            ratio                  decimal ratio   cents

(3/4)*(256/243)*((2187/2048)^(1/2))   0.816497   ~-350.978   hemiolic chromatic lichanos
(9/8)/((2187/2048)^(1/2))             1.088662   ~ 147.067   pyknon = tone - 1/2-apotome
(256/243)*((2187/2048)^(1/2))         1.088662   ~ 147.067   pyknon = limma + 1/2-apotome



We can thus see that 'one and a half times the enharmonic diesis'
is an approximate measure, doing double duty in dividing both sizes
of semitone.




Armed with a good measurement for _lichanos_, let us revisit this
statement:


[1.25.30-31]
> apo de tes hemioliou epi ten enarmonion diesis,
>
>> from the _lichanos_ of the hemiolic chromatic ... to the enharmonic
>> _lichanos_ is a diesis ...



Eliminating (3/4)*(256/243) from the above calculation for _lichanos_
leaves an 'enharmonic diesis' which equals 1/2-apotome:


     ratio       decimal ratio   cents

(2187/2048)^(1/2)   1.033378   ~56.843   1/2-apotome


providing further evidence that Aristoxenus meant to imply a division
of the 'other' Pythagorean semitone here.



His statement that 'each of [the hemiolic chromatic] dieses is one
and a half times the corresponding enharmonic diesis' is troubling
because it seems to imply an enharmonic _pyknon_ which spans a
'semitone' composed of 1/2-limma plus 1/2-apotome, an interval
totally uncharacteristic of any other Greek music-theory, and one
which in fact divides the 9/8 tone exactly in half - which is something
that most Greek theorists argued was 'impossible':


((256/243)^(1/2)) * ((2187/2048)^(1/2)) = (9/8)^(1/2)


It seems best to assume that here Aristoxenus used the word
'corresponding' also in an approximate way.



In any case, the location of _parhypate_ poses a problem because
it is unclear whether Aristoxenus really meant to divide this _pyknon_
exactly in half, and if he did not, exactly what division he did
intend.



Dividing the hemiolic chromatic _pyknon_ exactly in half gives:


              ratio             decimal ratio  cents

((9/8)/((2187/2048)^(1/2)))^(1/2)   1.04339   ~73.534



Considering his equation of this diesis with 1.5 * the 'enharmonic
diesis', calculating 2/3 of this hemiolic chromatic diesis gives
an 'enharmonic diesis' of:


                  ratio                    decimal ratio  cents

(((9/8)/((2187/2048)^(1/2)))^(1/2))^(1/1.5)   1.028721   ~49.022


which is ~1.955 cents less than an exact 1/4-tone, ~7.820 cents less
than 1/2-apotome, and ~3.910 cents more than 1/2-limma.



Lacking any more concrete means of determining a measurement, we will
assume that this particular 'enharmonic diesis' will be multiplied
by 1.5 to give the 'hemiolic chromatic diesis' calculated immediately
above, thus locating the _parhypate_ and giving the tetrachord:


1                                               1              0.000   mese
(3/4)*(256/243)*((2187/2048)^(1/2))             0.816497   ~-350.978   hemiolic chromatic lichanos
(3/4)*(((256/243)*((2187/2048)^(1/2)))^(1/2))   0.7825423  ~-424.511   hemiolic chromatic parhypate
3/4                                             0.75       ~-498.045   hypate



--------------



The 'relaxed' shade of the chromatic genus - part 2
---------------------------------------------------


<< we still have not determined with certainty the relaxed chromatic _lichanos_ >>


[1.26.1-2]
> from the lowest chromatic to the hemiolic _lichanos_ is a twelfth part
> of a tone.


                   ratio                            dectimal ratio   cents

((3/4)*(256/243)*((2187/2048)^(1/2)))/((9/8)^(1/12))   0.8085217   ~-367.970   lowest chromatic lichanos


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

'pyknon = tone - diesis';  diesis = limma/2;  2.51 RULES OUT ALL OTHERS
1/((9/8)*(256/243)*((256/243)^(1/2)))       0.822047551   -339.247  

'pyknon = tone - diesis';  diesis=tone/4
1/((9/8)*(256/243)*((9/8)^(1/4)))           0.819267365   -345.112   

'pyknon = 1.5 enh. pyknon' = tone^(2/3);  SAME AS ABOVE - NOTE HOW CLOSE TO NEXT ONE
1/(((9/8)^2)*(256/243))*((9/8)^(2/3))       0.811265383   -362.105

'pyknon = 1.5 enh. pyknon' = limma^(3/2);
'pyknon = tone - diesis';  diesis = hemiolic chr. Diesis = ~limma^(3/4)
1/(((9/8)^2)*(256/243))*((256/243)^(3/2))   0.810983094   -362.708





1                                           1                0.000   mese

lichanos - 'pyknon=tone-diesis' - diesis=limma/2
2.51 RULES OUT ALL OTHERS
1/((9/8)*(256/243)*((256/243)^(1/2)))       0.822047551   -339.2471/((9/8)*(256/243)*((9/8)^(1/4)))           0.819267365   -345.112   hemiolic chromatic lichanos - 'pyknon=tone-diesis' - diesis=tone/4

lichanos - 'pyknon=1.5 enh. pyknon' = tone^(2/3)
SAME AS ABOVE - NOTE HOW CLOSE TO NEXT ONE
1/(((9/8)^2)*(256/243))*((9/8)^(2/3))       0.811265383   -362.105

lichanos - 'pyknon=1.5 enh. pyknon' = limma^(3/2) - 'pyknon=tone-diesis'
diesis=hemiolic chr. Diesis=~limma^(3/4)
1/(((9/8)^2)*(256/243))*((256/243)^(3/2))   0.810983094   -362.708

parhypate = 'diesis = 1.5 * enh.d.' = limma^(3/4)
1/(((9/8)^2)*(256/243))*((256/243)^(3/4))   0.779895711   -430.376

1/(4/3)                                     0.75          -498.045   hypate




The 'tonic' shade of the chromatic genus
----------------------------------------


[1.24.27]
> tetarton  eilephtho pyknon toniaion;
>
>> consider ... a fourth [_pyknon_, in the series of six],
>> which is a tone ...


[2.51.8-11]
> toniaiou de chromatos diaipesis estin en he to men pyknon
> ex hemi|tonion duo synkeitai to de loipon triemitonion estin.
>
>> The division of the tonic chromatic is that in which the
>> _pyknon_ consists of two semitones and the remainder is
>> three semitones.



Aristoxenus presents this genus quite simply, without elaborate
explanation.  Even tho he claims that the _pyknon_ is divided into
two 'equal semitones'   <<<<< DOES HE ??? >>>>, it is likely that he means to imply the
usual Pythagorean chromatic division of the tetrachord, employing
both the 'limma' and 'apotome' semitones in the _pyknon_.



If we calculate the 'characteristic interval' or 'remainder' of
'three semitones' at the top of the tetrachord as either a tone
plus an exact 1/2-tone or a tone plus an 'apotome', our _pyknon_
is smaller than a tone:


       ratio                 decimal ratio   cents

1/((9/8)^(3/2))                 0.838052   ~-305.865   lichanos = tone * 1.5 below mese
(1/((9/8)^(3/2)))/(3/4)         1.117403   ~ 192.180   pyknon; CI = tone * 1.5

1/((9/8)*(2187/2048))           0.832393   ~-317.595   lichanos = tone + apotome below mese
(1/((9/8)*(2187/2048)))/(3/4)   1.109858   ~ 180.450   pyknon; CI = tone + apotome



Thus neither of these values will work.



Applying the usual Pythagorean values, however, fits Aristoxenus's
description exactly <<<< (except that he called the semitones equal): >>>>>>


        ratio              decimal ratio   cents

1/((9/8)*(256/243))           0.84375   ~-294.135   lichanos = tone + limma below mese
(1/((9/8)*(256/243)))/(3/4)   1.125     ~ 203.910   pyknon; CI = tone + limma


with a _pyknon_ which is exactly a 9/8 tone.



Thus we may fairly confidently locate the notes in Aristoxenus's
tonic chromatic genus:


       ratio                     decimal ratio   cents

1                                 1                0.000   mese
1/((9/8)*(256/243))               0.84375       -294.135   tonic chromatic lichanos
1/((9/8)*(256/243)*(2187/2048))   0.790123457   -407.820   tonic chromatic parhypate
3/4                               0.75          -498.045   hypate




The diatonic genera
-------------------



1                                           1                0.000   mese

					
lichanos - pythagorean - tone+limma
1/((9/8)*(256/243))                         0.84375       -294.135

lichanos - equal division of tone
***RULED OUT BY 1.24 'pyknon=tone'
1/((9/8)^(3/2))                             0.838052481   -305.865

lichanos - pythagorean - tone+apotome
***RULED OUT BY 1.24 'pyknon=tone'
1/((9/8)*(2187/2048))                       0.832393436   -317.595
					

parhypate - pythagorean					
1/((9/8)*(256/243)*(2187/2048))             0.790123457   -407.820

					
1/(4/3)                                     0.75          -498.045   hypate





The 'relaxed' shade of the diatonic genus
--------------------------------------



[1.24.28-30]
> pempton de pros toi autoi, to ex hemitoniou kai hemioliou
> diastem|matos synestekos systema eilephtho;
>
>> fifthly, from the same note [_hypate_] take the _systema_ composed
>> of a semitone and an interval one and a half times as great, ...



[1.25.26-28]
> He de barytate diatonos tes barytates chromatikos hemitonioi
> kai dodekatemorioi tonou oxytera estin. 
>
>> The lowest diatonic _lichanos_ is higher than the lowest chromatic
>> by a semitone and a twelfth part of a tone.


                ratio                                   decimal ratio   cents

((1/((9/8)^2))*((9/8)^(1/6)))*((256/243)*((9/8)^(1/12)))   0.857268   ~-266.618	



[1.25.26-30]
> He de barytate diatonos ... epi men gar ten tou hemioliou chromatos
> lichanon | hemitonion en ap' autes,
>
>> The lowest diatonic _lichanos_ ... is a semitone from the _lichanos_
>> of the hemiolic chromatic



Since we have located the hemiolic chromatic _lichanos_ more confidently
than that of the relaxed chromatic genus, let us make use of it in
our calculation of the relaxed diatonic and measure both of the
Pythagorean semitones above it:


               ratio                             decimal ratio   cents

((3/4)*(256/243)*((2187/2048)^(1/2)))*(256/243)     0.8601775   ~-260.753   hemiolic chromatic lichanos + limma
((3/4)*(256/243)*((2187/2048)^(1/2)))*(2187/2048)   0.8719131   ~-237.292   hemiolic chromatic lichanos + apotome




> [2.51.11 ...] Mechri men oun tautes tes di-[12]aipeseos
> amphoteroi kinountai hoi phthon-[13]goi.
> meta tauta d' he men parypate me-[14]nei;
> dieleluthe gar ton hautes topon;
> he de [15] lichanos kineitai diesin enarmonion.
> kai [16] gignetai to lichanou kai hypates
> diaste-[17]ma ison toi lichanou kai meses.
> hoste me-[18]keti gignesthai pyknon en tautei tei diai-[19]pesei.
> symbainei d' hama payesthai to py-[20]knon,
> synistamenon en tei ton tetrachor-[21]don diairesei
> kai archesthai gignomenon to [22] diatonon genos.
>
>> [2.51.11 ...] Up to [the tonic chromatic] division,
>> [12] both the notes [_lichanos_ and _parhypate_] move,
>> [13] but after this _parhypate_ stays still,
>> [14] since it has travelled through its whole range,
>> while [15] _lichanos_ moves through an enharmonic diesis,
>> and [16] the interval between _lichanos_ and _hypate_
>> [17] becomes equal to that between _lichanos_ and _mese_,
>> so that [18] in this division the _pyknon_ no longer occurs.
>> [19] The _pyknon_ disappears [20] in the division of the tetrachord [21] simultaneously
>> with the first occurrence of the [22] diatonic genus.


> [2.51.24 ...] malakou men oun esti diatonou diai-[25]pesis,
> en he to men hypates kai parypa-[26]tes hemitoniaion esti;
> to de parypates kai [27] lichanou trion dieseon enarmonion;
> [28] to de lichanou kai meses, pente dieseon.
>
>> [2.51.24 ...] The division of the soft diatonic is that
>> [25] in which the interval between _hypate_ and _parhypate_ [26] is a semitone,
>> that between _parhypate_ and [27] _lichanos_ is three enharmonic dieses,
>> and [28] that between _lichanos_ and _mese_ is five dieses.



As in the hemiolic chromatic, the 'three enharmonic diesis' is here
likely to be some combination of 1/2-limmas and 1/2-apotomes.


We will probably do best to measure downward from _mese_, assuming
'four dieses' to mean a 9/8 tone, and locating _lichanos_ either
1/2-limma or 1/2-apotome from that:


1/(9/8)/((256/243)^(1/2))     0.8660254   ~-249.022   mese - tone - 1/2-limma
1/(9/8)/((2187/2048)^(1/2))   0.8601775   ~-260.753   mese - tone - 1/2-apotome


The latter value here matches the one calculated above as an 'apotome'
higher than the hemiolic chromatic _lichanos_.


1                                           1                0.000   mese
					
1/((9/8)^2)*(2187/2048)*((256/243)^(1/2))   0.866025404   -249.022
   relaxed diatonic lichanos - pyknon=s+1.5s - 1.5s=apotome+d - d=limma/2

1/((9/8)^(5/4))                             0.863096483   -254.888
   relaxed diatonic lichanos - diesis=tone/4

1/((9/8)^2)*((256/243)^(3/2))               0.854369021   -272.483
   relaxed diatonic lichanos - pyknon=s+1.5s - 1.5s=limma^(3/2)

1/(((9/8)^(5/4))*((9/8)^(3/4)))             0.790123457   -407.820   parhypate

1/(4/3)                                     0.75          -498.045   hypate




The 'tense' shade of the diatonic genus
---------------------------------------


[1.24.31]
> hekton de to ex hemitoniou kai tonou.
>
>> sixthly, [consider] that [_systema_] composed of a semitone and a tone.



> [2.51.29] syntonou de,
> en he to men hypates kai pa-[30]rypates hemitoniaion;
> ton de loipon to-[31]niaion hekateron estin.
>
>> [2.51.29] [The division] of the tense diatonic is that
>> in which the interval between _hypate_ and [30] _parhypate_ is a semitone,
>> and each of the others [31] is a tone.



There is no reason to assume that Aristoxenus had in mind any
other division than the usual Pythagorean diatonic, for this genus.
This puts the 'limma' semitone at the bottom and two regular
9/8 tones above it:


  ratio      decimal ratio   cents

1             1                0.000   mese
1/(9/8)       0.888888889   -203.910   tense diatonic lichanos
1/((9/8)^2)   0.790123457   -407.820   tense diatonic parhypate
3/4           0.75          -498.045   hypate




The complete set of intervals deduced
-------------------------------------


(decimal and cents rounded off to several decimal places)


	all	2                                          2            1200	    mese

	6	2/(9/8)                                    1.777777778   996.0899983   tense diatonic lichanos

***	5	2/((9/8)^2)*(2187/2048)*((256/243)^(1/2))  1.732050808   950.9775004   relaxed diatonic lichanos - pyknon=s+1.5s - 1.5s=apotome+d - d=limma/2
		2/((9/8)^(5/4))                            1.726192966   945.1124978   relaxed diatonic lichanos - diesis=tone/4
		2/((9/8)^2)*((256/243)^(3/2))              1.708738042   927.51749     relaxed diatonic lichanos - pyknon=s+1.5s - 1.5s=limma^(3/2)

***	4	2/((9/8)*(256/243))                        1.6875        905.8650026   tonic chromatic lichanos - pythagorean - tone+limma
		2/((9/8)^(3/2))                            1.676104963   894.1349974   tonic chromatic lichanos - equal division of tone ***RULED OUT BY 1.24 'pyknon=tone'
		2/((9/8)*(2187/2048))                      1.664786872   882.4049922   tonic chromatic lichanos - pythagorean - tone+apotome ***RULED OUT BY 1.24 'pyknon=tone'

***	3	2/((9/8)*(256/243)*((256/243)^(1/2)))      1.644095102   860.7525048   hemiolic chromatic lichanos - 'pyknon=tone-diesis' - diesis=limma/2 - 2.51 RULES OUT ALL OTHERS
		2/((9/8)*(256/243)*((9/8)^(1/4)))          1.63853473    854.8875022   hemiolic chromatic lichanos - 'pyknon=tone-diesis' - diesis=tone/4
	2	2/((9/8)*(256/243)*((9/8)^(1/3)))          1.622530767   837.895002    relaxed chromatic lichanos - 'pyknon=tone-diesis' - diesis=tone/3
		2/(((9/8)^2)*(256/243))*((9/8)^(2/3))      1.622530767   837.895002    hemiolic chromatic lichanos - 'pyknon=1.5 enh. pyknon' = tone^(2/3) - SAME AS ABOVE - NOTE HOW CLOSE TO NEXT ONE
		2/(((9/8)^2)*(256/243))*((256/243)^(3/2))  1.621966188   837.2924944   hemiolic chromatic lichanos - 'pyknon=1.5 enh. pyknon' = limma^(3/2) - 'pyknon=tone-diesis' - diesis=hemiolic chr. Diesis=~limma^(3/4)

	1	2/((9/8)^2)                                1.580246914   792.1799965   enharmonic lichanos

***	6	2/((9/8)^2)                                1.580246914   792.1799965   tense diatonic parhypate

***	5	2/(((9/8)^(5/4))*((9/8)^(3/4)))            1.580246914   792.1799965   relaxed diatonic parhypate

***	4	2/((9/8)*(256/243)*(2187/2048))            1.580246914   792.1799965   tonic chromatic parhypate - pythagorean

	2	2/(((9/8)^2)*(256/243))*((9/8)^(1/3))      1.560062867   769.9250014   relaxed chromatic parhypate = 'diesis=1.5*enh.d.' = tone^(1/3) - NOTE HOW CLOSE TO limma^(3/4)
***	3	2/(((9/8)^2)*(256/243))*((256/243)^(3/4))  1.559791423   769.6237476   hemiolic chromatic parhypate = 'diesis=1.5*enh.d.' = limma^(3/4)

	1	2/(((9/8)^2)*((256/243)^(1/2)))            1.539600718   747.0674987   enharmonic parhypate
	all	2/(((9/8)^2)*(256/243))                    1.5           701.9550009   hypate


--------------------------


the above, reckoned downward from mese					


      1   1                                           1                0.000	mese
					
      6   1/(9/8)                                     0.888888889   -203.910   tense diatonic lichanos
					
***   5   1/((9/8)^2)*(2187/2048)*((256/243)^(1/2))   0.866025404   -249.022   relaxed diatonic lichanos - pyknon=s+1.5s - 1.5s=apotome+d - d=limma/2
          1/((9/8)^(5/4))                             0.863096483   -254.888   relaxed diatonic lichanos - diesis=tone/4
          1/((9/8)^2)*((256/243)^(3/2))               0.854369021   -272.483   relaxed diatonic lichanos - pyknon=s+1.5s - 1.5s=limma^(3/2)
					
***   4   1/((9/8)*(256/243))                         0.84375       -294.135   tonic chromatic lichanos - pythagorean - tone+limma
          1/((9/8)^(3/2))                             0.838052481   -305.865   tonic chromatic lichanos - equal division of tone ***RULED OUT BY 1.24 'pyknon=tone'
          1/((9/8)*(2187/2048))                       0.832393436   -317.595   tonic chromatic lichanos - pythagorean - tone+apotome ***RULED OUT BY 1.24 'pyknon=tone'
					
***   3   1/((9/8)*(256/243)*((256/243)^(1/2)))       0.822047551   -339.247   hemiolic chromatic lichanos - 'pyknon=tone-diesis' - diesis=limma/2 - 2.51 RULES OUT ALL OTHERS
          1/((9/8)*(256/243)*((9/8)^(1/4)))           0.819267365   -345.112   hemiolic chromatic lichanos - 'pyknon=tone-diesis' - diesis=tone/4
      2   1/((9/8)*(256/243)*((9/8)^(1/3)))           0.811265383   -362.105   relaxed chromatic lichanos - 'pyknon=tone-diesis' - diesis=tone/3
          1/(((9/8)^2)*(256/243))*((9/8)^(2/3))       0.811265383   -362.105   hemiolic chromatic lichanos - 'pyknon=1.5 enh. pyknon' = tone^(2/3) - SAME AS ABOVE - NOTE HOW CLOSE TO NEXT ONE
          1/(((9/8)^2)*(256/243))*((256/243)^(3/2))   0.810983094   -362.708   hemiolic chromatic lichanos - 'pyknon=1.5 enh. pyknon' = limma^(3/2) - 'pyknon=tone-diesis' - diesis=hemiolic chr. Diesis=~limma^(3/4)
					
      1   1/((9/8)^2)                                 0.790123457   -407.820   enharmonic lichanos
					
***   6   1/((9/8)^2)                                 0.790123457   -407.820   tense diatonic parhypate
					
***   5   1/(((9/8)^(5/4))*((9/8)^(3/4)))             0.790123457   -407.820   relaxed diatonic parhypate
					
***   4   1/((9/8)*(256/243)*(2187/2048))             0.790123457   -407.820   tonic chromatic parhypate - pythagorean
					
      2   1/(((9/8)^2)*(256/243))*((9/8)^(1/3))       0.780031434   -430.075   relaxed chromatic parhypate = 'diesis=1.5*enh.d.' = tone^(1/3) - NOTE HOW CLOSE TO limma^(3/4)
***   3   1/(((9/8)^2)*(256/243))*((256/243)^(3/4))   0.779895711   -430.376   hemiolic chromatic parhypate = 'diesis=1.5*enh.d.' = limma^(3/4)
					
      1   1/(((9/8)^2)*((256/243)^(1/2)))             0.769800359   -452.933   enharmonic parhypate
      6   1/(((9/8)^2)*(256/243))                     0.75          -498.045   hypate
      5   1/(4/3)                                     0.75          -498.045   hypate


----------------------------------


I have attempted to make use of the limma and apotome wherever
possible, since Aristoxenus always describes tones and semitones
in Pythagorean terms - see 2.55.





Monzo's measurements for Aristoxenus's genera
---------------------------------------------------



Here is a table of the intervals in cents, measured downward
from _mese_ (cents values rounded off to the nearest cent):


            ------ chromatic ------   -- diatonic --
enharmonic  relaxed hemiolic  tonic   relaxed  tense

     0         0        0        0        0      0    mese

                                              -204 \
                                       -249         |
                              -294                  |
                     -351                           | lichanos
            -370                                    |
  -408                        -408     -408   -408 <
                     -430                           |
            -438                                    | parhypate
  -453                                             /

  -498      -498     -498     -498     -498   -498    hypate



possible close ratios for Aristoxenus's irrational intervals:


7/6	1.166666667	267
11/9	1.222222222	347
8/7	1.142857143	231







And the sizes of the intervals in cents:
(cents values rounded off to the nearest cent)


                       ------ chromatic -------   -- diatonic --
           enharmonic  relaxed  hemiolic  tonic   relaxed  tense

mese
              408        370      351      294     249     204
lichanos
               45         68       79      114     159     204
parhypate
               45         60       68       90      90      90
hypate



||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||



Problem in Aristoxenus's 'hemiolic chromatic genus'
---------------------------------------------------



It is obvious from this table, where the 'hemiolic chromatic diesis'
is slightly larger than the 'semitone' ('limma'), that Aristoxenus's
calculations break down in describing the shades of chromatic,
particularly confusing the 'relaxed' and 'hemiolic'.


He states at 2.51 that the hemiolic chromatic pyknon is an
enharmonic diesis less than a tone above hypate:  in cents,
906 - 45 = 861.


At 1.25 he says that the lowest diatonic lichanos is a semitone
higher than the hemiolic chromatic lichanos:  861 + 90 = 951 cents.


[1.25.26-30]
> He de barytate diatonos ... epi men gar ten tou hemioliou chromatos
> lichanon | hemitonion en ap' autes,
>
>> The lowest diatonic _lichanos_ ... is a semitone from the _lichanos_
>> of the hemiolic chromatic



So this works, if by 'semitone' he meant the 'limma' version,
which seems likely to me.



The problem is where he says, again at 2.51, that the hemiolic
chromatic diesis equals 1.5 times the enharmonic diesis.



<<< 
One of Aristoxenus's most brilliant discoveries is that 
1.5 * enharmonic diesis [= limma^(3/4)] is almost exactly
the same as '1/3-tone' [= tone^(1/3)] (less than 1/3-cent difference):
>>>


    ratio              decimal ratio   cents     cents difference

(9/8)^(1/3)              ~1.040041912   ~67.970
(256/243)^(3/4)          ~1.039860949   ~67.669    ~0.301





But when he says at 2.51


> It is easy to see that the hemiolic _pyknon_ is greater than that
> of the soft chromatic, for the former falls short of being a tone
> by an enharmonic diesis, the latter by a chromatic diesis.



Let's tear that apart:


> ... the hemiolic _pyknon_ ... falls short of being a tone
> by an enharmonic diesis ...

(9/8)/((256/243)^(1/2))  ~1.096063402  ~158.798	  hemiolic chromatic _pyknon_



> ... the ... _pyknon_ ... of the soft chromatic ... falls
> short of being a tone ... by a chromatic diesis.


(9/8)/((9/8)^(1/3))	~1.081687178	~135.940  relaxed chromatic _pyknon_





he lessens the importance of his finding, because this
statement is self-contradictory:  (256/243)^(3/2) cannot equal
(9/8)/((256/243)^(1/2)), but is rather exactly a Pythagorean comma
smaller:


    ratio              decimal ratio   cents     cents difference
					
(9/8)/((256/243)^(1/2))  ~1.096063402  ~158.798	
(256/243)^(3/2)          ~1.081310792  ~135.337   ~23.460

(3^12)*(2^-19)           ~1.013643265   ~23.460	




In fact, 1.5 * enharmonic diesis [= (256/243)^(3/4)] gives the
*relaxed* chromatic diesis and not the hemiolic:


  ratio           decimal ratio   cents    cents difference

(256/243)^(3/4)   ~1.039860949   ~67.669	
(9/8)^(1/3)       ~1.040041912   ~67.970    -0.301




As we have already seen above, this equals almost exactly the
relaxed chromatic diesis of '1/3-tone'.





There is thus no possible way that the hemiolic chromatic can
have two equal dieses in its pyknon which are 1.5 * enharmonic diesis,
and still place the lichanos in its proper position 'an
enharmonic diesis less than a tone above hypate'.





||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||



ARISTOXENUS'S CONCEPTION OF TETRACHORDAL DIVISION
(c)1999 by Joe Monzo



r. = relaxed
t. = tense or tonic
h. = hemiolic


d. = diatonic
c. = chromatic
e. = enharmonic


l. = lichanos
p. = parhypate




Hemiolic chromatic


=======  pitches tuned in the normal Pythagorean way
*******  the hemiolic chromatic moveable notes
-------  notes from other genera used for comparison
- - - -  subdivision of the semitones into 4 parts, i.e., '1/8-tones'


        /   8:9 ======= t.d.l.
       |        - - - -
limma  |        ------- r.d.l.                                            \
       |        - - - -                                                    |semitone 
        > 27:32 ======= t.c.l                 \ enharmonic diesis          |= 1/2-limma 
       |        - - - -                        |= 1/2-apotome              |  + 1/2-apotome
apotome|        ******* h.c.l.                <                           < 
       |        - - - -                        |1&1/2 enharmonic dieses    |diesis 
        > 64:81 ======= e.l., r/t.d.p., t.c.p. |= 1/2-apotome + 1/4-limma / = 1/2-apotome
       |        ******* h.c.p.                <
limma  |        ------- e.p.                   |1&1/2 enharmonic dieses
       |        - - - -                        |= 3/4-limma
        \  3:4  ======= Hypate                /




Relaxed chromatic


=======  pitches tuned in the normal Pythagorean way
*******  the relaxed chromatic moveable notes
-------  notes from other genera used for comparison
- - - -  subdivision of the semitones into 6 parts, i.e., '1/12-tones'


        /  8:9  ======= t.d.l.
       |        - - - -
       |        - - - -
limma  |        ------- r.d.l.  \
       |        - - - -          |semitone (1/2-limma + 1/2-apotome) + 1/6-apotome
       |        - - - -          |
        > 27:32 ======= t.c.l.   |                \
       |        - - - -          |                 |
       |        - - - -          |                 |chromatic diesis = 2/3-apotome
apotome|        ------- h.c.l.   | \1/12-tone      |  
       |        ******* r.c.l.  /  /= 1/6-apotome <                   \
       |        - - - -                            |chromatic diesis   |1/6-tone 
        > 64:81 ======= e.l., r/t.d.p., t.c.p.     |= 1/3-apotome     / = 1/3-apotome
       |        - - - -                            |  + 1/3-limma
       |        ******* r.c.p.     \1/12-tone     <      
limma  |        ------- e.p.       /= 1/6-limma    |    
       |        - - - -                            |
       |        - - - -                            |chromatic diesis = 2/3-limma
        \  3:4  ======= Hypate                    /



It is apparent from the diagram that each chromatic diesis could also
be thought of as '1/3-tone'.





CONCLUSIONS
-----------


If our examination of Aristoxenus's tetrachordal divisions is correct,
we see that his 'enharmonic diesis' is usually considered to be
1/2-limma, but is sometimes 1/2-apotome, which are less than 6 cents
smaller and larger, respectively, than an exact 1/4-tone, and in
Aristoxenus's theory must be considered to be aurally equivalent to it:


     ratio       decimal ratio  cents

(2187/2048)^(1/2)  ~1.033378   ~56.843   1/2-apotome =  1/4-tone + ~5.865 cents
(9/8)^(1/4)        ~1.029886   ~50.978   1/4-tone
(256/243)^(1/2)    ~1.026400   ~45.112   1/2-limma   =  1/4-tone - ~5.865 cents





=====================================================



POSSIBILITIES FOR FURTHER INVESTIGATION
---------------------------------------



Appendix 1:  Aristoxenus's fractional divisions
-----------------------------------------------


					
Comparison of various fractional divisions of the
Pythagorean tone and both sizes of Pythagorean semitones:


					
     ratio         decimal ratio  cents   difference in cents from largest


'1/24-tone'

(2187/2048)^(1/12)  1.005487253   ~9.474	
(9/8)^(1/24)        1.004919689   ~8.496   ~0.978
(256/243)^(1/12)    1.004352445   ~7.519   ~1.955


'1/12-tone'

(2187/2048)^(1/6)   1.011004616  ~18.948
(9/8)^(1/12)        1.009863581  ~16.993   ~1.955
(256/243)^(1/6)     1.008723833  ~15.037   ~3.910


'1/6-tone'

(2187/2048)^(1/3)   1.022130333  ~37.895	
(9/8)^(1/6)         1.019824451  ~33.985   ~3.910
(256/243)^(1/3)     1.017523771  ~30.075   ~7.820


'1/4-tone'

(2187/2048)^(1/2)   1.033378485  ~56.843	
(9/8)^(1/4)         1.029883572  ~50.978   ~5.865
(256/243)^(1/2)     1.026400479  ~45.112  ~11.730







Appendix 2: Cleonides's (4/3)^(1/60)
------------------------------------



Later followers of Aristoxenus, particularly Cleonides, made use
of a division of the 'perfect 4th' into 30 equal parts in order
to explain Aristoxenus's divisions.  This was a result of combining
Aristoxenus's statement that 'the 4th is made up of 2&1/2 tones'
with his explicit use of '1/12-tones':  2.5 * 12 = 30.



[Cleonides, section 7; Strunk 1950, p 39-40]
> The tone is assumed to be divided into twelve least parts,
> of which each one is called a twelfth-tone.  The remaining
> intervals are also assumed to be divided in the same proportion,
> the semitone into six twelfths, the diesis equivalent to a
> quarter-tone into three twelfths, the diesis equivalent to a
> third-tone into four twelfths, the whole diatessaron into
> thirty twelfths.


I think that Cleonides most likely also intended regular Pythagorean
tuning for the basic intervals, following Aristoxenus closely, but
earlier in section 7 he mixes the use of 'equal' and 'equivalent'
in his description of the shades of genera, and blurs our reception
of his concept very successfully.


His frequent use of 'equivalent' in describing the interval sizes
seems to indicate that the division of the 4/3 into 30 parts is
to be taken very loosely.



Simply for the sake of analyzing this division, let us see what results
if we assume that Cleonides meant for the 4/3 '4th' to be divided evenly
into 30 parts, and tabulate his statements:


    interval         1/12-tones     (4/3)^(x/30) decimal ratio  cents

 tone              (12) 1/12-tones  (4/3)^(12/30)   1.121955   ~199.218
 semitone          6/12-tone        (4/3)^( 6/30)   1.059224   ~ 99.609
'1/4-tone diesis'  3/12-tone,       (4/3)^( 3/30)   1.029186   ~ 49.804
'1/3-tone diesis'  4/12-tone,       (4/3)^( 4/30)   1.039103   ~ 66.406
'4th'              (30) 1/12-tones  (4/3)^(30/30)   1.333333   ~498.045



[Cleonides, section 7; Strunk 1950, p 40]
> In terms of quantity, then, the enharmonic will be sung by 3, 3,
> and 24 twelfths, the soft chromatic by 4, 4, and 22, the hemiolic
> chromatic by 4&1/2, 4&1/2, and 21, the tonic chromatic by 6, 6,
> and 18, the soft diatonic by 6, 9, and 15, the syntonic diatonic
> by 6, 12, and 12.



                     1/12-tones

enharmonic           3, 3, 24
soft chromatic       4, 4, 22
hemiolic chromatic   4&1/2, 4&1/2, 21
tonic chromatic      6, 6, 18
soft diatonic        6, 9, 15
syntonic diatonic    6, 12, 12



Because of his use of 4&1/2 1/12-tones, Cleonides is really using
a conceptual division here of the '4th' into 60 parts, (4/3)^(x/60).
We will thus double all of our division-numbers.


    interval         1/24-tones     (4/3)^(x/60) decimal ratio  cents

 tone              (24) 1/24-tones  (4/3)^(24/60)   1.121955   ~199.218
 semitone          12/24-tone       (4/3)^(12/60)   1.059224   ~ 99.609
'1/4-tone diesis'  6/24-tone,       (4/3)^( 6/60)   1.029186   ~ 49.804
'1/3-tone diesis'  8/24-tone,       (4/3)^( 8/60)   1.039103   ~ 66.406
'4th'              (60) 1/24-tones  (4/3)^(60/60)   1.333333   ~498.045



                    1/24-tones

enharmonic           6, 6, 48
soft chromatic       8, 8, 44
hemiolic chromatic   9, 9, 42
tonic chromatic      12, 12, 36
soft diatonic        12, 18, 30
syntonic diatonic    12, 24, 24




     ratio        decimal ratio   cents 

1/((4/3)^(24/60))   0.891301   ~-199.218   tense diatonic lichanos
1/((4/3)^(30/60))   0.866025   ~-249.022   relaxed diatonic lichanos
1/((4/3)^(36/60))   0.841466   ~-298.827   tonic chromatic lichanos
1/((4/3)^(42/60))   0.817604   ~-348.631   hemiolic chromatic lichanos
1/((4/3)^(44/60))   0.809801   ~-365.233   relaxed chromatic lichanos
1/((4/3)^(48/60))   0.794418   ~-398.436   enharmonic lichanos, diatonic parhypate, tonic chromatic parhypate
1/((4/3)^(51/60))   0.783073   ~-423.338   hemiolic chromatic parhypate
1/((4/3)^(52/60))   0.779327   ~-431.639   relaxed chromatic parhypate
1/((4/3)^(54/60))   0.77189    ~-448.240   enharmonic parhypate

for comparison:

1/(9/8)             0.888889   ~-203.910
1/(81/64)           0.790123   ~-407.820
1/(9/8)/(256/243)   0.84375    ~-294.135
1/(5/4)             0.8        ~-386.314



Cleonides's measurements of Aristoxenus's genera
------------------------------------------------

            ------ chromatic ------   -- diatonic --
enharmonic  relaxed hemiolic  tonic   relaxed  tense

     0         0        0        0        0      0    mese

                                              -199 \  
                                                    |            -204  9:8
                                       -249         |
                                                    |            -294 32:27
                              -299                  | lichanos
                     -349                           |
            -365                                    |
                                                    |            -386  5:4
 -398                         -398     -398   -398 <
                                                    |            -408 81:64
                     -423                           | parhypate 
            -432                                    |
 -448                                              /

 -498       -498     -498     -498     -498   -498    hypate




The importance of Cleonides's concept in the history of music-theory
should not be underestimated:


[Strunk 1950, p 34]
> [Valla 1497] thus became one of the sources from which the
> musicians of the Renaissance drew their information about the
> music of Classical Antiquity.




Appendix 3: Some good rational interpretations of Aristoxenus
-------------------------------------------------------------


<<>>




Appendix 4:  Going completely the other way: Aristoxenus in 144-ET
------------------------------------------------------------------



This paper describing rational interpretations of the most non-rational
of theorists originated in communications to the Tuning List refuting
the validity of giving Aristoxenus credit for the approximations
leading to temperament.


Ending in a total 'about-face', I'd like to present a version of
Aristoxenus's tetrachords in 144-tone equal-temperament ['144-ET']
which I think gives a rather close approximation to what Aristoxenus
insisted were just a few examples of the infinite variety of shading
in tetrachord tuning, in a notation which makes it easy to understand
Aristoxenus's intervals in relation to the familiar 12-ET scale.



<<< see final section below, under date 2003.07.20 >>>





My conclusion


ARISTOXENUS'S CONCEPTION OF TETRACHORDAL DIVISION

(c)1999 by Joe Monzo


If Aristoxenus could be said to be a precursor of any equal division of the 'octave', it would be most accurate to claim
him as an advocate of 318-tET, since this temperament can quite accurately represent his implied tetrachord divisions.


- each tick-mark designates 2^(1/318) [= ~3.774 cents]

- every 6th tick-mark [notated 'xx'] designates 2^(1/53), the smallest 'octave'-based temperament which accurately
  represents the Pythagorean comma, and thus clearly separates the two different sizes of Pythagorean semitone: 2^(4/53)
  represents 256/243 [~0.341 cent too large] and 2^(5/53) represents 2187/2048 [~0.477 cent too small].


Aristoxenus's measurements:  e = enharmonic, c = chromatic, d = diesis, s = semitone, t = tone





                                                                                         relaxed   hemiolic    tonic      relaxed     tense
                                                                              enharmonic  chromatic  chromatic chromatic    diatonic   diatonic

2^(  0/318)     0.0     == 1/1================================================= MESE =====  MESE ===== MESE ==== MESE =====  MESE ===== MESE
.                                                                                 ^           ^          ^         ^           ^          ^
.                                                                                 |           |          |         |           |          |
.                                                                              (ditone)   (3s + cd)   [3s + ed]  (3s)      (5d, 2.5s)    (t)
.                                                                                 |           |          |         |           |          |
2^( 54/318)   203.774   xx 8/9 ===================================================|===========|==========|=========|===========|====  lichanos   203.910
                        -                                                         |           |          |         |           |      / | |
                        -                                                         |           |          |         |           |    /   | |
                        -                                                         |           |          |         |           |  /     | |
                        -----(8/9)/((256/243)^(1/6))-------2/24 = 1/12-tone       |           |          |         |           |/      /  |
                        -                                                         |           |          |         |           |      /   |
                        xx                                                        |           |          |         |          /|    (ed)  |
                        -                                                         |           |          |         |         / |    /     |
                        -----(8/9)/((256/243)^(2/6))-------4/24 = 1/6-tone        |           |          |         |        /  |    |     |
                        -                                                         |           |          |         |       /   |    |     |
                        -                                                         |           |          |         |      /    |    |     |
                        -                                                         |           |          |         |     /     |    |     |
2^( 66/318)   249.057   xx---(8/9)/((256/243)^(3/6))-------6/24 = 1/4-tone        |           |          |         |    /    lichanos     |      249.022
                        -                                                         |           |          |         |   /     / |  |       |
                        -                                                         |           |          |         |  /    /   |  |       |
                        -                                                         |           |          |         | /   /     |  |       |
                        -----(8/9)/((256/243)^(4/6))-------8/24 = 1/3-tone        |           |          |         |/   /      |  |       |
                        -                                                         |           |          |         |   /       |  /       |
                        xx                                                        |           |          |        /|  /        | /        |
                        -                                                         |           |          |       / | /         |/         |
                        -----(8/9)/((256/243)^(5/6))------10/24 = 5/12-tone       |           |          |      /  |/          |          |
                        -                                                         |           |          |     /   |          /|          |
                        -                                                         |           |          |    /   /|         / |          |
                        -                                                         |           |          |   /   / |        /  |          |
2^( 78/318)   294.340   xx 27/32 =================================================|===========|==========| (t)  /lichanos (s) =|=======  (t)     294.135
                        -                                                         |           |          | /   /   |      /    |          |
                        -                                                         |           |          |/   /    |     /     |          |
                        -                                                         |           |          |   /     |    /      |          |
                        -                                                         |           |         /|(s + )   |   /       |          |
                        ------(27/32)/((2048/2187)^(1/6))----2/24 = 1/12-tone     |           |        / |(1/12-t) |  /        |          |
                        xx                                                        |           |       /  |         | /         |          |
                        -                                                         |           |      /  /|         |/      (3ed, 1.5s)    |
                        -                                                         |           |     /  / |         |           |          |
                        -                                                         |           |    /  /  |        /|           |          |
                        ------(27/32)/((2048/2187)^(2/6))----4/24 = 1/6-tone      |           |   /  /   |       / |           |          |
                        -                                                         |           |  /  /    |      /  |           |          |
                        xx                                                        |           | /  /     |     /   |           |          |
                        -                                                         |           |/  /      |    /    |           |          |
                        -                                                         |           |  /       |   /     |           |          |
2^( 93/318)   350.943   ------(27/32)/((2048/2187)^(3/6))----6/24 = 1/4-tone      |          /| /    lichanos     (s)          |          |      350.978
                        -                                                         |         / ||     /  ||         |           |          |
                        -                                                         |        /  ||(1/12-t)||         |           |          |
                        xx                                                        |       /   ||   /    ||         |           |          |
                        -                                                         |      /    ||  /    / |         |           |          |
2^( 98/318)   369.811   ------(27/32)/((2048/2187)^(4/6))----8/24 = 1/3-tone      |     /  lichanos   /  |         |           |          |      369.925
                        -                                                         |    /    | |      /   |         |           |          |
                        -                                                         |   /     | |   (d)    |         |           |          |
                        -                                                         |  /     /  |   /      |         |           |          |
                        xx                                                        | /     /   |  /       |         |           |          |
                        ------(27/32)/((2048/2187)^(5/6))---10/24 = 5/12-tone     || (1/6-t)  | /      (1.5 ed)    |           |          |
                        -                                                         ||   /      |          |         |           |          |
                        -                                                         ||  /     / |          |         |           |          |
                        -                                                         || |    /   |          |         |           |          |
                        -                                                         || |  /   (cd)         |         |           |          |
2^(108/318)   407.547   xx 64/81 =============================================  lichanos  ====|==========|====  parhypate  parhypate  parhypate  407.820
                        -                                                         |           |          |         |           |          |
                        ------(64/81)/((256/243)^( 1/12))------1/24-tone          |           |          |         |           |          |
                        -                                                         |           |          |         |           |          |
                        ------(64/81)/((256/243)^( 2/12))------2/24 = 1/12-tone   |           |          |         |           |          |
                        -                                                         |           |          |         |           |          |
2^(114/318)   430.189   xx----(64/81)/((256/243)^( 3/12))------3/24 = 1/8-tone  (ed)          |       parhypate    |           |          |      430.376
                        -                                                         |           |          |         |           |          |
2^(116/318)   437.736   ------(64/81)/((256/243)^( 4/12))------4/24 = 1/6-tone    |        parhypate     |         |           |          |      437.895
                        -                                                         |       /   |          |         |           |          |
                        ------(64/81)/((256/243)^( 5/12))------5/24-tone          |  (1/12-t) |          |         |           |          |
                        -                                                         |     /     |          |         |           |          |
2^(120/318)   452.830   xx----(64/81)/((256/243)^( 6/12))------6/24 = 1/4-tone  parhypate     |          |        (s)         (s)        (s)     452.933
                        -                                                         |           |          |         |           |          |
                        ------(64/81)/((256/243)^( 7/12))------7/24-tone          |           |          |         |           |          |
                        -                                                         |           |       (1.5 ed)     |           |          |
                        ------(64/81)/((256/243)^( 8/12))------8/24 = 1/3-tone    |         (cd)         |         |           |          |
                        -                                                         |           |          |         |           |          |
                        xx----(64/81)/((256/243)^( 9/12))------9/24 = 3/8-tone  (ed)          |          |         |           |          |
                        -                                                         |           |          |         |           |          |
                        ------(64/81)/((256/243)^(10/12))-----10/24 = 5/12-tone   |           |          |         |           |          |
                        -                                                         |           |          |         |           |          |
                        ------(64/81)/((256/243)^(11/12))-----11/24-tone          |           |          |         |           |          |
                        -                                                         |           |          |         |           |          |
2^(132/318)   498.113   xx 3/4 ==============================================   HYPATE  ==  HYPATE === HYPATE == HYPATE  ==  HYPATE === HYPATE   498.045


                                                                                           relaxed    hemiolic   tonic       relaxed     tense
                                                                               enharmonic  chromatic  chromatic chromatic    diatonic   diatonic





listen to Aristoxenus's enharmonic genus
listen to Aristoxenus's relaxed chromatic genus
listen to Aristoxenus's hemiolic chromatic genus
listen to Aristoxenus's tonic chromatic genus
listen to Aristoxenus's relaxed diatonic genus
listen to Aristoxenus's tense diatonic genus


So if my above analysis is correct,
 the complete table of intervals discussed by Aristoxenus is as follows:




  interval 
 cents	
 instance
  1/12-tone 
  15.037 
  relaxed chromatic parhypate : enharmonic parhypate
  1/12-tone  
   18.948   
  hemiolic chromatic lichanos : relaxed chromatic lichanos	
  1/6-tone   
  37.895   
  relaxed chromatic lichanos : enharmonic lichanos
  enharmonic diesis  
   45.112   
  enharmonic parhypate : hypate
  enharmonic diesis   
  45.112   
  enharmonic lichanos : enharmonic parhypate
  enharmonic diesis  
   45.112   
  tense diatonic lichanos : relaxed diatonic lichanos
  diesis  
   56.843  
   hemiolic chromatic lichanos - enharmonic lichanos
  chromatic diesis   
  60.150   
  relaxed chromatic parhypate : hypate
  1.5 enharmonic dieses  
   67.669   
  hemiolic chromatic parhypate : hypate
  chromatic diesis   
  67.970   
  relaxed chromatic lichanos : relaxed chromatic parhypate
  1.5 enharmonic dieses  
   79.399  
   hemiolic chromatic lichanos : hemiolic chromatic parhypate
 semitone   
  90.225    
  diatonic & tonic chromatic parhypate : hypate
  semitone   
  101.955   
  relaxed diatonic lichanos : hemiolic chromatic lichanos
 semitone  
   113.685   
  tonic chromatic lichanos : tonic chromatic parhypate
  semitone + 1/12-tone   
  120.903  
   relaxed diatonic lichanos : relaxed chromatic lichanos
  3 enharmonic dieses   
  158.798   
  relaxed diatonic lichanos - relaxed diatonic parhypate
  tone   
  203.910   
  mese : tense diatonic lichanos
  5 dieses   
  249.022   
  mese - relaxed diatonic lichanos
  3 semitones   
  294.135   
  mese - tonic chromatic lichos
  3 semitones + enharmonic diesis   
  350.978  
   mese - hemiolic chromatic lichanos
  3 semitones + chromatic diesis   
  369.925  
   mese - relaxed chromatic lichanos
  ditone  
   407.820   
  mese - enharmonic lichanos






I've realized that
if one assumes Aristoxenus did *not* mean for his system to
be tempered, probably the most accurate interpretation would
be to assume that he used geometry to make equal divisions of
small sections of a *length of string* on the monochord!
This is a bit different from my assumption (in my 'big diagram')
that he used geometry to divide an abstract 'pitch space'.
However, the results of the two approaches are within a few cents.












I think these values are the most likely of all for Aristoxenus,
unless one assumes that he intended his system to be tempered.
It is interesting to note that the enharmonic parhypate 
I derived here is
precisely the same string-length given some 835 years later by Boethius.





ARISTOXENUS'S METHOD OF 'TUNING BY CONCORDS'

(c)1999 by Joe Monzo


- tick-marks designate Pythagorean commas = 1/9-tones





                                                                                       relaxed   hemiolic    tonic      relaxed     tense



- D            D 4:3 498
-                 / \
-                 | |
-                /   \
- C#             |   |                                      C# 81:64 408
-               /     \                                         | | 
-               |     |                                        /   \
-              /       \                                       |   |
-              |       |                                      /     \
- C           /         \                 C 32:27 294         |     |
-             |         |                     | |            /       \
-           (4:3)        \                   /   \           |       |
-            |           |                   |   |          /         \
- B         /             \    B 9:8 204    /     \         |         |
-           |             |       / \       |     |        /           \
-          /             (3:2)    | |      /       \       |          (4:3)
-          |               |     /   \     |       |      /             \
-         /                 \    |   |    /         \     |             |
- Bb      |                 |   /     \   |         |    /               \Bb 256:243 90 ---- Bb ---------- Bb
-        /                   \  |     |  /           \ (3:2)             | /                 |             |
-        |                   | /       \(4:3)        |  /                 \|                 |             |
-       /                     \|       |/             \ |                 |                  |             |
- A   MESE                    /         |             |/                  |\                 |             |
-       |                     |\       / |             \                 / |                 |             |
-        \                   / |       |(4:3)         /|                 |  \                |             |
-        |                   |  \     /   |           |(3:2)            /   |                |             |
- G#      \                 /   |     |    \         /  |               |G# 243:256 -90      |             |
-         |                 |    \   /     |         |   \             /      \              |             |
-          \              (3:2)  |   |      \       /    |             |      |              |             |
-          |               |      \ /       |       |     \          (4:3)     \             |             |
-           \             /       | |        \     /      |           |        |             |             |
- G         |             |   G 8:9 -204     |     |       \         /          \            |             |
-            \           /                    \   /        |         |          |     (262144:177147 )     |
-           (4:3)        |                    |   |         \       /            \           |           (3:2)
-             \         /                      \ /          |       |            |           |             |
- F#          |         |                 F# 27:32 -294      \     /              \          |             |
-              \       /                                     |     |              |          |             |
-              |       |                                      \   /              (4:3)       |             |
-               \     /                                       |   |                |         |             |
-               |     |                                        \ /                  \        |             |
- F              \   /                                     F 64:81 -408             |        |             |   
-                |   |                                                               \       |             |
-                 \ /                                                                |       |             |
-                 | |                                                                 \      |             |
- E           E 3:4 -498                                                              |      |             |
-                                                                                      \     |             |
-                                                                                      |     |             |
-                                                                                       \    |             |
- D#                                                                               D# 729:1024 -588        |
- Eb                                                                                                 Eb 512:729 -612






Aristoxenus claims that the resulting 262144:177147 is a '5th', but it is a very narrow one
of ~678 cents, and would not sound like a 'perfect 5th'.


Ingeniously, the resulting scale is not only composed of 12 tones, but is also perfectly
symmetrical around mese.


This tuning scheme also provides two 'tones' that are made up of 8 commas rather than 9:
(256/243)/(243/256) and (64/81)/(729/1024), both ~180.45 cents.  These both play a part
in Aristoxenus's 'proof' that 2&1/2 tones make up a '4th'.


He is explicitly ignoring the Pythagorean comma which he has bumped up against here.





1999-12-1  new thoughts


Altho he shunned the use of ratios, what Aristoxenus intended in his theories can
still be quantified numerically, based on the fact that his prescribed method of
tuning is patently Pythagorean.



318-tET can be used very successfully to convey the entire extent of Aristoxenus's
harmonic theory, and altho he almost certainly did not have in mind an equal division
of the 'octave' like this, the mathematics of his approach to tuning ultimately
do allow 318-tET to be used as the simplest accurate metric by which to measure 
his system.


<<< closeup graph of 2^(x/318) vs (256/243)^(x/24) & (2187/2048)^(x/30)  >>>




Additions 1999-12-3


Altho an examination of Aristoxenus's descriptions shows that
a 1/24-tone interval separates the hemiolic chromatic parhypate
from that of the relaxed chromatic, Aristoxenus himself never
mentions any pitch-space smaller than a 1/12-tone.


Aristides appears to be the first author to specifically state
that Aristoxenus divided the tone into 24 parts.


Boethius accepted this, and thereby extrapolated Cleonides's
(4/3)^(30/30) (with which he may not have been familiar) to
(4/3)^(60/60), of which 24 divisions made up a 'tone' and 12
a 'semitone'.  This division of course means that the 'tone'
is not the usual 9/8 ratio, but rather (4/3)^(24/60) = (4/3)^(2/5)
[= ~199.218 cents].


Westphal 1883 ignores the Aristoxenian tradition of these divisions
of the 4/3, and considers Aristoxenus's 'quarter-tones' to represent
2^(24/24) [= 24-ET], and uses the equivalent mathematical notation
(with root signs) in his illustrations of Aristoxenus's divisions.
This of course implies a '4th' with not the rational interval 4/3,
but rather the irrational 2^(10/24), which is the same as the
familar '4th' of 2^(5/12) [= 500 cents] in the 12-ET system.


Extrapolating Westphal's interpretation to include the 1/24-tones
gives us 2^(144/144) [= 144-ET].





APPENDICES - other mathematical possibilities for measuring Aristoxenus's genera


1. Cleonides's (4/3)^(1/30)


2. 144-tET - a useful approximation to Aristoxenus's system, much less sophisticated than 318-tET


3. very extended Pythagorean - I was speculating that this was a possibility for Aristoxenus,
   but Macran 1902 [p 247] gives Plutarch [33.1145] quoting Aristoxenus:

>> the magnitude of this interval (i.e., the quarter-tone) cannot be determined by concord,

>> as can the semitone, the tone, and the like.


This passage clearly indicates that Aristoxenus did not carry the Pythagorean tuning out far
enough to derive the quarter-tones from it; it just as clearly indicates, however, that
Aristoxenus expected to be able to derive his Pythagorean semitones from the method of
Tuning by Concords.


<< give extent of system: to 12 tones as described in A's Tuning by Concords >>


4. Rational - Ptolemy, me






Amendment from 2003.07.20



Aristoxenus's description of the "perfect 5th" tuned by concords,
as diagrammed above, has always bothered me.


I have finally decided that the best interpretation of his
tetrachord divisions would take into account both his descriptions
of "tuning by concords" and his equivocation of the "perfect 4th"
with "2 + 1/2 tones".  This yields the descriptions given by
Cleonides and Boethius, and in fact is very close to 144edo
as described by Westphal.


My thinking is that Aristoxenus would have tuned the "ditones"
first by means of concords (4/3 ratios), then adjusted them
to be (4/3)^(4/5).  Then, find two more notes a 4/3 on either
side of those "ditones" yields a "5th" of ~697.2629988 cents,
which is not so far off from the "pure" 3/2 "5th" as to be
objectionable -- in fact, it is a typical meantone "5th".







First, a series of concords A:D:G:C:F is tuned by 4/3s and 3/2s:



note ratio                                      ~cents

A   1                                           0
D   1*(4/3)                                   498.0449991
G   (1*(4/3))/(3/2)                          -203.9100017
C   ((1*(4/3))/(3/2))*(4/3)                   294.1349974
F   (((1*(4/3))/(3/2))*(4/3))/(3/2)          -407.8200035








Next, another series A:E:B:F#:C#:G# is tuned:


note ratio                                      ~cents

A   1                                           0
E   1/(4/3)                                  -498.0449991
B   (1/(4/3))*(3/2)                           203.9100017
F#  ((1/(4/3))*(3/2))/(4/3)                  -294.1349974
C#  (((1/(4/3))*(3/2))/(4/3))*(3/2)           407.8200035
G#  ((((1/(4/3))*(3/2))/(4/3))*(3/2))/(4/3)   -90.22499567

    








Next, Aristoxenus would temper the final two notes according
to the formula (4/3)^(x/5) (i.e., 5 semitones per "4th"), then
calculate one more concord from each of those:


F   (((1*(4/3))/(3/2))*(4/3))/(3/2) = ~ -407.8200035 cents
tempered to F  1/((4/3)^(4/5))  = ~ -398.4359993 cents,
then Bb =(1/((4/3)^(4/5)))*(4/3) = ~ 99.60899983 cents:







and G#  ((((1/(4/3))*(3/2))/(4/3))*(3/2))/(4/3) = ~ -90.22499567 cents
tempered to =(1/(4/3))*((4/3)^(4/5))  = ~ -99.60899983 cents,
then D# ((1/(4/3))*((4/3)^(4/5)))/(4/3) = ~ -597.653999 cents:






The difference between those two notes is
((1/((4/3)^(4/5)))*(4/3))/(((1/(4/3))*((4/3)^(4/5)))/(4/3))
= ~ 697.2629988 cents. So that the whole procedure yields this:





This is the only method whereby one may obtain a final interval
from this procedure which sounds reasonably like a "5th", by
following only the statements actually made by Aristoxenus.
This is the primary reason why i now consider this calculation
to be the one which best represents Aristoxenus's admittedly
approximative descriptions.


Making use of this temperament gives a "tone" of
(4/3)^(2/5) = ~ 199.2179997 cents, a "semitone"
of (4/3)^(1/5) = ~ 99.60899983 cents, and all of
the small intervals described above in Appendix 2.


144edo does indeed provide a very close approximation to
these calculations, as shown here:

















The measurements of Aristoxenus's tetrachord divisions
in 144edo is thus:


144edo  decimal ratio  cents                 name
degrees

   24    0.890898718    -200        tense diatonic lichanos
   30    0.865536561    -250        relaxed diatonic lichanos
   36    0.840896415    -300        tonic chromatic lichanos
   42    0.816957727    -350        hemiolic chromatic lichanos
   44    0.809130575    -366+(2/3)  relaxed chromatic lichanos
   48    0.793700526    -400        enharmonic lichanos, diatonic parhypate, tonic chromatic parhypate
   51    0.782321399    -425        hemiolic chromatic parhypate
   52    0.778564727    -433+(1/3)  relaxed chromatic parhypate
   54    0.771105413    -450        enharmnic parhypate




These agree exactly with the somewhat more convoluted version 
(fractional divisions of quarter-tones) given by Westphal 1883.



Westphal's measurements of Aristoxenus's genera (144edo)
--------------------------------------------------------

            -------- chromatic -------   -- diatonic --
enharmonic  relaxed    hemiolic  tonic   relaxed  tense

     0         0           0        0        0      0    mese

                                                 -200 \  
                                                       |            -204  9:8
                                          -250         |
                                                       |            -294 32:27
                                 -300                  | lichanos
                        -350                           |
            -366+(2/3)                                 |
                                                       |            -386  5:4
 -400                            -400     -400   -400 <
                                                       |            -408 81:64
                        -425                           | parhypate 
            -433+(1/3)                                 |
 -450                                                 /

 -500       -500        -500     -500     -500   -500    hypate






Below is a tabulation of all the notes necessary for
all six of the genera described by Aristoxenus, with
the fractions indicating EDO degrees taken from subsets
of 144edo, using the lowest-possible-cardinality EDO
(listed from the top down, one 8ve only; the notes are
in my 144edo HEWM notation) :


enharmonic: 24edo

A       24/24	
F       16/24	
Fv      15/24	
E       14/24	
    D          10/24
C        6/24	
Cv       5/24	
B        4/24	
    Bb          2/24
    Bbv         1/24
A   A    0/24   0/24







relaxed chromatic: 36edo

A        36/36	
F>       25/36	
F<       23/36	
E        21/36	
    D           15/36
C>       10/36	
C<        8/36	
B         6/36	
    Bb>          4/36
    Bb<          2/36
A   A     0/36   0/36







hemiolic chromatic: 48edo

A          48/48	
F^         34/48	
F~<        31/48	
E          28/48	
     D            20/48
C^         14/48	
C~<        11/48	
B           8/48	
     Bv            6/48
     Bb~<          3/48
A    A      0/48   0/48







tonic chromatic: 12edo

A         12/12	
F#         9/12	
F          8/12	
E          7/12	
    D             5/12
C#         4/12	
C          3/12	
B   B      2/12   2/12
    Bb            1/12
A   A      0/12   0/12







relaxed diatonic: 24edo

A       24/24	
Gv      19/24	
F       16/24	
E       14/24	
    D          10/24
Dv       9/24	
C        6/24	
    Cv          5/24
B        4/24	
    Bb          2/24
A   A    0/24   0/24







tense diatonic: 12edo

A       12/12	
G       10/12	
F        8/12	
E        7/12	
D   D    5/12   5/12
C   C    3/12   3/12
B        2/12	
    Bb          1/12
A   A    0/12   0/12













===========================================================



REFERENCES
----------



Aristoxenus.  c 330 BC.  _Harmonika stoicheia_.
  Athens?



Cleonides.  c 100 AD.  _Eisagoge_.
  [English translation in Strunk 1950.]



Plutarch.  _de Musica_.
  [English translation in Volume 1 of Barker 1989]


Boethius, Anicius Manlius Severinus.  c 505 AD.
  [English translation in Bower 1989]



M - Codex Venetus. c 1150. Constantinople. (with corrections from many hands)
  Ma - the original script
  Mb - corrections before 1300
  Mc - corrections 1300 or later
  Mx - unidentified corrections
  In Library of St Mark, Venice.



V - Codex Vaticanus. 1200-1400.
  Va - original script
  Vb - corrections by another hand



H - Codex from Protestant Seminary, Strassburg.  1400s?
  Destroyed in war 1870.



Valla, Georgius.  1497.  Cleonides: _Eisagoge_.
  Latin translation of Cleonides.
  Venice.



S - Codex Seldenianus. c 1500.
  In Bodleian Library, Oxford.



R - Codex Riccardianus, 1500-1600.  Florence.



B - Codex Berberinus. 1500-1550.  Bibliotheca Berberina, Rome.



Meibom, Marcus.  1652.
  _Antiquae musicae auctores septem, Graece et Latine_.
  Apud Ludovicum Elzevirium, Amsterdam.
  (Contents: Vol. 1:
    I. Aristoxeni Harmonicorum elementorum libri III.
   II. Euclidis Introductio harmonica.
  III. Nicomachi Geraseni, Pythagorici, Harmonices manuale.
   IV. Alypii Introductio musica.
    V. Gaudentii Philosophi, Introductio harmonica.
   VI. Bacchii senioris Introductio artis musicae
  Vol. 2: Aristidis Quintiliani De musica libri III
    & Martiani Capellae de musica liber IX.)
  [Contains Greek texts and Latin translations.]



Marquard, Paul.  1868.
  _Die Harmonischen Fragmente des Aristoxenus_
  Greek text and German translation.
  With critical notes and explanatory commentary.
  Berlin.




Westphal, H.  1883 (volume 1), 1893 (volume 2).
  _Aristoxenos von Tarent:  Melik und Rhythmik des Classischen Hellenentums_.
  Translated and explained.
  [Attempted reconstruction of Aristoxenus's alleged original work]
  Leipzig.
  Reprint: 1965, Georg Olms Verlagsbuchhandlung Hildesheim.



Macran, Henry Stewart.  1902.
  _The Harmonics of Aristonexus_.
  Edited with translation, notes, introduction, and index of words.
  Clarendon Press, Oxford.
  Reprinted 1974, Hildesheim; New York: G. Olms Verlag.
  [Contains complete English translation.]



Laloy, L. 1904.
  _Aristoxène de Tarente et la musique de l'Antiquité_.
  (Includes _Lexique d'Aristoxène_.)
  Paris.



Strunk, Oliver.  1950.  _Source Readings in Music History_.
  Selected and annotated [and translated].
  W. W. Norton.  New York.
  [English translation of Cleonides on p 34-46.]



da Rios, R.  1954. 
  _Aristoxeni Elementa harmonica_, edited.
  Rome.
  [Includes Latin introduction and Italian translation.]
  [Text of Aristoxenus used by Barker 1989.]



Crocker, Richard L.  1966.
  'Aristoxenus and Greek Mathematics'.
  In _Aspects of Medieval and Renaissance Music:
     A Birthday Offering to Gustave Reese_.
  Ed. Jan LaRue.
  W. W. Norton, New York.  p 96-110.



Burkert, Walter.  1972.
  _Lore and Science in Ancient Pythagoreanism_.
  English translation by Edwin L. Minar, Jr.
  Cambridge, Harvard University Press.
  (original German edition 1962)



Mathiesen, Thomas J.  1976.
  'Problems of Terminology in Ancient Greek Theory: `APMONIA',
  in _Festival Essays for Pauline Alderman: a musicological tribute_.
  Ed. Burton L. Karson.
  Brigham Young University Press; Provo, Utah. p 3-17.



Barker, Andrew. 1978.
  'Music and Perception: A Study in Aristoxenus'.
  _Journal of Hellenistic Studies_, v 98, p 9-16.



Litchfield, Malcolm.  1988.
  'Aristoxenus and Empiricism: A Reevalutation Based on his Theories'.
  _Journal of Music Theory_, v 32 # 1, p 51-73.



Barker, Andrew.  1989.
  _Greek Musical Writings_,
  volume 1:
  volume 2: 'Harmonic and Acoustic Theory'.
  Translated and edited.
  Cambridge University Press, New York.
  [Contains complete English translation of Aristoxenus
  _Elementa harmonica_ in vol 2, p 126-184.]



Bower, Calvin M.  1989.  Boethius: _Fundamentals of Music_.
  English translation of Boethius.
  Yale University Press, New Haven.



Landels, John G.  1999.
  _Music in ancient Greece and Rome_.
  Routledge, London and New York.










Updated:

2003.07.20 -- added final section on 144edo approximation of (4/3)^(x/60)

2000.10.13

1999.12.3

1999.11.25

1999.11.22

1999.11.20



By Joe Monzo





















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