
© 2004 Tonalsoft Inc. 
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 musictheory that was distinctively new, a result of the formulation of classic geometry around this time by Euclid. Aristoxenus 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 Greekspeaking areas.
When his work was rediscovered by the Italians around 1450, the idea of 12tone equaltemperament (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, or at any rate not entirely correct, 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 (1/4tones, 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 fairly 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 127) 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.
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 119125), 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 moreorless 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 selfconscious. 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 mid300s 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 musictheory 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.
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 stringlength, 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 musictheory was radically different from anything that had come before in ancient Greece.
As Andrew Barker says [Barker 1989, p 119120]:
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 musictheory 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 musictheorists 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 pitchspace, 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 pitchspace. 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.
[2.55.212]
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 pisteiei 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 stringlengths, 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.
Aristoxenus explains [1.22.521 and 2.46.2022] 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 notenames, 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 [1.2223, 2.4647].
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 4652]: 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).
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' [1.24.410, 2.46.2], and gives an elaborate 'proof' that this is so [2.56.1458.4]. Litchfield addresses this claim:
[Litchfield 1988, p 6465]
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 144tET 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 ~498.045results in the same ratio of 3/4 for _hypate_.
These notes may be rendered in letternames 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_ indicates 'compression', and refers to the grouping together of small intervals at the bottom of the tetrachord.
[1.24.1114]
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).
[1.24.1517] 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.
The 'tone' and its subdivisions
Aristoxenus defines his basic units of measurement:
[1.21.2223] 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:
[1.21.2430] Diaireistho d' eis treis diaireseis; meloideistho gar  autou to te hemisu kai to triton meros kaitetarton; 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_.
. . . . . . .
[2.46.38] 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 quartertone '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 'halftone' semitone (9/8)^(1/3) ~1.040041912 ~67.970 'thirdtone' least chromatic diesis (9/8)^(1/4) ~1.029883572 ~50.978 'quartertone' 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':
[2.55.1323] 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, letternames, 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
[2.55.2426] ean d' epi tounantion prostachthe labein to diaphonon, 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:
[2.55.2731] 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 tessaronsymphonias; 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:
[2.55.3156.12] delon de hoti hoi ten hyperochen periechontes he to dia tessaron hyperechei tou ditonou dia symphonias esontai pros allelous eilemmenou; hyparchousi men gar hoi tou dia tessaron horoi symphonoi; apo de tou oxyterou auton lambanetai phthongos symphonos epi to oxy dia tessaron, apo de tou lephthentos 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 tonhyperochen horizonton, host' einai phaneron, 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]ounton 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.
. . . . . .
[1:23.1222]
12 ..... Ο'ι μεν γαρ τηι νυν κατεχουσηι
13 μελοποιιαι συνηθεις μονον οντες εικο
14 τως την διτονον λιχανον εξοριζουσι•
15 συντονωτεραις γαρ χρωνται σχεδον ο'ι
16 πλειστοι των νυν. τουτου δ' αιτιον το βουλε
17 σθαι γλυκαινειν αει. σημειον δ' 'οτι του
18 του στοχαζονται. μαλιστα μεν γαρ και
19 πλειστον χρονον εν τωι χρωματι δια
20 τριβουσιν. 'οταν δ' αφικωνται ποτε εις
21 την 'αρμονιαν, εγγυς του χρωματος
22 προσαγουσι συνεπισπωμενου του ηθους.
....... Hoi men gar tei nun katechousei
melopoiiai synetheis monon ontes eiko
tos ten ditonon lichanon exorizousi;
syntonoterais gar chrontai schedon hoi
pleistoi ton nun. toutou d' aition to boules
thai glukainein aei, semeion d' hoti tou
tou 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 semiditone  tone (5/4)/(32/27) 1.0546875 ~ 92.179 'sweet' ditone  semiditone (81/64)/(32/27) 1.0678711 ~113.685 apotome = true ditone  semiditone (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 'semiditone' (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 5limit '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 5limit '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 1/10tone. Aristoxenus explicity says in his descriptions that he expects us to perceive a 1/12tone difference between _lichanoi_ of two of his genera, and implies that we should perceive a 1/24tone difference between two others [see below].
We have already seen how he stipulated and preferred the 81/64 'ditone'. This blurring of distinctions between functionallyidentical 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 'quartertone' 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 'quartertone'. 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 geometricallyequal division, the irrational interval (256/243)^(1/2), for 'enharmonic diesis' or 'quartertone'.
The 'thirdtone' 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 '1/3tone' [= tone^(1/3)], with less than 1/3cent 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 'thirdtone' and 'least chromatic diesis', altho this will cause problems upon closer analysis, which will be encountered later in this paper.
Aristoxenus describes the ranges for the two moveable notes in the tetrachord as follows:
The range of the _lichanoi_
[1.24.2225] katholou gar barytatai men hai enarmonioi lichanoi esan, echomenai d' hai chromatikai, syntonotatai d' hai diatonoi. taken overall, the enharmonic _lichanoi_ are the lowest, the chromatic next, and the diatonic the highest. [1.22.2730] Lichanou men oun esti toniaios ho sympas topos en hoi kineitai, oute gar elatton aphistatai meses toniaiou diastematos 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. [2.46.2833] 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_:
[1.26.1327] 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. [1.26.2427] 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. [1.26.1418] 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_
[1.23.2729] 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.[1.23.2729] let the range of ... _parhypate_ [be agreed to be] the smallest diesis [i.e., '1/4tone'], 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 '1/2tone' and '1/4tone':
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 '1/4tone' 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.
[1.23.3024.1] 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_. [2.46.3447.8] 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:
[1.26.3033, 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 1.26.3033] 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 [1.24.1632] 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_.
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[1.24.1525.11]
[1.24.1525.11] > [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 lichanoi eilemmenai duo genon barytatai, > he men harmonias he de chromatos. katholou gar barytatai men > hai enarmonioi lichanoi esan, echomenai d' hai chromatikai, > syntonotatai 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 > diastemmatos 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. [1.25.1126.7] [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 chromatike 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:
^{[25.11 ...]} He men oun ^{[12]} barytate chromatike lichanos tes ^{[13]} enarmoniou barytates hektoi merei to^{[14]}nou oxytera estin.  ^{[25.11 ...]}Thus the ^{[12]} lowest chromatic lichanos from ^{[13]} the lowest enharmonic a sixth part of a tone ^{[14]} is higher. 
epeideper he chro^{[15]}matike diesis tes enarmoniou dieseos ^{[16]} dodekatemorioi tonou meizon esti.  since the chromatic ^{[15]} diesis from the enharmonic diesis ^{[16]} by a twelfth part of a tone is greater. 
Dei ^{[17]} gar to tou autou tritemorion tou tetartou ^{[18]} merous dodekatemorioi hyperechein.  A ^{[17]} third part of anything from a quarter of the ^{[18]} same thing a twelfthpart must exceed. 
^{[19]} hai de duo chromatikai ton duo ^{[20]} enarmonion delon hos toi diplasioi.  ^{[19]} and the two chromatics from the two ^{[20]} enharmonics evidently must double that. 
^{[21]} touto de estin hektemorion elatton dia^{[22]}stema tou elachistou ton meloidoume^{[23]}non.  ^{[21]} This is a sixth part, an interval smaller ^{[22]} than the least of the melodic ^{[23]} ones. 
Ta de toiauta ameloideta ^{[24]} estin. ameloideton gar legomen, ho me ^{[25]} tattetai kath' heauto en systemati.  Such intervals are unmelodic, ^{[24]} since we call 'unmelodic' any ^{[25]} that is not placed in a systema. 
He de ^{[26]} barytate diatonos tes barytates ^{[27]} chromatikos hemitonioi kai dodekate^{[28]}morioi tonou oxytera estin.  The ^{[26]} lowest diatonic (lichanos) from the lowest ^{[27]} chromatic a semitone and a twelfth ^{[28]} part of a tone is higher. 
epi men gar ^{[29]} ten tou hemioliou chromatos lichanon ^{[30]} hemitonion en ep' autes.  We said that ^{[29]} from the hemiolic chromatic lichanos, ^{[30]} it is a semitone. 
apo de tes hemio^{[31]}liou epi ten enarmonion, diesis.  and from the hemiolic ^{[31]} to the enharmonic is a diesis. 
apo ^{[32]} de tes enarmoniou epi ten baryta^{[33]}ten chromatiken, hektemorion.  from ^{[32]} the enharmonic to the lowest ^{[33]} chromatic is a sixth part. 
apo de ^{[34]} tes barytates chromatikes epi ten hemiolion, dodekatemorion tonou.  from ^{[34]} the lowest chromatic to the hemiolic is a twelfth part of a tone. 
to ^{[26.1]} de tetartemorion ek trion dodeka^{[2]}temorion synkeitai.  Now ^{[26.1]} a quarter of three twelfth^{[2]}parts is composed. 
host' einai phane^{[3]}ron,  so that it is clear ^{[3]} that 
hoti to eiremenon diastema estin ^{[4]} apo tes barytates diatonou, epi ten ^{[5]} barytaten chromatiken.  it is the interval stated ^{[4]} from the lowest diatonic, to the ^{[5]} lowest chromatic. 
he de syn^{[6]}tonotate diatonOS tes barytates ^{[7]} diatonou, diesei esti syntonotera.  The tensest ^{[6]} diatonic (lichanos) from the lowest ^{[7]} diatonic is a diesis tenser. 
> Ek touton de phaneroi gignontai hoi topoi ton lichanon hekastes;
==========
The enharmonic genus 
^{[1.24.1521]}
> Touton
Extracting what is relevant to this genus:
^{[1.24.1521]} >> 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
[2.50.2324]
> Mia men oun
> [2.50.22 ...] Mia men [23] oun
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 hypateOther 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'. [1.23.510]
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:
[1.24.1521] > 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 1/3tone. Assuming an exact 3part 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 = 1/3tone ('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.
[1.25.1114] > 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, ...
[1.25.3233] > apo de tes enarmoniou epi ten barytaten chromatiken hektemorion, > >> from the enharmonic to the lowest chromatic _lichanos_ is a >> sixth part of a tone, ...
[1.25.1416] > epeideper he chromatike 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 '1/6tone' and '1/12tone' 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 ~17cent size of the latter to be perceptible.
He explains by simple mathematics that since the 'chromatic diesis' is 1/3tone and the 'enharmonic diesis' is 1/4tone, the difference between them must be 1/3 [= 4/12]  1/4 [= 3/12] = 1/12tone. 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 1/12tones 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 = 1/2limmaTwo of these dieses do indeed comprise a _pyknon_ which is exactly 1/6tone higher than the 'ditone' enharmonic _lichanos_, i.e., exactly 1/6tone larger than a 'limma' semitone.
Comparing this with his earlier definition of the 'chromatic diesis' as '1/3tone', 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 '1/4tone'. It seems thus far that Aristoxenus's characterizations of the smaller dieses as 1/4 and 1/3tones is simply an approximation.
[2.51.48] > [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 1/2limma plus 1/12tone, and thus the _pyknon_ to equal the 'limma' semitone plus 1/6tone, 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 1/12tone larger than), this chromatic diesis:
ratio decimal ratio cents (9/8)/((256/243)*((9/8)^(1/6))) 1.047113 ~79.700 tone  (limma + 1/6tone)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  (1/2limma + 1/12tone)These values therefore do not work out.
Let us now reexamine Aristoxenus's definitions of the 1/6 and 1/12tone based on our earlier value for the 'chromatic diesis' as 1/3tone.
If we subtract the 'enharmonic diesis' (1/2limma) from the 'chromatic diesis' (1/3tone), we get an interval that is actually much closer to 1/9tone:
ratio decimal ratio cents ((9/8)^(1/3))/((256/243)^(1/2)) 1.013291 ~22.858 '1/12tone' = 1/3tone  1/2limmaAnd doubling this interval (i.e., subtracting the enharmonic _pyknon_ composed of two 1/2limmas [= one 'limma' semitone] from the relaxed chromatic _pyknon_ spanning an interval of 2/3tone) gives a result that is nothing like 1/6tone, but rather is almost exactly the same as our 'enharmonic diesis' of 1/2limma:
ratio decimal ratio cents ((9/8)^(2/3))/(256/243) 1.026758 ~45.715 2/3tone  'limma' semitone (256/243)^(1/2) 1.0264 ~45.112 1/2limmaIf our value of 1/2limma for the 'enharmonic diesis' is correct, then an exact 1/3tone 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 redefine 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 1/3tone.
ratio decimal ratio cents (9/8)/((9/8)^(1/3)) ~1.081687178 ~135.940 pyknon = '1/3tone' less than a tone ((9/8)^(1/3))^2 1.081687 ~135.940 pyknon = two '1/3tones' 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 + 1/6tone (9/8)/((256/243)*((9/8)^(1/6))) 1.047113 79.700 tone  pyknon; pyknon = (limma + 1/6tone) > DOES NOT EQUAL CHR DIESIS (9/8)/(((256/243)^(1/2))*((9/8)^(1/12))) 1.085358 141.805 pyknon = tone  (1/2limma + 1/12tone)  (9/8)/((9/8)^(1/3)) 1.081687 135.940 pyknon = tone  1/3tone ((9/8)^(1/3))^2 1.081687 135.940 pyknon = two 1/3tones  this is the only soft chr that works ((9/8)^(1/3))/((256/243)^(1/2)) 1.013291 22.858 '1/12tone' = 1/3tone  1/2limma 9/8 1.125 203.910 8.920922 =this means that the above value is really close to a 1/9tone ((9/8)^(2/3))/(256/243) 1.026758 45.715 4.460461 =2/3tone  semitone DOES NOT EQUAL 1/6tone (256/243)^(1/2) 1.0264 45.112 1/2limma, for comparisonxxxxxxxxxxxxxxxxxxxxxx
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_]...
[1.25.3031] > 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:
[2.51.47] > the hemiolic _pyknon_ ... falls short of being a tone by > an enharmonic diesis
Assuming that the 'enharmonic diesis' is always 1/2limma does not agree with both of these propositions simultaneously:
ratio decimal ratio cents (256/243)^1.5 1.081311 ~135.337 1/2limma * 1.5 (9/8)/((256/243)^(1/2)) 1.096063 ~158.798 tone  1/2limmaIf, 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 1/2limma and in others equal to 1/2apotome.
If we make the 'enharmonic diesis' above the _pyknon_ equal to 1/2apotome, and make the size of the _pyknon_ itself, spanning an interval 'one and a half times the enharmonic _pyknon_', equal to the limma plus 1/2apotome (that is, three 'quartertones' distributed as two 1/2limmas and one 1/2apotome), 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  1/2apotome (256/243)*((2187/2048)^(1/2)) 1.088662 ~ 147.067 pyknon = limma + 1/2apotomeWe 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:
[1.25.3031] > 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 1/2apotome:
ratio decimal ratio cents (2187/2048)^(1/2) 1.033378 ~56.843 1/2apotomeproviding 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 1/2limma plus 1/2apotome, an interval totally uncharacteristic of any other Greek musictheory, 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 ~73.534Considering 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 ~49.022which is ~1.955 cents less than an exact 1/4tone, ~7.820 cents less than 1/2apotome, and ~3.910 cents more than 1/2limma.
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_ >>
[1.26.12] > 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=tonediesis'  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=tonediesis'  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=tonediesis' 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 hypateThe '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 ... [2.51.811] > toniaiou de chromatos diaipesis estin en he to men pyknon > ex hemitonion 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 1/2tone 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 + apotomeThus 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 + limmawith 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 hypateThe 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 hypateThe 'relaxed' shade of the diatonic genus 
[1.24.2830] > pempton de pros toi autoi, to ex hemitoniou kai hemioliou > diastemmatos 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, ...
[1.25.2628] > 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[1.25.2630] > 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 1/2limmas and 1/2apotomes.
We will probably do best to measure downward from _mese_, assuming 'four dieses' to mean a 9/8 tone, and locating _lichanos_ either 1/2limma or 1/2apotome from that:
1/(9/8)/((256/243)^(1/2)) 0.8660254 ~249.022 mese  tone  1/2limma 1/(9/8)/((2187/2048)^(1/2)) 0.8601775 ~260.753 mese  tone  1/2apotomeThe 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 hypateThe '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 hypateThe 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=tonediesis'  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=tonediesis'  diesis=tone/4 2 2/((9/8)*(256/243)*((9/8)^(1/3))) 1.622530767 837.895002 relaxed chromatic lichanos  'pyknon=tonediesis'  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=tonediesis'  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=tonediesis'  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=tonediesis'  diesis=tone/4 2 1/((9/8)*(256/243)*((9/8)^(1/3))) 0.811265383 362.105 relaxed chromatic lichanos  'pyknon=tonediesis'  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=tonediesis'  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 hypatepossible close ratios for Aristoxenus's irrational intervals:
7/6 1.166666667 267 11/9 1.222222222 347 8/7 1.142857143 231And 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.
[1.25.2630] > 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 '1/3tone' [= tone^(1/3)] (less than 1/3cent difference): >>>
ratio decimal ratio cents cents difference (9/8)^(1/3) ~1.040041912 ~67.970 (256/243)^(3/4) ~1.039860949 ~67.669 ~0.301But 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 selfcontradictory: (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 ~23.460In 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 0.301As we have already seen above, this equals almost exactly the relaxed chromatic diesis of '1/3tone'.
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., '1/8tones' / 8:9 ======= t.d.l.      limma   r.d.l. \      semitone > 27:32 ======= t.c.l \ enharmonic diesis = 1/2limma      = 1/2apotome  + 1/2apotome apotome ******* h.c.l. < <      1&1/2 enharmonic dieses diesis > 64:81 ======= e.l., r/t.d.p., t.c.p. = 1/2apotome + 1/4limma / = 1/2apotome  ******* h.c.p. < limma   e.p. 1&1/2 enharmonic dieses      = 3/4limma \ 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., '1/12tones' / 8:9 ======= t.d.l.           limma   r.d.l. \      semitone (1/2limma + 1/2apotome) + 1/6apotome       > 27:32 ======= t.c.l.  \              chromatic diesis = 2/3apotome apotome  h.c.l.  \1/12tone   ******* r.c.l. / /= 1/6apotome < \      chromatic diesis 1/6tone > 64:81 ======= e.l., r/t.d.p., t.c.p. = 1/3apotome / = 1/3apotome       + 1/3limma  ******* r.c.p. \1/12tone < limma   e.p. /= 1/6limma             chromatic diesis = 2/3limma \ 3:4 ======= Hypate /
It is apparent from the diagram that each chromatic diesis could also be thought of as '1/3tone'.
CONCLUSIONS 
If our examination of Aristoxenus's tetrachordal divisions is correct, we see that his 'enharmonic diesis' is usually considered to be 1/2limma, but is sometimes 1/2apotome, which are less than 6 cents smaller and larger, respectively, than an exact 1/4tone, 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 1/2apotome = 1/4tone + ~5.865 cents (9/8)^(1/4) ~1.029886 ~50.978 1/4tone (256/243)^(1/2) ~1.026400 ~45.112 1/2limma = 1/4tone  ~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 '1/24tone' (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 '1/12tone' (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 '1/6tone' (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 '1/4tone' (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 ~11.730Appendix 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 '1/12tones': 2.5 * 12 = 30.
[Cleonides, section 7; Strunk 1950, p 3940] > The tone is assumed to be divided into twelve least parts, > of which each one is called a twelfthtone. The remaining > intervals are also assumed to be divided in the same proportion, > the semitone into six twelfths, the diesis equivalent to a > quartertone into three twelfths, the diesis equivalent to a > thirdtone 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 1/12tones (4/3)^(x/30) decimal ratio cents tone (12) 1/12tones (4/3)^(12/30) 1.121955 ~199.218 semitone 6/12tone (4/3)^( 6/30) 1.059224 ~ 99.609 '1/4tone diesis' 3/12tone, (4/3)^( 3/30) 1.029186 ~ 49.804 '1/3tone diesis' 4/12tone, (4/3)^( 4/30) 1.039103 ~ 66.406 '4th' (30) 1/12tones (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.
1/12tones 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, 12Because of his use of 4&1/2 1/12tones, 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 divisionnumbers.
interval 1/24tones (4/3)^(x/60) decimal ratio cents tone (24) 1/24tones (4/3)^(24/60) 1.121955 ~199.218 semitone 12/24tone (4/3)^(12/60) 1.059224 ~ 99.609 '1/4tone diesis' 6/24tone, (4/3)^( 6/60) 1.029186 ~ 49.804 '1/3tone diesis' 8/24tone, (4/3)^( 8/60) 1.039103 ~ 66.406 '4th' (60) 1/24tones (4/3)^(60/60) 1.333333 ~498.045 1/24tones 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 hypateThe importance of Cleonides's concept in the history of musictheory 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 144ET 
This paper describing rational interpretations of the most nonrational 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 'aboutface', I'd like to present a version of Aristoxenus's tetrachords in 144tone equaltemperament ['144ET'] 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 12ET scale.
<<< see final section below, under date 2003.07.20 >>>
My conclusion
ARISTOXENUS'S CONCEPTION OF TETRACHORDAL DIVISION
(c)1999 by Joe MonzoIf 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 318tET, since this temperament can quite accurately represent his implied tetrachord divisions.
 each tickmark designates 2^(1/318) [= ~3.774 cents]
 every 6th tickmark [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 = 1/12tone     / /        /  xx     / (ed)       /  /  (8/9)/((256/243)^(2/6))4/24 = 1/6tone     /         /         /         /    2^( 66/318) 249.057 xx(8/9)/((256/243)^(3/6))6/24 = 1/4tone     / lichanos  249.022      / /         / /         / /    (8/9)/((256/243)^(4/6))8/24 = 1/3tone    / /         /  /  xx    / /  /      /  / /  (8/9)/((256/243)^(5/6))10/24 = 5/12tone    / /       /  /      / / /       / /  /   2^( 78/318) 294.340 xx 27/32 ====================================================================== (t) /lichanos (s) ======== (t) 294.135     / /  /      / /  /       /  /      /(s + )  /   (27/32)/((2048/2187)^(1/6))2/24 = 1/12tone   / (1/12t)  /   xx   /   /      / / / (3ed, 1.5s)     / /        / /  /   (27/32)/((2048/2187)^(2/6))4/24 = 1/6tone   / /  /       / /  /    xx   / /  /      / /  /       /  /    2^( 93/318) 350.943 (27/32)/((2048/2187)^(3/6))6/24 = 1/4tone  / / lichanos (s)   350.978   /  /       / (1/12t)    xx  /  /       /  / /     2^( 98/318) 369.811 (27/32)/((2048/2187)^(4/6))8/24 = 1/3tone  / lichanos /     369.925   /   /       /   (d)       / /  /     xx  / /  /     (27/32)/((2048/2187)^(5/6))10/24 = 5/12tone  (1/6t)  / (1.5 ed)      /        / /         /         / (cd)     2^(108/318) 407.547 xx 64/81 ============================================= lichanos ================== parhypate parhypate parhypate 407.820        (64/81)/((256/243)^( 1/12))1/24tone              (64/81)/((256/243)^( 2/12))2/24 = 1/12tone              2^(114/318) 430.189 xx(64/81)/((256/243)^( 3/12))3/24 = 1/8tone (ed)  parhypate    430.376        2^(116/318) 437.736 (64/81)/((256/243)^( 4/12))4/24 = 1/6tone  parhypate     437.895   /      (64/81)/((256/243)^( 5/12))5/24tone  (1/12t)        /      2^(120/318) 452.830 xx(64/81)/((256/243)^( 6/12))6/24 = 1/4tone parhypate   (s) (s) (s) 452.933        (64/81)/((256/243)^( 7/12))7/24tone          (1.5 ed)    (64/81)/((256/243)^( 8/12))8/24 = 1/3tone  (cd)            xx(64/81)/((256/243)^( 9/12))9/24 = 3/8tone (ed)             (64/81)/((256/243)^(10/12))10/24 = 5/12tone              (64/81)/((256/243)^(11/12))11/24tone              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 diatoniclisten 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 genusSo if my above analysis is correct, the complete table of intervals discussed by Aristoxenus is as follows:
interval cents instance 1/12tone 15.037 relaxed chromatic parhypate : enharmonic parhypate 1/12tone 18.948 hemiolic chromatic lichanos : relaxed chromatic lichanos 1/6tone 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 + 1/12tone 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 stringlength given some 835 years later by Boethius.
ARISTOXENUS'S METHOD OF 'TUNING BY CONCORDS'
(c)1999 by Joe Monzo tickmarks designate Pythagorean commas = 1/9tones
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 612Aristoxenus 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.
1999121 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.
318tET 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 318tET 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 1999123
Altho an examination of Aristoxenus's descriptions shows that a 1/24tone interval separates the hemiolic chromatic parhypate from that of the relaxed chromatic, Aristoxenus himself never mentions any pitchspace smaller than a 1/12tone.
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 'quartertones' to represent 2^(24/24) [= 24ET], 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 12ET system.
Extrapolating Westphal's interpretation to include the 1/24tones gives us 2^(144/144) [= 144ET].
APPENDICES  other mathematical possibilities for measuring Aristoxenus's genera
1. Cleonides's (4/3)^(1/30)
2. 144tET  a useful approximation to Aristoxenus's system, much less sophisticated than 318tET
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 quartertone) 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 quartertones 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 parhypateThese agree exactly with the somewhat more convoluted version (fractional divisions of quartertones) 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 hypateBelow 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 lowestpossiblecardinality 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 0/24relaxed 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. 12001400. 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, 15001600. Florence.
B  Codex Berberinus. 15001550. 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 3446.]
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 96110.
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 317.
Barker, Andrew. 1978. 'Music and Perception: A Study in Aristoxenus'. _Journal of Hellenistic Studies_, v 98, p 916.
Litchfield, Malcolm. 1988. 'Aristoxenus and Empiricism: A Reevalutation Based on his Theories'. _Journal of Music Theory_, v 32 # 1, p 5173.
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 126184.]
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.
concerning observations on the change of preference from the enharmonic genus to the chromatic, see: yahoo tuning group, message 6947 [Wed Dec 15, 1999 6:46 pm].
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
For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here
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