A Summary of W. S. B. Woolhouse's
'Essay on musical intervals'

© 1999 by Joe Monzo

(Originally posted to the Onelist Tuning List, appearing in Onelist Tuning Digest # 446, message 6. Revised and updated for this webpage.)

From: Joe Monzo
To: tuning@onelist.com
Subject: Woolhouse's Essay on Musical Intervals...

There has been discussion here over the past few days about the theories of Wesley S. B. Woolhouse. Here's a summary of his book.

Contents

Recent references to Woolhouse on the Tuning List

Woolhouse's book

Introductory
Sound
Musical intervals
5-limit JI
The 'mean semitone'
The basic JI intervals for scale construction
730-EDO as a basic unit of measurement
Problems with JI
Harmonics
Temperament
12-EDO
The optimal temperament: 7/26-comma meantone
50-EDO
31-EDO
19-EDO
53-EDO
Beats of Imperfect Concords
Miscellaneous Additions

A spokesman for the new view is the musical mathematician W. S. B. Woolhouse, who wrote in the 19th century: "It is very misleading to suppose that the necessity of temperament applies only to instruments which have fixed tones. Singers and performers on perfect instruments must all temper their intervals, or they could not keep in tune with each other, or even with themselves; and on arriving at the same notes by different routes, would be continually finding a want of agreement. The scale of equal temperament obviates all such inconveniences, and continues to be universally accepted with unqualified satisfaction by the most eminent vocalists; and equally so by the most renowned and accomplished performers on stringed instruments, although these instruments are capable of an indefinite variety of intonation. The high development of modern instrumental music would not have been possible, and could not have been acquired, without the manifold advantages of tempered intonation by equal semitones, and it has, in consequence, long become the established basis of tuning."

At present, although most of us see the belief in the absolute supremacy of integers as applied to music to be naive, it also seems, as Paul Erlich has said with regard to simultaneous notes, that integer ratios or very near integer ratios (closer than called for by 12-EQT) correspond to a psychological reality as regards our perception of music.

I believe that Mr. Woolhouse, in stating that equal temperament is rightly the long established basis of tuning, really went too far and overlooked the fact of experience that the deviations of equal temperament from the just ratios are so large that they really do have an appreciable effect on the sound of music performed in equal temperament.

Some of Mr. Woolhouse's contemporaries strongly disagreed with him, too.

[Paul Erlich, TD 439.11:]

Note that Wesley Woolhouse was a prominent advocate of 19 tone equal temperament.

[Paul Erlich, TD 439.12:]

First of all, my knowledge of Woolhouse's theories suggests that he viewed some form of meantone temperament to be ideal. According to Mandelbaum, Woolhouse derived an optimal meantone tuning (I believe it was the squared-error optimal tuning for the three 5-limit consonances, namely 7/26-comma meantone), and decided that 19-tone equal tempermant was a close enough approximation, and one which gave to the musician the desirable properties of a closed system which were giving 12-equal its rise to prominence at the time. In fact, 31- or 50-tone equal temperaments would be better approximations (though not as convenient from a practical point of view), and neither of those tunings commits any errors larger than 6 cents in any of the classic (5-limit) consonant intervals. 19-equal has major thirds and perfect fifths that are over 7 cents off. 12-equal, of course, has major thirds 14 cents off and minor thirds 16 cents off.

Second and more importantly, I think Dave Hill has missed the importance of the last quoted statement (from Woolhouse) above. Many if not most common-practice musical passages performed in just intonation would result in contradictory tunings for the same written pitch. Whether or not Mr. Woolhouse went "too far", Mr. Hill has failed to address Woolhouse's point here. Furthermore, far from overlooking the errors from just intonation, Woolhouse sought the best way to reduce them while preserving the musical meaning of the notes in the Western tradition.

[Dave Hill, TD 440.1:]

I took the quote from Woolhouse from the book: "Piano Tuning" by J. Cree Fischer. The book was originally published in 1907. I have the Dover edition of 1975. The author, Fischer, introduces the quote from Woolhouse as follows (p. 144 of the Dover edition):

"That the equal temperament is the only practical temperament, is confidently affirmed by Mr. W. S. B. Woolhouse, an eminent authority on musical mathematics, who says:- 'It is very misleading to suppose that the necessity of temperament applies only to instruments which have fixed tones...' - rest of longish quote -

Apart from the quotation in Fischer, I know nothing about W. S. B. Woolhouse excepting that he lived in the 19th century and that he is mentioned in Ellis' translation of Helmholtz in connection with a 19 tone equal temperament. In posting the quote to the tuning list, I assumed Mr. Fischer's reliability in conveying Mr. Woolhouse's point of view. Is it possible that Mr. Woolhouse had done studies on 19 tone equal temperament, but nevertheless emphatically advocated the more usual 12 equal temperament for practical musical performance?

[Carl Lumma, TD 440.7:]

[Paul Erlich wrote...]

Whether or not Mr. Woolhouse went "too far", Mr. Hill has failed to address Woolhouse's point here. Furthermore, far from overlooking the errors from just intonation, Woolhouse sought the best way to reduce them while preserving the musical meaning of the notes in the Western tradition.

This would unquestionably involve meantone. But there are two points I'd like to make here: [etc...snip]

[Paul Erlich, TD 441.16:]

Dave Hill wrote,

Is it possible that Mr. Woolhouse had done studies on 19 tone equal temperament, but nevertheless emphatically advocated the more usual 12 equal temperament for practical musical performance?

Quite possible, given what composers of his time were trying to do (by the 19th century, most composers thought of G# and Ab, etc., as entirely interchageable and used such <>enharmonic equivalency to modulate around fractions or the entirety of the circle of fifths). However, his advocacy of 19-tone equal tempermant, though it would not allow such practices, clearly had a very "practical" component, since the "optimal" meantone he derived (from 16th-18th century musical considerations) was essentially 50-tone equal temperament, and the only possible reason for suggesting 19 instead of 50 would be a practical one of actually getting all those notes onto instruments.

The second chapter begins [p 3-7] with definitions of important terms regarding vibrations, sonance, and measurement of intervals, and explains [p 8-9] how to measure the string-lengths of the 'important consonances' of the basic 5-limit JI scale.

5-limit JI

He makes a statement about his prime- or odd-limit (he doesn't specify which interpretation of 'limit'):

[Woolhouse 1835, p 8:]

It has been found by experience, that proportions exceeding the number 5 are generally discordant, as the coincidences become then so very seldom. Our consonances will thus be limited to the proportions -

4/5, 3/4, 2/3, 3/5, 1/2

[p 10] Woolhouse devises the following pentatonic scale from the basic 5-limit JI ratios:

note interval     ratio  string-length

C   octave          2/1   180/360
A   sixth-major     5/3   216/360
G   fifth           3/2   240/360
F   fourth          4/3   270/360
E   third-major     5/4   288/360
C   first/keynote   1/1   360/360

	

which can be illustrated by this lattice diagram:

      A---E
     / \ / \
    F---C---G

	

Woolhouse then observes that 240/180 = 360/270 (= 4/3) and divides this scale into 2 'tetrachords' (even tho they only have 3 notes at this point) which have dissimilar spacing, then adds 'D' and 'B' to the scale to 'equalize' each tetrachord (i.e., give them identical intervallic structure), using the following ratios:

note interval     ratio  string-length

B   seventh        15/8   192/360
C   second          9/8   320/360

	

which produces the usual 5-limit JI 'major' scale. Adding these two notes to our lattice gives us:

      A---E---B
     / \ / \ / \
    F---C---G---D

	

The 'mean semitone'

[p 12] Woolhouse defines 'mean semitone' as 2^(1/12). He shows the mean semitone values of the JI intervals, almost exactly the same way I use Semitones (the exception being that he carries the decimal part further than 2 places).

He explains [p 11-16] how to calculate string-lengths of the 12-EDO scale for comparison to the just ratios.

The basic JI intervals for scale construction

Then he defines [p 17] the following basic intervals (I have added the tilde [~] to indicate that a value is only approximate):

 interval    ratio  mean semitones

major-tone    9/8    ~2.0391
minor-tone   10/9    ~1.8240
limma        16/15   ~1.1173

	

(Note that this definition of limma is quite different from the usual pythagorean one, where it is equal to 256/243 [= ~0.90 Semitone]; Woolhouse's limma is in fact much closer to the larger Pythagorean semitone, the apotome [= ~1.12 Semitones]. His terminology is following the function of the interval in pythagorean tuning, where the limma is the diatonic-semitone. The relative sizes of the two different semitones are reversed between pythagorean and meantone.)

730-EDO as a basic unit

Towards the end of this chapter [p 18-19] comes Woolhouse's most original contribution: 730-tET, or as I prefer to call it, 730-EDO using 2^(1/730) as the basic unit of measurement for the comparison of different intervals - a precursor to Ellis's use of 2^(1/1200) = 1 cent in his 1875 translation of Helmholtz.

[Woolhouse 1835, p 18:]

It will be useful to divide the octave into such a number of equal divisions that each interval of the scale may comprise an integral number of them ... such as will render the major and minor-tones and limma whole numbers, since all other intervals result from the various combinations of these elemental ones.

Note the emphasis on his 5-limit JI conception here.

He examines some other notably accurate divisions of the 'octave':

[Woolhouse 1835, p 19:]

It has been proposed by some to divide the octave into 53 divisions, taking 9 of them for the major-tone, 8 for the minor-tone, and 5 for the limma [the 15:16 semitone], which furnishes a pretty accurate scale.

[p 20] Woolhouse mentions likening the octave to a circle, and using degrees, minutes, and seconds to measure the intervals. This amounts to 1296000-EDO (yes, that's over a million degrees: 360 * 60 * 60), which Woolhouse dismisses as 'of no advantage in musical computations'.

Then he mentions 301-EDO of Sir J. Herschel [paper on 'Sound', vol 2 of mixed sciences, Encyclopedia Metropolitana], and presents a table comparing the errors from JI for 301-, 53-, and 730-EDO, which shows that

[Woolhouse 1835, p 20:]

the last, which has been found by various trials, is that which differs less than any other from the true series, unless we ascend to very high numbers; and is the one which is therefore most to be recommended'.

(See EDO prime error, 730-edo entry)

Woolhouse chose this division precisely because he wished to avoid using decimals or fractions, and the integer values of 730-EDO designating the three basic intervals of 15:16, 9:10, and 8:9 are so close to the actual JI values:

 interval       ratio   mean semitones   abbrev.  log(ratio)*(730/log(2))

JI:

major-tone       9/8       ~2.0391          t      ~124.045
minor-tone      10/9       ~1.8240          t,     ~110.962
limma           16/15      ~1.1173          θ      ~ 67.970
comma           81/80      ~0.2151          c      ~ 13.083

730-EDO:

major-tone   2^(124/730)   ~2.0384          t        124.000
minor-tone   2^(111/730)   ~1.8247          t,       111.000
limma        2^( 68/730)   ~1.1178          θ         68.000
comma        2^( 13/730)   ~0.2137          c         13.000


(θ is the Greek letter theta.)

	

Woolhouse gives the more exact value (which I placed in the last column of the first table), then notes how closely the integer values (which I put in the second table) approximate them. He will use the abbreviations and Greek letters in his math equations.

Problems with JI

The conclusion of this second chapter:

[p 23] Woolhouse refers us back to where he selected an arbitrary measurement for 'D' and 'B', the '2nd' and '7th' of the scale, respectively, to 'equalize' the two tetrachords. Now he emphasizes the arbitrariness of that choice, and says that the '2nd' my be tuned either to 9/8, to fit it into the harmonic series over the 'keynote', or 10/9, which makes the intervallic structure of the tetrachords identical.

[p 24] Then follows Woolhouse's crucial statement concerning JI versus temperament, referring to, but not mentioning, commatic drift:

[Woolhouse 1835, p 24:]

This difference in the note D, which in theory is a comma, is entirely done away with in practice, as the harmonic advantages which could be derived from the true theoretical scale, as directed by nature, would by no means compensate the difficulties of its performance.

The third chapter [p 25-35] is an explanation of harmonics, about which most here on the List know plenty already. Woolhouse gives tables [p 30-31] showing harmonics up to the 20th, and the 'error' of the harmonics from the 'true diatonic' (i.e., 5-limit JI).

Without making any claims for their use, he gives measurements [p 33-35] which show how to get the set of 'natural harmonics' on a violin, thus encouraging the reader to experiment.

The fourth chapter is Woolhouse's exploration of temperament, which really forms the climax of the book.

He starts out [p 36] with an emphasis on 'a very approximate diatonic series on the assumption of any one of them as a key-note', whose 'practical formation' demands temperament.

12-EDO

First is an examination of 12-EDO, noting that its 'greatest imperfections are those of the 3rd and 6th', with a maximum of 9&1/2 'degrees' deviation [(6/5) / 2^(3/12) {or 2^(9/12) / (5/3)} = ~15.641 cents = 2^(~9.515/730)], and after tabulating all the 'errors' from JI, he says that

[Woolhouse 1835, p 38:]

These deviations, however, which are considerably less than a comma or 13 degrees, are too small to affect, in a very sensible degree, the melody of the intervals.

(Woolhouse's "13 degrees" = 2^(13/730) = ~21.370 cents; the actual syntonic-comma = (81/64) / (5/4) = ~21.506 cents = 2^(~13.083/730). By 'considerably less' Woolhouse is referring to the ~5.865 cent difference between the syntonic comma and the ~15.641 cents maximum deviation of 12-EDO from JI cited above; in other words, the maximum deviations in 12-EDO do not distort melodies as badly as the commatic deviations that would be encountered in true JI performance.)

Note that here he emphasizes the perception of melody and not of harmony. He also says [p 38-39] that any human tuner is bound to commit small errors because of the imperfections of his hearing, and that because of this, it is futile to argue that 12-EDO produces a blandness which the unequal temperaments don't have. (That reminds me of Johnny Reinhard saying "people aren't perfect...12-equal is microtonal!...")

Woolhouse explains very briefly [p 40-41] that 12-EDO can be tuned by tempering the '5ths' 'about one degree [= 2^(1/730) = ~1.64 cents] flat', because 'each fifth must contain 425&5/6 degrees':

(730 degrees / 12 semitones) * 7 semitones [ a '5th'] = 425&5/6 degrees.

3/2 = 2^(~427.023/730)

2^(7/12) = 2^(425&5/6 / 730)

The exact figure to flatten the '5th' is (3/2) / 2^(7/12) = ~1.955 cents =
2^(~1.189/730), or quite close to 1&1/5 (and even closer to 1&3/16) 730-EDO degrees.

This may be done either with an ascending series of '5ths' (C to A#, with the break in notation between A# and F, from where the cycle repeats) or a descending series (which Woolhouse again, this time strangely, notates as sharps), the latter being the usual method used.

Finally on 12-EDO he concludes:

[Woolhouse 1835, p 41:]

This scale [12-EDO] is, without doubt, the best one for such instruments as the common pianoforte, organ, &c. which must necessarily have but one sound for both a sharp and the flat of the next upper note.

I don't really understand why he says that, because all thru the rest of the book he stresses that the goal of his work is to find a good tempered approximation to 5-limit JI which will give a practicable closed system, and he will go on to choose a meantone and several ETs which all fulfill these wishes better than 12-EDO. But there it is. (I suppose it is good to remember that when this book was published Beethoven had not even been dead for 10 years yet, and the great experiments in more-than-12-per-octave keyboards were still a few decades in the future. It will be seen later in this paper that Woolhouse does not think highly of those experimental keyboards which had been produced with many tones per octave.)

I think examining that paragraph alone like this is a perfect example of taking something out of context.

It seems to me at first like he's making a great praise that ultimately turns out to be pretty empty. In what instance must a keyboard 'necessarily have but one sound for both a sharp and the flat of the next upper note'?

An emphasis on that sentence seems to me to betray a slyly indignant way of implying that if one chooses to have an instrument specially built with more than 12 keys, then there are other temperaments that are better. And indeed, the rest of the book will pretty much bear this out.

The optimal temperament: 7/26-comma meantone

He immediately continues by saying that a system of temperament that gives two different sizes of semitone provides a much better approximation to JI. Note that he wants only one size for the tone, so that the comma will vanish, thus preventing commatic drift.

[p 42] With echoes of Marchetto, Woolhouse discusses the 'major' or 'diatonic semitone', the 'minor' or 'chromatic semitone', and the 'enharmonic diesis', the latter being the difference between the two former. Their relationships can be summarized thus:

diatonic semitone > (9/8)^(1/2)
chromatic semitone < (9/8)^(1/2)
enharmonic diesis = diatonic - [minus] chromatic semitones

And here he utters the most important sentence in the book (and the basis of Paul's admiration of his work), where he sets out:

[Woolhouse 1835, p 45:

... to ascertain the particular values which must be assigned to the *tone* and *diatonic semitone*, so that all the concords shall be affected with the least possible imperfections; and this we shall effect by the principle of least squares. We must first observe, that the *third-minor*, *third-major* and *fourth*, are the only concords necessary to be considered, because the others are merely the inversions of these, and we know that any error which may increase or diminish a concord, will have precisely the same effect in decreasing or increasing its inversion, as the octave, which is composed of them both, is unchangeable.

(Note that here Woolhouse is explicitly invoking the concept of the octave-complement. This "least squares" is the same method Paul Erlich used to independently discover the same 7/26-comma meantone as an optimal meantone tuning.)

Putting that on a lattice for a geometric view of the situation:

           E
          / \
        M3   m3
        m6   M6
       /       \
      C -P4/P5- G

	

It should be obvious that the 5-limit lattice can be extended infinitely in both dimensions by adding on additional cells just like this one (or parts of it), thus proving Woolhouse's statement.

Then [p 43-45] he uses algebra to find the difference between JI and this ideal temperament with τ (the Greek letter tau) to represent the tempered tone and σ (sigma) to represent the tempered diatonic semitone, and the other abbreviations presented earlier for the JI intervals; these will make up the six intervals on my lattice.

The result is:

[Woolhouse 1835, p 45:]

 τ  =   t, + (6/13)c  =  117 degrees

 σ  =   θ  + (9/26)c  =  72&1/2 degrees

		

Or in English and math:

tempered tone     = (10/9)  * ((81/80)^(6/13)) = 2^(~117.001/730)
tempered semitone = (16/15) * ((81/80)^(9/26)) = 2^(~ 72.499/730)

	

He notes that the relationship between these two is 'very nearly in the ratio of 8 to 5'. More precisely:

   ~117.001       : ~72.449
=  ~  1.613832172 :   1
=  ~  8.06916086  :   5

	

Woolhouse then presents a simplified tuning which is a good approximation to this:

50-EDO

[Woolhouse 1835, p 45:]

We may therefore divide the octave into 50 equal divisions, and appropriate 8 of them to the tone and 5 to the diatonic semitone.

Woolhouse is assuming a diatonic scale with a basic interval mapping of 5L,2s; that is, 5 large 'steps' and 2 small ones (see L&s mapping in my Tuning Dictionary).

Solving the simple algebraic equation 5L + 2s = 50 gives

L = 2s - 2

s = L/2 + 1

L = 8 and s = 5

This produces a basic scale of 21-out-of-50 tones per octave, which Woolhouse illustrates as follows with the number of 50-EDO degrees between both the Diatonic and Chromatic scale members (I have added ratios and Semitones, and for comparison, the cents-values for his optimal 7/26-comma meantone, which I had erroneously previously called "6/13-&-9/26-comma meantone"):

                     50-EDO          7/26-comma meantone

                                            cents difference
                 ratio    cents     cents      of 50-EDO

   /  C   \     2^(50/50)  1200    1200.000      0.000
  |        2
  |   B#  <     2^(48/50)  1152   ~1153.978      1.978
  5        1
  |   Cb  <     2^(47/50)  1128   ~1126.846     -1.154
  |        2
   >  B   <     2^(45/50)  1080   ~1080.824      0.824
  |        3
  |   Bb  <     2^(42/50)  1008   ~1007.670     -0.330
  8        2
  |   A#  <     2^(40/50)   960   ~961.648       1.648
  |        3
   >  A   <     2^(37/50)   888   ~888.495       0.495
  |        3
  |   Ab  <     2^(34/50)   816   ~815.341      -0.659
  8        2
  |   G#  <     2^(32/50)   768   ~769.319       1.319
  |        3
   >  G   <     2^(29/50)   696   ~696.165       0.165
  |        3
  |   Gb  <     2^(26/50)   624   ~623.011      -0.989
  8        2
  |   F#  <     2^(24/50)   576   ~576.989       0.989
  |        3
   >  F   <     2^(21/50)   504   ~503.835      -0.165
  |        2
  |   E#  <     2^(19/50)   456   ~457.813       1.813
  5        1
  |   Fb  <     2^(18/50)   432   ~430.681      -1.319
  |        2
   >  E   <     2^(16/50)   384   ~384.659       0.659
  |        3
  |   Eb  <     2^(13/50)   312   ~311.505      -0.495
  8        2
  |   D#  <     2^(11/50)   264   ~265.484       1.484
  |        3
   >  D   <     2^( 8/50)   192   ~192.330       0.330
  |        3
  |   Db  <     2^( 5/50)   120   ~119.176      -0.824
  8        2
  |   C#  <     2^( 3/50)    72   ~ 73.154       1.154
  |        3
   \  C   /     2^( 0/50)     0      0.000       0.000


	

With deviations all under 2 cents, it is apparent that 50-EDO is indeed a very close approximation to Woolhouse's calculated optimal meantone.

[Woolhouse 1835, p 46:]

This system is precisely the same as that which Dr. Smith, in his Treatise on harmonics [Smith 1759], calls the scale of equal harmony. It is decidedly the most perfect of any systems in which the tones are all alike (i.e., the most perfect meantone).

Based on this scale, Woolhouse gives a list [p 47] of 'all the major-keys which are necessary in music', which can be summarized as:

2^(x/50)

      / (C)
  5
      >  B
  3
      >  Bb
  5
      >  A
  3
      >  Ab
  5
      >  G
  3
      >  Gb
  5
      >  F
  5
      >  E
  3
      >  Eb
  5
      >  D
  3
      >  Db
  5
      \  C

	

and he proposes [p 48] to eliminate the keys of F#, C# and Cb major as superfluous.

31-EDO

Woolhouse then gives [p 49] the same kind of description of Huygens's 31-EDO, about which he says:

[Woolhouse 1835, p 49:]

This scale, therefore, has the greatest temperament in the minor-third, and its inversion the major-sixth, which is the principal objection to it, as it is known that these concords are most readily put out of tune, and consequently should have the least temperament. However, taking all into account, it must be acknowledged to be a very good scale.

19-EDO

Next he considers 19-EDO:

[Woolhouse 1835, p 50:]

For the practical tuning of a keyed instrument, such as the organ, in which the full enharmonic scale is to be introduced, perhaps the best method after all would be to divide the octave into 19 equal intervals by 20 keys.

and he gives a list of all the keys and a diagram of how it could be produced on a keyboard.

Here is a table which provides the cents-values for 19-EDO and the deviation of it from Woolhouse's optimal meantone tuning:

degree   cents   cents deviation from optimal meantone

  19    1200.000      0.000
  18   ~1136.842    ~17.136
  18   ~1136.842   -~ 9.996
  17   ~1073.684    ~ 7.140
  16   ~1010.526   -~ 2.856
  15   ~ 947.368    ~14.280
  14   ~ 884.211    ~ 4.284
  13   ~ 821.053   -~ 5.712
  12   ~ 757.895    ~11.424
  11   ~ 694.737    ~ 1.428
  10   ~ 631.579   -~ 8.568
   9   ~ 568.421    ~ 8.568
   8   ~ 505.263   -~ 1.428
   7   ~ 442.105    ~15.708
   7   ~ 442.105   -~11.424
   6   ~ 378.947    ~ 5.712
   5   ~ 315.789   -~ 4.284
   4   ~ 252.632    ~12.852
   3   ~ 189.474    ~ 2.856
   2   ~ 126.316   -~ 7.140
   1   ~  63.158    ~ 9.996
   0       0.000      0.000

	

It is clear from this table that 50-EDO provides a much closer approximation to the optimal meantone than 19-EDO, which indicates that Woolhouse advocated this temperament purely for the practical reason of instrument-construction limitations.

53-EDO

Woolhouse then analyzes the resources of a 53-EDO 'enharmonic organ', built by J. Robson and Son, St. Martin's-lane, but says that the number of keys is too much to be practicable, and settles again on 19-EDO.

Woolhouse concludes this chapter [p 56] with a table providing exact monochord-string measurements for 12-, 19-, 31-, 50-, and 53-EDO.

So Woolhouse expresses his limit of accuracy here in Hz as about 1/4 percent, which is about 4 cents in this frequency range.

He gives a table [p 65] showing the beat divisors for 12-, 19-, 31-, 50-, and 53-EDO, and at the end of this chapter [p 66-68] gives the beats for all '5ths' in the 12- and 19-EDO systems.

I had only scanned thru Mandelbaum 1961 once at Johnny Reinhard's, reading certain small parts of it more deeply because they piqued my interest. Most of what I remember was my first encounter with Fokker's work, but if I did read about Woolhouse and don't remember, I can say now that based on what Paul wrote in TDs 439.12 and 441.16, Mandelbaum gives an excellent summary of Woolhouse's book. (If Paul did the summarizing, then kudos to you Paul!)

Citing Mandelbaum on Woolhouse, these statements are absolutely correct:

[Paul Erlich, TD 439.12]

First of all, my knowledge of Woolhouse's theories suggests that he viewed some form of meantone temperament to be ideal.

Absolutely.

According to Mandelbaum, Woolhouse derived an optimal meantone tuning (I believe it was the squared-error optimal tuning for the three 5-limit consonances,

Indeed it was - Woolhouse's description led me to believe originally that I'd be discussing two different tunings right here...

:) (more on that below)

namely 7/26-comma meantone),

In a private email he's given me permission to share, Paul said:

It seems he was about an eyelash away from discovering my cherished 7/26-comma meantone.

I was going to say: it's still 'no cigar', but I'd say it's closer than an eyelash away. [Well, it turns out that they are the same - see below.] Take a look at this analysis:

The total difference between the sum of ratios in JI and those in the meantone is 3 commas. This must be spread over all notes in the scale to make the 'octave' be 1:2, which is specified by Woolhouse.

The amounts added to the JI 'tone' and 'semitone' are respectively 6/13- [= 12/26] and 9/26-comma.

3 commas, each divided into 26 parts = (3*26)/26 = 78/26 commas.

What I find most interesting about Woolhouse's approach is that it is more along the lines of what I've just recently gotten very involved in, via Dan Stearns: L&s mapping.

Noticing that Woolhouse based his meantone not on a constant 7/26-comma applied to all intervals as Paul did, but rather on the *two* basic intervals of the scale, the 'tone' and 'semitone', which are respectively the L and s intervals in a '5L,2s' mapping, applying different amounts of tempering to the two different intervals:

                         ratio                 Semitones

tempered tone     = (10/ 9)*((81/80)^(6/13)) = ~1.92
tempered semitone = (16/15)*((81/80)^(9/26)) = ~1.19

	

I saw that the total scale was:

(5*(10/9)*((81/80)^(6/13))) + (2*(16/15)*((81/80)^(9/26)))

Focusing on the addition of just the commatic parts shows how Woolhouse divided the 3-comma discrepancy:

     3/1  commas
=   78/26
=   60/26  +   18/26
=   30/13  +   18/26
= (5*6/13) + (2*9/26)  commas

	

which (I don't know if there's any significance) can be further reduced to:

= 5*(3*4) + 2*(3*3) / 26
= ((2^2)*3*5) + (2*(3^2)) / (2*13)  commas

	

(Paul can manipulate these numbers better, and perhaps show some new insight into what Woolhouse is doing.)

But the 5L,2s mapping puts the '7' in there... don't know if that means anything either, in regard to 7/26-comma meantone.

Now, the reason why I wrote earlier that 'I was going to say...':

Paul sent me a table of cents-values of his 7/26-comma meantone diatonic 'major' scale carried out to three decimal places, and it was exactly the same as Woolhouse's temperament at that accuracy level:

    50-EDO    7/26-comma meantone

C    1200       1200.000
B    1080      ~1080.824
A     888      ~ 888.495
G     696      ~ 696.165
F     504      ~ 503.835
E     384      ~ 384.659
D     192      ~ 192.330
C       0          0.000

	

I was intrigued by this and decided to do the calculation myself and see how far I had to take the decimal place to find a difference. After trying 12 decimal places, I finally found a discrepancy of one digit at the end of only one of the numbers, which I think was probably due to rounding in my spreadsheet anyway.

So it turns out that Woolhouse actually *did* describe Paul's 7/26-comma meantone, but not in that way. Woolhouse described it in terms of a '5L,2s' mapping, whereas Paul describes it based on a cycle of tempered 2:3s. Hmmm...

Perhaps someone will bother to go thru the drudgery to show us exactly why it works out both ways; I'm certainly interested, especially in how 12/26 and 9/26 averages out to 7/26!

[back to Paul]

and decided that 19-tone equal tempermant was a close enough approximation,

Absolutely.

and one which gave to the musician the desirable properties of a closed system which were giving 12-equal its rise to prominence at the time.

Again, 100% true.

In fact, 31- or 50-tone equal temperaments would be better approximations (though not as convenient from a practical point of view)

Both points made by Woolhouse.

[Paul, TD 441.16]

the "optimal" meantone he derived (from 16th-18th century musical considerations)

As far as I recall, Woolhouse doesn't specifically mention anything about repertoire.

was essentially 50-tone equal temperament, and the only possible reason for suggesting 19 instead of 50 would be a practical one of actually getting all those notes onto instruments.

This is all absolutely true, except that I would say his optimal tuning was the 7/26-comma meantone, and even advocating 50-EDO was a bit of backsliding; due to what, I don't know: he could have given beat-counting charts for the meantone as well as the ETs. Perhaps this gives a bit of weight to Dave Hill's hypothesis that Woolhouse was more willing to accept the tuning status-quo than the book makes him seem. But Paul (or Mandelbaum) is correct in that 50-EDO is aurally virtually identical to the optimal meantone.

A very intriguing little treatise. I'll be making a webpage out of this in the next few days, and can probably make the whole book into a set of webpages without too much work. I'm pretty sure there's no copyright restriction on it by now. Stay 'tuned' for updates.

Another tidbit: in his Introduction to Bosanquet 1987 [p 40 and 45], Rasch shows that earlier versions of Bosanquet's book included summaries of Woolhouse's, which were deleted from the published edition, thus proving that Bosanquet was quite familiar with Woolhouse 1835.

And here's a challenge to Paul or any others of the mathematically-inclined out there: since we've all heard so much about how well 53-EDO approximates both 3- and 5-limit JI and relatively little about 50-EDO, how about showing us exactly why 50-EDO is better. Or if it's not better, then tell us what descriptions best characterize the comparison of these two temperaments. The most important difference I can discern is that the syntonic comma vanishes in 50-EDO, while 53-EDO is the smallest EDO that clearly differentiates it.

[Paul Erlich, Onelist Tuning Digest # 447, message: 17]

Bless you, Joe, for digging this up!

[Monzo:]

He makes a statement about his prime- or odd-limit (he doesn't specify which interpretation of 'limit'):

[Woolhouse 1835, p 8]

It has been found by experience, that proportions exceeding the number 5 are generally discordant, as the coincidences become then so very seldom. Our consonances will thus be limited to the proportions -

4/5, 3/4, 2/3, 3/5, 1/2

Joe, clearly Woolhouse means an integer limit of 5, not a prime- or odd-limit.

[Monzo:]

[p 24] Then follows Woolhouse's crucial statement concerning JI versus temperament, without mentioning commatic drift:

[Woolhouse 1835, p 24]

This difference in the note D, which in theory is a comma, is entirely done away with in practice, as the harmonic advantages which could be derived from the true theoretical scale, as directed by nature, would by no means compensate the difficulties of its performance.

Commatic drift or shift would be two ways of addressing these difficulties.

[Woolhouse 1835, p 41]

This scale [12-tET] is, without doubt, the best one for such instruments as the common pianoforte, organ, &c. which must necessarily have but one sound for both a sharp and the flat of the next upper note.

I don't really understand why he says that, because all thru the rest of the book he stresses that the goal of his work is to find a good tempered approximation to 5-limit JI which will give a practicable closed system, and he will go on to choose a meantone and several ETs which all fulfill these wishes better than 12-tET. But there it is.

Joe, it's not too hard to understand what Woolhouse is saying. Read it again.

(a) most meantones are not closed systems

(b) ETs other than 12-tET do not have "but one sound for both a sharp and the flat of the next upper note". In other ETs, Ab is different from G#, for example.

[Monzo:]

In what instance must a keyboard 'necessarily have but one sound for both a sharp and the flat of the next upper note'?

If it has only 12 notes per octave!

[Monzo:]

An emphasis on that sentence seems to me to betray a slyly indignant way of implying that if one chooses to have an instrument specially built with more than 12 keys, then there are other temperaments that are better.

So you do understand? But what's slyly indignant about it?

[Monzo:]

And here he utters the most important sentence in the book (and the basis of Paul's admiration of his work), where he sets out:

[Woolhouse 1835, p 45]

... to ascertain the particular values which must be assigned to the *tone* and *diatonic semitone*, so that all the concords shall be affected with the least possible imperfections; and this we shall effect by the principle of least squares. We must first observe, that the *third-minor*, *third-major* and *fourth*, are the only concords necessary to be considered, because the others are merely the inversions of these, and we know that any error which may increase or diminish a concord, will have precisely the same effect in decreasing or increasing its inversion, as the octave, which is composed of them both, is unchangeable.

(This is the same method Paul used to discover his 7/26-comma meantone tuning.)

Right.

[Monzo:]

Putting that on a lattice for a geometric view of the situation:

           E
          / \
        M3   m3
        m6   M6
       /       \
      C -P4/P5- G

		

You could just as easily have depicted a minor triad instead.

[Monzo:]

It should be obvious that the 5-limit lattice can be extended infinitely in both dimensions by adding on additional cells just like this one (or parts of it), thus proving Woolhouse's statement.

Not sure what it proves.

[Monzo:]

(I have added ratios and Semitones, and for comparison, the cents-values for his optimal 6/13-&-9/26-comma):

                     50-EDO          6/13-&-9/26-comma meantone

		

Joe, I don't know why you're using such a strange name for Woolhouse's optimal temperament (is that what Woolhouse called it? [ed-note: No, Woolhouse actually did not give it a specific name.]). In 1/6-comma meantone temperament, the 10/9 is tempered by 2/3-comma, and the 16/15 by negative 1/6-comma. Do we call it "2/3-&-negative 1/6-comma meantone"? No, we simply call it 1/6-comma meantone, after the amount by which the fifth is tempered. Similarly, in 7/26-comma meantone, the 10/9 is tempered by

1 - 2*(7/26) comma
= 1 - 7/13 comma
= 6/13 comma

and the 16/15 is tempered by

5*(7/26) - 1 comma
= 35/26 - 1 comma
= 9/26 comma

So what Woolhouse discovered was simply 7/26-comma meantone. I'm glad he and I agree!

[Woolhouse 1835, p 46]

This system is precisely the same as that which Dr. Smith, in his Treatise on harmonics [Smith 1759], calls the scale of equal harmony. It is decidedly the most perfect of any systems in which the tones are all alike.

Is Woolhouse here referring to 50-tET or 7/26-comma meantone? Smith's ideal tuning was almost exactly 5/18-comma meantone, which is close to 50-tET but on the other side of it relative to 7/26-comma meantone.

[Monzo:]

He then analyzes the resources of a 53-tET 'enharmonic organ', built by J. Robson and Son, St. Martin's-lane, but says that the number of keys is too much to be practicable, and settles again on 19-tET.

53-tET would also be subject to the same objection he levelled on JI, while 19-, 31-, and 50-tET would not.

[Monzo:]

He certainly noted the disadvantages of strict 5-limit JI, yet he sought ultimately a temperament which would approximate it better than 12-tET.

And one that would eliminate the comma, which 12-, 19-, 31-, 43-, 50-, and 55-tET do, but 53-tET does not.

[Monzo:]

Noticing that Woolhouse based his meantone not on a constant 7/26-comma applied to all intervals as Paul did, but rather on the *two* basic intervals of the scale, the 'tone' and 'semitone',

It of course amounts to the same thing.

[Monzo:]

So it turns out that Woolhouse actually *did* describe Paul's 7/26-comma meantone, but not in that way. Woolhouse described it in terms of a '5L,2s' mapping, whereas Paul describes it based on a cycle of tempered 2:3s. Hmmm...

Perhaps someone will bother to go thru the drudgery to show us exactly why it works out both ways; I'm certainly interested, especially in how 12/26 and 9/26 averages out to 7/26!

Again, in 7/26-comma meantone, the 10/9 is tempered by

1 - 2*(7/26) comma [because a 10/9 is two fifths up and a comma down, and the direction of the tempering is opposite that of the fifth]
= 1 - 7/13 comma
= 6/13 comma

and the 16/15 is tempered by

5*(7/26) - 1 comma [because the 16/15 is five fifths up and a comma down]
= 35/26 - 1 comma
= 9/26 comma

[Paul Erlich, in private email:]

the "optimal" meantone he derived (from 16th-18th century musical considerations)

[Monzo:]

As far as I recall, Woolhouse doesn't specifically mention anything about repertoire.

The construction of the diatonic scale, and the statement about the difficulty in practice introduced by the two theoretical values of the note D, are both considerations highly specific to Western commom practice repertoire (including, I would say, most pop music).

[Monzo:]

since we've all heard so much about how well 53-tET approximates both 3- and 5-limit JI and relatively little about 50-tET, how about showing us exactly why 50-tET is better.

Again, it's the diatonic scale, and the fact that 53-tET (like 15-, 22-, 27-, 34-, and 41-tET) has two different versions of "D" while 50-tET (like 12-, ) does not. So everything Woolhouse says to admonish JI applies to 53-tET as well.

[Paul Erlich, >Onelist Tuning Digest # 447, message: 24]

Joe Monzo wrote,

[Monzo:]

730-tET as a basic unit

Towards the end of this chapter [p 18-19] comes Woolhouse's most original contribution: 730-tET, using 2^(1/730) as the basic unit of measurement for the comparison of different intervals - a precursor to Ellis's use of 2^(1/1200) = 1 cent in his 1875 translation of Helmholtz.

[Woolhouse 1835, p 18]

It will be useful to divide the octave into such a number of equal divisions that each interval of the scale may comprise an integral number of them ... such as will render the major and minor-tones and limma whole numbers, since all other intervals result from the various combinations of these elemental ones.

Note the emphasis on his 5-limit JI conception here.

730-tET can be found in Paul Hahn's http://library.wustl.edu/~manynote/consist2.txt (This chart only shows those ETs which have a higher consistency level at some harmonic limit than all lower-numbered ETs, and goes up to 10000TET.)

In Paul Hahn's terminology, 730-tET is consistent at the 5-limit to level 22, meaning that any interval constructed of any combination of up to 22 consonant 5-limit intervals (minor thirds, major thirds, perfect fourths) will be represented the same way whether the interval is first computed in JI and then rounded to 730-tET, or if the consonant intervals are rounded to 730-tET first and then combined. By comparison, 12-tET is consistent at the 5-limit only to level 3: the ratio 648:625 (often called the greater diesis) can be constructed from four 6:5 minor thirds; while four minor thirds (minus an octave) add up to a unison in 12-tET, the actual size of 648:625 in JI is 62.565 cents, so it rounds up to one step in 12-tET. 53-tET is consistent at the 5-limit to level 8.

Bosanquet's 612-tET is almost as good, as it is consistent at the 5-limit to level 21. In order to improve on the consistency level of 730-tET in the 5-limit, one would have to go up to 1783-tET, which is consistent to level 42 (heh, I just said level 42), and then to improve on that, 4296-tET is 5-limit consistent to level 119. Note that 4296 is a multiple of 12. So if you wanted to get a 12-tET-based synthesizer to provide ridiculously exact JI (5-limit consonances within 0.002 cents of just), you could design it to divide the semitone into 4296/12 = 358 equal parts . . . Being a little more realistic, note that 612 is also divisible by 12 and gives 5-limit consonances within 0.093 cents of just . . .

(posted by me to the Tuning List by me, monz, answering Paul:)

Thanks to Dave Hill and Paul Erlich for expressing their appreciation to me here for doing the summary on Woolhouse. I'm certainly very glad I went thru the trouble of getting a copy of this book - it's a small but very valuable addition to my library!

And thanks very much to Paul for his valuable clarifications and additions on what I had to say about Woolhouse. Following are some further comments.

[Paul Erlich, TD 447.17, in his initial response to the question of what type of limit Woolhouse had in mind]

Joe, clearly Woolhouse means an integer limit of 5, not a prime- or odd-limit.

My initial response to that was:

Thanks for clearing that up, Paul. I think that's an important point in his theory. Woolhouse blithely skims over many things that your or I (or lots of others) would probably elaborate much more fully.

But after reading Paul's subsequent observation, where Woolhouse is about to set out his 'optimal meantone' tuning:

[Paul Erlich, TD 447.23]

I'd have to say that here (which is where it matters), Woolhouse is thinking of 5 as the odd limit, since in an odd limit all intervals are considered equivalently to their inversions.

I'll say that I have to agree. This is clearly a 5-odd-limit. Hmmm... then again, it *could* be a 5-prime-limit too, couldn't it?... But in any case, it's *not* a 5-integer-limit.

[me, monz, TD 446.6]

[p 24] Then follows Woolhouse's crucial statement concerning JI versus temperament, without mentioning commatic drift:

[Woolhouse 1835, p 24]

This difference in the note D, which in theory is a comma, is entirely done away with in practice, as the harmonic advantages which could be derived from the true theoretical scale, as directed by nature, would by no means compensate the difficulties of its performance.

[Paul Erlich, TD 447.17]

Commatic drift or shift would be two ways of addressing these difficulties.

It's obvious that he's talking about commatic drift or shift - I think the difference is negligible in Woolhouse's case, since his point is to get rid of it -, and he does explain the retunings that would be necessary on the second degree of the scale (the 'D' in C major here) to clear up the problems, but, at least in *this* book, he never actually comes right out and *says* that commatic drift is the problem.

In the posting from Dave Hill that originally started all this, where he quotes Fischer quoting Woolhouse, Woolhouse says about JI:

Singers and performers on perfect instruments (i.e., not fixed pitch: violins, etc.) ... on arriving at the same notes by different routes, would be continually finding a want of agreement.

So here he clearly expresses the musical movement that would cause commatic drift. But this quote is not in the Essay on Musical Intervals... - at least I never found it there. It must appear in something else Woolhouse wrote.

[Woolhouse 1835, p 41]

This scale [12-tET] is, without doubt, the best one for such instruments as the common pianoforte, organ, &c. which must necessarily have but one sound for both a sharp and the flat of the next upper note.

[me, monz]

An emphasis on that sentence seems to me to betray a slyly indignant way of implying that if one chooses to have an instrument specially built with more than 12 keys, then there are other temperaments that are better.

[Paul, TD 447.17]

... what's slyly indignant about it?

Well... the whole point of Woolhouse's book is to present and explain tunings that he feels are better than 12-tET (or 12-EDO), and considering the fact that he goes thru all the trouble of determining 7/26-comma meantone, and finding the more practical 50-EDO close approximation to it, and then finally accepts 19-EDO as more practical still, it seems awfully strange that he uses such strong language when he says 'This scale [12-tET] is, *WITHOUT DOUBT* [emphasis mine], the best one for such instruments...'.

Sounds to me like, just for that one moment, he was succumbing to the status-quo simply because of the improbability that anyone would go thru the trouble to build an instrument to his specifications. Maybe it's just the way I'm reading it.

But I do think that it points out one of the things Dave was originally discussing: that this book emerged at a time (1835) when 12-tET had still not fully 'conquered' the musical scene. Woolhouse clearly felt the need to express a very positive opinion about 12-tET *given certain conditions*, even tho he knew there were other, better alternatives which required different conditions.

In fact, 12-EDO may not yet have been the status-quo in England. From what I've read, England seems to have been the last country in Europe to generally adopt 12-EDO, and so perhaps he was really making a legitimate push for it here. But he certainly would have appreciated a more inventive approach to building instruments, because he clearly preferred 19-EDO.

[me, monz]

Putting that on a lattice for a geometric view of the situation:

           E
          / \
        M3   m3
        m6   M6
       /       \
      C -P4/P5- G


			

[Paul, TD 447.17]

You could just as easily have depicted a minor triad instead.

Absolutely true - but all of Woolhouse's JI descriptions are based on the 5-limit 'major' scale and 'major' chords. He certainly implicity included 'minor', but doesn't *explicity* say anything about it, except concerning the commatic problems of the 'D' second degree. I was just presenting his own explanations in diagrammatic form.

[me, monz]

It should be obvious that the 5-limit lattice can be extended infinitely in both dimensions by adding on additional cells just like this one (or parts of it), thus proving Woolhouse's statement.

[Paul, TD 447.17]

Not sure what it proves.

It proves what he said in the part I quoted just before that:

[Woolhouse 1835, p 45]

the *third-minor*, *third-major* and *fourth*, are the only concords necessary to be considered, because the others are merely the inversions of these, and we know that any error which may increase or diminish a concord, will have precisely the same effect in decreasing or increasing its inversion

You can see plainly on the lattice that all these intervals and their inversions can be added on infinitely in any direction without involving any other intervals, which was his point. He uses these three intervals, and no others, in his derivation of the 7/26-comma meantone.

Tell you what... I'll make a separate post quoting his mathematics for this derivation.

[me, monz]

his optimal 6/13-&-9/26-comma

[Paul, TD 447.17]

Joe, I don't know why you're using such a strange name for Woolhouse's optimal temperament (is that what Woolhouse called it?).

No - he actually didn't call it anything at all. He says, by way of introducing it:

[Woolhouse 1835, p 41]

A system of sounds may, however, be formed, in which all the keys are more nearly approximate than that of equal semitones ['equal semitones' means 12-EDO] ...

And then, after he's described the math for his "optimal meantone" (that's either your name or Mandelbaum's), he calls its close cousin 50-EDO [p 46] 'decidedly the most perfect of any systems in which the tones are all alike' [i.e., in which the 81/80 (syntonic comma) vanishes]. But he never really puts a name on that "optimal meantone" tuning, which we know is 7/26-comma.

Woolhouse segues *very* quickly after that into 50-EDO, and never mentions that meantone again anywhere else in the book, so I really don't think he ever actually envisioned its use at all in practice. It's just a mathematically optimum tuning, which he certainly thought could be well enough represented by 50-EDO (with less than 2-cents error from the meantone) if anyone actually wanted to try to build an instrument in *that* tuning, and to which he clearly felt 19-EDO (with up to ~16 cents error from the meantone) came close enough for practical purposes.

(I've added to my webpage a table of cents values and error from the 7/26-comma meantone for 19-EDO.)

I didn't realize, until I carried the calculations of both his scale and your 7/26-comma meantone out to 12 decimal places, that they were indeed exactly the same tuning. So that weird name is just my earlier description of it, which needs to be changed to '7/26-comma'. I certainly appreciate your mathematical explanation that shows that they are both the same. Thanks!

[Woolhouse 1835, p 46]

This system is precisely the same as that which Dr. Smith, in his Treatise on harmonics [Smith 1759], calls the scale of equal harmony. It is decidedly the most perfect of any systems in which the tones are all alike.

[Paul, TD 447.17]

Is Woolhouse here referring to 50-tET or 7/26-comma meantone?

He's referring here to 50-tET (which I'd rather call 50-EDO), as Dr. Smith's 'scale of equal harmony'.

A side note: those who have looked at my webpage of this will see that I've changed 'tET' to 'EDO' in every case. I did that because Woolhouse specifically states that the 'octave' is always to be a 1:2 ratio.

[Paul Erlich, TD 447.17]

Smith's ideal tuning was almost exactly 5/18-comma meantone, which is close to 50-tET but on the other side of it relative to 7/26-comma meantone.

How do you know that?! Do you have access to Smith's book?! Tell us more!!

[me, monz]

He certainly noted the disadvantages of strict 5-limit JI, yet he sought ultimately a temperament which would approximate it better than 12-tET.

[Paul Erlich, TD 447.17]

And one that would eliminate the comma, which 12-, 19-, 31-, 43-, 50-, and 55-tET do, but 53-tET does not.

Yes, I realized after I put the posting together that Woolhouse's objection to 53-tET would be the differentiation of the comma.

But still, he likes its close approximations to JI well enough to keep including it (and 31-EDO) in his tables thru-out the rest of the book.

. . .

I'd love to be able to figure out an accurate lattice diagram of the actual Woolhouse/Erlich 7/26-comma meantone, to accompany the simple one of its 5-limit JI implications. Actually, my lattice formula can probably already do it...

Woolhouse Monzo  name                 ratio

variables not known

 tau       T   tempered tone             ?
 sigma     S   tempered semitone         ?

variables known

 t         t   just major-tone          9/8
 t,        t,  just minor-tone         10/9
 theta     s   just diatonic semitone  16/15
 c         c   syntonic comma          81/80

	

t  = t, + c        major-tone = minor-tone + comma
t, = t  - c        minor-tone = major-tone + comma
c  = t  - t,       comma      = major-tone - minor-tone

	

degree:  I   II   III   IV   V   VI   VII   (I)
          \ /  \ /   \ /  \ / \ /  \ /   \ /
 tempered  T    T     S    T   T    T     S
 just      t    t,    s    t   t,   t     s

	

  'octave'
= 5T              + 2S          tempered intervals
= 3t        + 2t, + 2s          just intervals
= 3(t, + c) + 2t, + 2s          just intervals
= 5t,             + 2s + 3c     just intervals

	

  5T        + 2s                   tempered intervals
- 5t,       + 2s       + 3c        just intervals
----------------------------
= 5(T - t,) + 2(S - s) - 3c
= 0

	

  5(T - t,) + 2(S - s) - 3c  =  0
=             2(S - s) - 3c  =  -5(T - t,)
=             2(S - s)       =  -5(T - t,) + 3c
=              (S - s)       =  (-(5/2)*(T-t,))+((3/2)*c)

	

tempered minor 3rd = T        + S

just minor 3rd     = t        + s
                   = (t, + c) + s
                   = t,       + s  + c

	

    T       + S                tempered minor 3rd
-   t,      + s       + c      just minor 3rd
-------------------------
=  (T - t,) + (S - s) - c      minor 3rd error

	

   (T - t,) + (S - s)                   - c
=  (T - t,) + (-(5/2)*(T-t,))+((3/2)*c) - c
=  (-(3/2) * (T - t,))  +  ((1/2) * c)

	

tempered major 3rd = 2T

just major 3rd     = t        + t,
                   = (t, + c) + t,
                   = 2t,            + c

	

   2T                 tempered major 3rd
-  2t,       + c      just major 3rd
----------------
=  2(T - t,) - c      major 3rd error

	

tempered 4th       = 2T            + S

just 4th           = t        + t, + s
                   = (t, + c) + t, + s
                   = 2t,           + s  + c

	

   2T        + S                tempered 4th
-  2t,       + s       + c      just 4th
--------------------------
=  2(T - t,) + (S - s) - c      4th error

	

   2(T - t,) + (S - s)                   - c
=  2(T - t,) + (-(5/2)*(T-t,))+((3/2)*c) - c
=  (-(1/2) * (T - t,))  +  ((1/2) * c)

	

minor 3rd error =  (-(3/2) * (T - t,))  +  ((1/2) * c)
major 3rd error =  2       * (T - t,)   -           c
4th error       =  (-(1/2) * (T - t,))  +  ((1/2) * c)

	

Now to find to RMS [= 'root mean square'] total error of the three intervals,

[Woolhouse 1835, p 44]

To determine the value of (T - t,), so that the sum of the squares of these three errors shall be the least possible, multiply them by the respective coefficients -(3/2), +2, -(1/2), and the sum of the products will be

(13/2)(T - t,) - 3c.

By putting this = 0, we find

(T - t,) = (6/13)c,

which used in the above value of (S - s), gives also

(S - s) = (9/26)c

Paul Erlich explained this crucial step to me in more detail:

[Paul Erlich, private communication]

You can't just use algebra to solve an optimization problem You need calculus.

Let's look at the expression for the errors:

 minor 3rd error =  (-(3/2) * (T - t,))  +  ((1/2) * c)
 major 3rd error =  2       * (T - t,)   -           c
 4th error       =  (-(1/2) * (T - t,))  +  ((1/2) * c)

			

Let's simplify the notation by calling (T - t,) "x":

 minor 3rd error =  (-(3/2) * x)  +  ((1/2) * c)
 major 3rd error =  2       * x   -           c
 4th error       =  (-(1/2) * x)  +  ((1/2) * c)

		

The sum of the squared error is thus:

   ((-(3/2) * x)  +  ((1/2) * c))^2
 + (2       * x   -           c )^2
 + ((-(1/2) * x)  +  ((1/2) * c))^2

		

To minimize this sum-of-squared errors as a function of x, you set its "derivative" or rate of change with respect to x equal to zero. That means that the rate at which the sum-of-squared error is changing when x changes is zero, which can only happen at a local minimum or a local maximum. In this case we know it will be a local minimum.

The derivative of the sum-of-squared error with respect to x is:

   2*((-(3/2) * x)  +  ((1/2) * c))*(-(3/2))
 + 2*(2       * x   -           c )*2
 + 2*((-(1/2) * x)  +  ((1/2) * c))*(-(1/2))

		

Since we are setting this equal to 0, we can eliminate the three "2*"'s:

   ((-(3/2) * x)  +  ((1/2) * c))*(-(3/2))
 + (2       * x   -           c )*2
 + ((-(1/2) * x)  +  ((1/2) * c))*(-(1/2))
 = 0

		

Simplifying:

(9/4)*x - (3/4)*c + 4*x - 2*c + (1/4)*x - (1/4)*c = 0

or

(13/2)*x - 3*c = 0

= (13/2) * (T - t,) - 3c = 0

[Paul Erlich, private communication]

What the equation above actually represents is not the sum of the squared errors, but a relationship between the amount of tempering and the comma when the amount of tempering is such as to minimize the sum of the squares of the errors. And that relationship comes from setting the derivative (or rate of change) of the sum-of-squared error, with respect to the amount of tempering, equal to zero, because that is a condition that only occurs at the minimum.

Now we can calculate the amount of temperament for the tone.

Solving for (T - t,) gives:

(13/2) * (T - t,) - 3c   =   0
(13/2) * (T - t,)        =   3c
 13    * (T - t,)        =   6c
         (T - t,)        =   (6/13) * c

	

There's our error, or the amount of tempering, for the tone.

So the tempered tone is:

T = (t, + ((6/13) * c))
tempered tone = just minor-tone + 6/13 comma

For the semitone:

Solving for (S - s) gives:

  (S - s)
= (-(5/2) * (T - t,)    )  +  ((3/2)   * c)
= (-(5/2) * ((6/13) * c))  +  ((3/2)   * c)
= (-(15/13)         * c )  +  ((3/2)   * c)
= (-(30/26)         * c )  +  ((39/26) * c)
= (9/26) * c

	

There's our error, or the amount of tempering, for the semitone.

S = (s + ((9/26) * c))
tempered semitone = just diatonic semitone + 9/26 comma

In addition to knowing the tempering from the 'minor-tone' as calculated above, it will be convenient to know the amount of tempering for the tone from the 'major-tone':

t = (t, + ((13/13) * c))
T = (t, + (( 6/13) * c))
T = (t - (( 7/13) * c))

tempered tone = just major-tone - 7/13 comma

Now, proving that this temperament is indeed Paul Erlich's 7/26-comma meantone, let's calculate the tempered '5th':

  '5th'
= 3T + s
= 2(t -  ( 7/13) * c) + (t, + ( 6/13) * c) + (s + (9/26) * c)
= 2(t -  (14/26) * c) + (t, + (12/26) * c) + (s + (9/26) * c)
= 2t  - ((28/26) * c) +  t, + (12/26) * c) +  s + (9/26) * c)
= 2t - t, + s - ((7/26) * c)
= just '5th'  - ((7/26) * c)
= just '5th'  -  7/26 comma

	

Which in numbers, of course, is:

(3/2) / ( (81/80)^(7/26) ) = ~696.165 cents

There it is.

I asked Paul Erlich to factor that as far as could be done, and his answer was:

  ((2^-1)*(3^1)) / (((2^-4)*(3^4)*(5^-1))^(7/26))

= ((2^-1)*(3^1)) / (((2^-(28/26))*(3^(28/26))*(5^-(7/26))))

= 2^(2/26)*3^(-2/26)*5^(7/26)

= 2^(1/13)*3^(-1/13)*5^(7/26)

	

Comparing this with the "5th" of the 50-EDO recommended by Woolhouse, we find a deviation of:

       2^      3^    5^

  |   1/13   -1/13  7/26 |   7/26-comma "5th"
- |  29/50    0     0    |   50-EDO "5th"
--------------------------
  |-327/650  -1/13  7/26 |   difference


= ~27 14mus = ~1/6 cent.


	

Updated:

1999.12.19
2000.02.04
2000.02.06
2001.05.30
2001.07.08
2005.05.05
-- conversion to new website 2020.05.28 -- corrected broken URLs to internal links, minor editing

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