On the Expansion of the Musician's Realm of Harmony

Introduction. Nowadays there is a growing number of musicians who feel that there is a need of more values, new harmonic values, and the problem is where to find them and how to make them available. History has established a convention to express the language of music with twelve notes only, in each octave. On keyboards these are the seven white and five black keys. That duodecimal system is now granted once for ever. Even members of some advanced avant-garde groups retain these twelve notes as a foundation of their novel creations. Yet one has to realise, that the duodecimal temperament is not a gift from heaven, no pre-established structure of fact. It is wholly man-made. It is a compromise between the finite technical power of man and his inmost desire to realise the highest musical truths attainable in his songs and instrumental playing. For singing voices the field of possibilities is unrestricted. The voice is able to choose the pitch of its notes from an innumerable continuous multitude of sounds.
For keyboard instruments there is no space for an endless array of notes. Presently they do not afford room for more than twelve notes in the octave. Accordingly only a restricted number of chords can be provided for, the chords based on the numerical relations of the small integer numbers 1, 2, 3, 4, 5 and 6. The smallest degrees, in former times, were major and minor semitones. During a couple of centuries there were controversies between composers. Impetuous composers wanted a full cycle of chords in the keyboard instruments, and they were ready to sacrifice the pure perfect intervals. On the other side the lovers of perfect harmony in the chords with major thirds refused to give up the beauty of the pure relations. Since a century the present rule prevails to tune the instruments with twelve equal semitones in the octave. A majority of musicians are not aware of the deficiencies of that duodecimal equal temperament.
It will be clear that if one wants to find a way to introduce new harmonies, harmony related with one more prime number with 7, say, it will be necessary to review the organisation of the keyboard and to tackle the problem of equal temperament anew.

Early endeavours. Even without wanting a new harmony, in our desire to produce a cycle of chords with perfect fifths and major thirds we are led to improve the keyboard by multiplying the number of keys. This was an urgent need felt by great masters of the Renaissance in the sixteenth century. The Italian Nicola Vicentino proclaimed in 1555 that the octave should be divided into thirty-one degrees. A century later in Paris Christiaan Huygens confirmed this statement, and he computed the exact figures for this division. After Vicentino several artists tried to construct a harpsichord, an archicembalo with 31 keys in the octave. Alas, they failed. The technique of that epoch was unable to cope with the difficulties. For vocal music, about 1600 Carlo Gesualdo di Venosa composed unparalleled five-part madrigals with utmost refinery of perfect chords. But vocal art had no support by instruments and Gesualdo's achievements sank into oblivion.
In our own age technicians have the ability to take up the construction problem which has been dropped by the Renaissance builders of the archicembalo. And mathematicians are ready to serve the musicians to clear up the situation. It is well known that from the arithmetical point of view musical intervals are like irrational numbers. Trying to represent them as integer multiples of elementary degrees would be looking for common divisors of these irrational numbers. Mathematic shows that it is impossible to find any exact common divisor for such numbers. Therefore we have to be content with approximations.

The fundamental intervals. After Ellis we measure intervals in cents. The octave, by definition of the cent, will have 1200 cents. The octave has the ratio 2/1.

The perfect fifth or the ratio 3/2, has Q = 701,9550 . . . cents
the perfect major third or the ratio 5/4, has T = 386,3137 . . . cents
the perfect seventh or the ratio 7/4, has S = 968,8259 . . . cents
the perfect eleventh or the ratio 11/8, has U = 551,3179 . . . cents
the perfect thirteenth or the ratio 13/8, has D = 840,5277 . . . cents

Our problem is how to find suitable approximate common divisors of these irrational numbers.

Approximations. In the case of two irrational numbers there is a straight-forward method to find the approximate common divisors. But the case of three or more such numbers offered an unsolved problem, until some ten years ago in Norway professor Viggo Brun gave a method which he called antanairesis 2), using the Greek word for cognate operations employed by Archimedes.
For the octave and the perfect fifth, for O = 1200,0000 and Q = 701,9550 . . the successive approximations of their relation are

5 : 3 = 1200 : 720
12 : 7 = 1200 : 700
41 : 24 = 1200 : 702,44
53 : 31 = 1200 : 701,89
306 : 179 = 1200 : 701,96

It is obvious that the last figures oscillate round the given number. For the octave O and for the perfect major third (T = 386,3137 . .) successive approximations to their relation are

28 : 9 = 1200 : 385,714
59 : 19 = 1200 : 386,441
146 : 47 = 1200 : 386,301
643 : 207 = 1200 : 386,314

Here again there is a gradual improvement of the approximation. We need not stop to investigate the approximation for the ratios of O and S, of O and U, of O and D. We proceed to the intricate problem of three intervals, octave, fifth and major third. From daily experience we know that

12 : 7 : 4 = 1200 : 700 : 400.

That is the approximation used on the present keyboard instruments. Now by the antanairesis method of Viggo Brun we get the approximations

19 : 11 : 6 = 1200 : 694,74 : 378,95
31 : 18 : 10 = 1200 : 696,77 : 387,10
34 : 20 : 11 = 1200 : 705,88 : 388,24
53 : 31 : 17 = 1200 : 701,89 : 384,91
87 : 51 : 28 = 1200 : 703,45 : 386,21
118 : 69 : 38 = 1200 : 701,69 : 386,44

I will not lose time to find approximations for other combinations of three intervals, such as O, Q, S or O, Q, U, a. s. o. I take at once the case of four intervals, octave, fifth, major third and perfect seventh, S = 968,83. On the usual keyboard one is met by

12 : 10 : 7 : 4 = 1200 : 1000 : 700 : 400.

This is a poor approximation indeed. From the results of antanairesis we collect the approximations

15 : 12 : 9 : 5 = 1200 : 960 : 720 : 400
31 : 25 : 18 : 10 = 1200 : 967,74 : 696,77 : 387,10
53 : 43 : 31 : 17 = 1200 : 973,85 : 701,89 : 384,91
68 : 55 : 40 : 22 = 1200 : 970,59 : 705,88 : 388,24

Next we want suitable approximations if we include the perfect eleventh among the principal intervals we want to work with. The eleventh, undecimus, measures U = 551,32 cent. Among the approximations suggested by antanairesis I collect the following

22 : 18 : 13 : 10 : 7 = 1200 : 981,82 : 709,09 : 545,45 : 381,82
37 : 30 : 22 : 17 : 12 = 1200 : 972,97 : 713,51 : 551,35 : 389,19
41 : 33 : 24 : 19 : 13 = 1200 : 965,85 : 702,44 : 556,10 : 380,49
63 : 51 : 37 : 29 : 20 = 1200 : 971,43 : 704,76 : 552,38 : 380,95
72 : 58 : 42 : 33 : 23 = 1200 : 966,67 : 700,00 : 550,00 : 383,33

Finally I wish to give some approximations where the perfect thirteenth too, D = 840,53 cent, is among the desired intervals to be included. Of course we now want a greater number of degrees. Here are two of these temperaments.

87 : 70 : 61 : 51 : 40 : 28 = 1200 : 965,52 : 841,38 : 703,45 : 551,72 : 386,21
94 : 76 : 66 : 55 : 43 : 30 = 1200 : 970,21 : 842,55 : 702,13 : 548,94 : 382,98

Misfit numbers. Obviously in every temperament every basic interval will be afflicted with an error, the octave, by definition, excepted. The suitability of the temperament will be judged by means of these errors. Errors may be positive or negative. We want an overall information considering all errors together. We should be mistaken by summing up the errors because positive and negative errors might cancel and thus flatter the judgment. Mathematicians use to adjust a curve to a series of measured points by what they call the method of least squares. Accordingly I will take the sum of the squares of the errors. I will take such a sum of squares as a measure for the deficiency of the temperament and I shall call it the misfit number, M.
From the above collection I choose the divisions which are best known. Some of them have been explored experimentally.

  • There is the usual division with 12 equal semitones.
  • There is Woodhouse's (and Yasser's) division with 19 tritotones.
  • There is the division of 22 sruti's (which in India do not have all the same size). They are between tritotones and quartertones.
  • There is the division of Huygens with 31 diŽses, fifths of a tone.
  • There is the division of Von Janků, with 41 supracommas.
  • There is the division of Mercator (17th century) with 53 commas.

I shall add some more divisions, with 63, with 72 infracommas, with 87 and 94 semi-commas. The errors will be denoted by (eQ) for the error of the fifth, by (eT) for the error of the major third, and consequently by (eS), by (eU) and (eD) for the other basic intervals.
If we confine ourselves to the use of the fifth only, the misfit number will be M1 = (eQ)2. If the temperament is intended to be used for music availing itself of fifths and major thirds the misfit number will be M2 = (eQ)2 + (eT)2. Likewise M3 = (eQ)2 + (eT)2 + (eS)2, then M4 = (eQ)2 + (eT)2 + (eS)2 + (eU)2 and at last M5 = (eQ)2 + (eT)2 + (eS)2 + (eU)2 + (eD)2.

Table of misfit numbers, in cent squared

Number of
degrees
N =12 19 22 31 41 53 63 72 87 94
(eQ)2 = 4 52 50 27 0,25 0,00 8 4 2,25 0,02
M1 = 4 52 50 27 0 0 8 4 2 0
(eT)2 = 188 55 20 0,64 33,6 2 29 9 0,01 11
M2 = 192 107 70 28 34 2 37 13 2 11
(eS)2 = 973 441 169 1,16 9 23 7 5 11 2
M3 = 1165 548 239 29 43 25 44 18 13 13
(eU)2 = 2371 289 33 86 23 63 1,12 2 0,16 6
M4 = 3536 837 272 115 66 88 45 20 13 19
(eD)2 = 1640 361 484 121 68 8 6 51 0,72 4
M5 = 5176 1198 756 236 134 96 51 71 14 23

It is quite obvious that with finer grain, owing to higher division numbers, the misfit must become less and less.
The misfit for fifths only is least in Mercator's 53 comrnas. The 94-division does extremely well, and the 41 Janků supracommas are very good. Next come the temperaments with multiples of twelve.
The best fit for fifths and thirds together is realised by Mercator's commas. These are better still than the 87 subcommas. The 87 lie very far off, so to say beyond and below the horizon. Apart these superfine divisions, the next better to Mercator's 53 are the Huygens 31 diŽses. These are better than Janků's 41 supracommas. The Woodhouse tritotones and the common 12 semitones are very far behind.
For fifths, major thirds and sevenths together -apart the higher divisions- the Mercator commas are again leading (M3 = 25). The next to follow are again Huygens's 31 diŽses (M3 = 29). The extreme good major thirds and sevenths make them preferable to Janků's 41 supracommas.
If we want the eleventh to come in, and if we for a moment disregard the higher division numbers, we see that Janků's 41 supracommas are leading with M4 = 66. Because of their good approximative eleventh they surpass even Mercator's 53. If we want to have a better fit, we must not turn to 53, but to 72, or at least to 63, to twelfths of a tone, or tenths of a tone.
Finally, for including the thirteenth we must refer to divisions beyond 53. By far the best are the 87 subcommas, with M5 = 14.

Complication factors. From the practical point of view, larger division numbers imply increasing difficulties. The keyboard must become a much more complicated structure. The proper notation of the notes is still an unsolved problem. It is quite impossible to make comparative experiments, not to mention measurements. Sure the difficulties entailed by a greater division number will not be simply proportional to that number. The increase of intricacy is not the addition of a certain amount tor each additional note. The intricacy will rather increase by a certain factor for each additional note. As a working guess I will assume that each additional note implies a multiplication factor of 100 cent, say 18/17. If the complication of the 12-note temperament is considered to be unity, then the complication of 24-note temperament would be taken to be 1200 cent, that is 2 times more, and therefore 2. The 19-tone temperament, 7 notes more, would be 700 cent, or 3/2 times more complicated than the 12-note temperament. This I will call the complication factor C19 = 1,5. I recall that 500 cents (the fourth) is the ratio 4/3. For 400 cent we put the ratio of the major third, i.e. 5/4, and 300 cent is the ratio 6/5. Owing to my working guess the complication factors for the various temperaments will be considered to be as listed in the table below.

Table of complication factors

Division N 12 19 22 31 41 53 63 72 87 94
factor in cent 0 700 1000 1900 2900 4100 5100 6000 7500 8200
CN 1 1,5 1,6 3,0 5,3 10,7 19,2 32 77 115

Comparison by deficiency numbers. If a temperament, owing to a finer grain, thanks to a greater division number shows less misfit, but a greater complication factor, it is quite plausible to multiply misfit and complication numbers in order to estimate the value of the temperament in practice. The resulting deficiency number CM for different choices of the fundamental intervals is shown in the following table.

Table of deficiency numbers

N 121922 31 41 53 63 72 87 94
CM1 4 78 90 81 1 0 154 128 173 2
CM2 192 161 126 84 180 21 710 416 174 1265
CM3 1165822 430 87 228 268 845 576 1001 1495
CM4 3536 1256 490 445 350 942 864 640 1016 2385
CM5 5176 1797 1360 708 710 1027 979 2272 1078 2645

On inspecting this table we shall remain aware of the fact, that the information contained rests on a working guess, a reasonable guess concerning the growing intricacies as the grain becomes finer. It is not based on experimental truth. Still it leads to some suggestions worth considering. 3)
For the use of fifths only, Mercator's 53 commas are superior, with their deficiency number 0,052. They are followed by Janků's 41 supracommas, by the 94 semicommas and by the common duodecimal temperament.
If the major thirds come in, Mercator's 53 commas again are first, with CM2 = 21. This time they are followed not by Janků's 41, but by Huygens's 31 diŽses (CM2 = 84). Janků now is on a level with the usual 12 semitones. Aiming at high fidelity for sevenths, fifths and major thirds we see Huygens's 31 diŽses as the most recommendable (CM3 = 87). Now Mercator stands back (268) even behind Janků (228).
The elevenths, with fifths, major thirds and sevenths are best served by Janků's supra-commas (CM4 = 350), and not by Mercator's commas (942). Next to Janků we find Huygens (445). The finer divisions 63, 72 and 87 have far better realisations of the eleventh, but their overwhelming complication factors frustrate their suitability. It is worthwhile to notice that the diŽses do not come far behind the supracommas.
For accommodating the thirteenth besides the other four intervals there is a close competition between the 31 diŽses and the 41 supracommas, 708 against 710. Within the errors of the working guess this is a dead heat.

Conclusion. There is great need for using a novel variety of harmonic intervals and chords, and to expand the realm of harmony.
It goes without saying that the very first thing to do will be adding the seventh to the common chord so as to use the primary tetrad with harmonic numbers 4 : 5 : 6 : 7. That was a demand of Bťla Bartůk.
In vocal music this is readily done. Here no special provision for an equal temperament is necessary. But in order to support vocal music with the aid of instruments and especially with keyboard instruments tuned according to an equal temperament it is important to adapt the temperament to the new needs. Here it is good to know that an equal temperament of 31 diŽses in the octave promises well. For the reproduction of the primary tetrad 4 : 5 : 6 : 7 it is conspicuously the most suitable. For the nearest future it seems to be the best policy to switch over from twelve to thirty-one. Music today seems ready for that.
In times to come the eleventh too is liable to rise to the status of a recognised and desirable concord. In that future situation the equal temperament with 41 supracommas might show some advantages. Until those distant days the 31 diŽses will offer a fair approximation to the eleventh. The error here is -9,4 cent only. That error compares favourably with the +13,7 cent error of the major third in the usual duodecimal temperament of semitones. Many people are prepared to accept the latter error without grudging, as daily experience shows.
The use of the thirteenth as a concord is still farther off in a more remote future. We saw that the 31 diŽses and the 41 supracommas offered equal advantages for balancing the interests of the eleventh and the thirteenth.
Hence for many reasons a recommendation of the choice of thirty-one diŽses for an equal temperament seems sound and safe.

Footnotes

1) This paper is a preparatory study for a report on the problem of the notation of microtones, which will be presented to the general assembly of the International Musicological Society in 1967.

2) The antanairesis can be illustrated by taking the irrational numbers to represent the components of a vector. The direction coefficients of that vector consequently will show irrational proportions. The gist of the method is to combine and regroup the components so as to replace them by new sets of components. The direction coefficients of these components are systematically made to keep rational proportions between one another, while they each are made gradually to approach the direction of the resultant vector. Thus the initial irrational proportions are gradually approximated by rational proportions. See references in the article A.D. Fokker, Multiple antanairesis, in: Koninkl. Nederl. Akademie van Wetenschappen, Amsterdam, Proceedings, Series A, 66, 1963.

3) In view of the frequent use of a temperament of quarter tones (N = 24) a few words may be added. The error in the major third here is the same as in duodecimal temperament. The eleventh is much better. The error in the seventh is reduced with 40%. The figures for the deficiency numbers are 8, 384, 1106, 1110, and 1290 respectively. These compare unfavourably with the figures given in the table.

Adriaan D. Fokker, 1967