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 avantgarde 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 preestablished
structure of fact. It is wholly manmade. 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 thirtyone 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 fivepart 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 straightforward 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 diëses, fifths of a tone.
 There is the division of Von Jankó, 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
semicommas. 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 94division does extremely
well, and the 41 Jankó 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 diëses.
These are better than Jankó'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 diëses
(M3 = 29). The extreme good major thirds and sevenths make them preferable to
Jankó'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 Jankó'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 12note temperament is considered to be unity, then the complication
of 24note temperament would be taken to be 1200 cent, that is 2 times more, and therefore
2. The 19tone temperament, 7 notes more, would be 700 cent, or 3/2 times more complicated
than the 12note 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
 C_{N}  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
 12  19  22  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  1165  822  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 Jankó'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 Jankó's 41, but by Huygens's 31
diëses (CM2 = 84). Jankó 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 diëses as the most recommendable (CM3 = 87). Now Mercator stands back (268)
even behind Jankó (228).
The elevenths, with fifths, major thirds and sevenths are best served by
Jankó's supracommas (CM4 = 350), and not by Mercator's commas (942). Next to
Jankó 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 diëses do not
come far behind the supracommas.
For accommodating the thirteenth besides the other four intervals
there is a close competition between the 31 diëses 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 Béla Bartók.
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 diëses 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 thirtyone. 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 diëses 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 diëses 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 thirtyone diëses 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
