Equal Temperament and the Thirty-one-keyed organ

There are a great many musics among the people inhabiting our planet, and they show differences as great as the languages spoken by the various races and nations. In addition, there is the music of the birds, and there are melodic lines of human speech. It is a very difficult task to represent all this music by written symbols. This task requires centuries of analyzing and systematizing work. Too soon one is eager to believe that one's own music is the most natural, and that one's own way of putting it on paper in notes on staves is the most straightforward and efficient way of recording it, and that there is no alternative. The fact is, however, that our subdivision of the interval of the octave into 12 equal semitones represents a solution for certain special needs only. It embodies a stage in the historical development of the art, a system with its virtues and its deficiencies, and one that is likely to be modified or even relinquished during the course of historical evolution. There are vast civilizations of autochtonic musics, not very remote from the idioms of Western civilization, that must remain behind the horizon of our understanding, unless the system of 12 equal semitones in the octave is disposed of, and a better approximation is substituted for it.

The word octave, which denotes the fundamental distance (and similarity) between the notes sung by men and women simultaneously when they join in a common hymn or song, testifies to a specified musical tradition with scales in which the eighth note is identified with the first. The Greeks did not pin down music in one scale of seven (8 - 1 = 7). They called this relation diapason, that is "through all", and by so doing they put into evidence the idea that the whole variety of notes is contained in a sequence between the first and the last note of a disapason. Nowadays the two notes separated by this interval are known to have frequencies in the relationship of 1:2, the higher note having double the frequency of the lower. Such notes, in spite of their differences, are identified as having the same musical value, nowadays as well as among the ancient Greeks.
In what follows I take for granted that the reader is acquainted with the names of the notes, A, C, D, E, F, G, A (or la, ti, do, re, mi, fa, so, la) and their lowering and raising by signs called flats () and sharps (): A-flat = A, C-sharp = C, and so forth. Where necessary I have added refinement.
Pythagoras is reported to have systematized the collection of notes by selecting a certain specified interval relationship and ordering a sequence of notes according to this relationship, with the rejection or exclusion of all notes not entering this order. The selected relationship was the one that is presently called the perfect fifth, the notes in question having frequencies in the proportion 2:3. Thus the number 3 comes in (in spite of the historical name fifth). The series of notes arranged by this interval relationship was
F : C : G : D : A : E : B
The sequence may be continued to the right, where sharps come in:
E : B : F : C : G...,
or to the left, where flats come in:
...D : A : E : B : F : C.
Obviously this was a one-dimensional series. The only true consonances to be found were in the octaves, in the perfect fifths and, in a lesser degree, in the fourths. The relationship C:E (in numbers 64:81) was considered to be a discord, and so it was in Pythagoras' linear system.
In the Renaissance period singers and composers of polyphonic music discovered that by a slight alteration of Pythagoras' E one arrived at a perfect consonance of C and E. This was the concord of notes with frequencies in the relationship 4:5 (64:80), where the number 5 comes in (the whim of history makes us call the interval 4:5 a perfect third!) Accordingly one is led to a two-dimensional specification of the legitimate notes with concordant relationships, as follows:

In this field one meets two kinds of semitones, major semitones - such as C : D and D : E - and minor semitones - such as D : D, and E : E. In the major semitones the frequency ratio is 15:16; in the minor semitones the ratio is 24:25. Obviously this procedure leads to an innumerable crowd of notes; and, although in the human voice there is no restriction whatever, on an instrument it is not possible to provide separate strings, pipes, or keys for all of them. There is the slight difference of notes on different lines bearing the same name. In an early age musicians agreed to discard this difference. In the 16th century Gioseffo Zarlino said that one should adjust the sequence of "fifths" in C : G : D : A : E in order to have a perfect concord between C and E. This was a very slight readjustment, perceptive to a very acute ear but not a disturbing nuisance.
Rather bigger are the differences between notes such as D and C or G and A, which are separated by three intervals in a vertical column. If one can afford only a string or a pipe for G, and not one for A, one can sound the perfect common chord E : G : B but not the chord A : C : E. The increasing demand that the chords be available in all keys led to a compromise. The present equal temperament is based on the decision to disregard these differences, thus identifying all notes bearing the same names and every fourth in the vertical columns. Thus one is left with a collection of 12 notes. (Notes that duplicate notes already put down in order places are in parentheses.)

All semitones are leveled, and the ratio is very approximately 17:18. This equal temperament was advocated by Vicenzio Galileo, in the 16th century. The exact figures to four decimal places were calculated by Simon Stevin early in the 17th century. Not before the middle of the 19th century was the equal temperament universally accepted.
No doubt this equal temperament with its perfect cyclic symmetry has rendered great services in the evolution of Western musical civilization. One must recognize this. On the other hand, many musicians are aware of the deficiency of the chords sound on the pianoforte and the organ that have been tuned according to the duodecimal temperament. Their complaint is the same as the objection of the 17th- and 18th-century musicians who refused to have their beautiful harmonic consonances spoiled by such a compromise. One of them was Christiaan Huygens, who lived in the 17th century. He was a very fine musician, but he also happened to be a mathematical genius. As a mathematician he discovered a new harmonic cycle (nouveau cycle harmonique) when, in attempting to preserve the difference between major and minor semitones, he took just this difference as an elementary unit for the even division of the octave. This same unit is the excess of an octave over three perfect thirds. It is about half the minor semitone and one-third of the major semitone. Thus it is one-fifth of a whole tones, and Huygens called it diesis. It follows that the octave is partitioned into 31 elementary steps.
The choice of 31 notes to represent the boundless meshwork of notes related by the ratio numbers 2, 3, and 5 evidently means the adoption of another identification of notes instead of the identification of flats and sharps found in the black keys of the ordinary keyboard (C and D, and so forth). Indeed, in Huygens' new cycle notes such as B-double-sharp and D-double-flat are taken to be identical. Other examples are found in A, and G, in E, and G, D and F. The mesh in the net occupied by the 31 notes is as follows:

One finds the seven basic notes A, B, C, D, E, F, G, their flats and sharps, their double flats and double sharps, ruling out two double flats that are identical to two double sharps, and vice versa.
Such a multitude of notes was far too much for the comprehensive faculties of Huygens' contemporaries, and his discovery was left to oblivion. Yet, he made another very significant observation. He remarked that, besides a faithful rendering of the perfect third, his harmonic cycle also provided a nearly exact reproduction of the perfect seventh. The perfect seventh is the interval of the fourth and the seventh harmonic notes of a string. It is a concord in which the number 7 comes in. Huygens claimed that it was just as valid and legitimate as the other perfect intervals used in music and a beautiful concord, but contemporary public opinion was against him. This repudiation of the perfect seventh was not unlike the disqualification of the perfect third as a concord by medieval people.

In the field of notes related by the numbers 2, 3, and 5, one finds pairs - such as C and A-sharp - related by the numbers 128 (= 27) and 225 (= 32 × 52). Owing to the complexity of the latter number, the interval might be taken to sound a discord. But next to 225 lies 224 (25×7), and 128:224 = 4:7. This simple ratio fully qualifies the interval as a concord, and Huygens' point was that if his cycle were adopted, the number 225 would be readjusted to a very near approach to 224. Huygens was aware of latest possibilities and beauties. As an example he pointed out the beauty of the chord B : E : G. This is a marvellous remark, and in his days it must been very shocking indeed, because every 17th-century professional musician would have denounced the chord as utterly discordant.

Nowadays there is a new demand that the system underlying our music be improved. Well known is the attempt by Alois Hába. He was aware that in order to preproduce native songs of his countrymen in Czechoslovakia he needed a finer grain, so to speak, and therefore he halved the equal semitones. He tried to use quarter-tones. However, it is clear that by simply adding 12 more notes between the existing 12 notes one cannot improve the harmonic relationships and the musical quality of the original 12 notes. What one needs for improvement is not quarter-tones but fifths of tones - the system Huygens used.
Again, Bela Bartók, the Hungarian composer who died in 1945 in the United States, recognizing the value of the Hungarian peasant music both as a living tradition and as a source of inspiration for modern music, stated that he over and over again met the interval of the harmonic seventh and that he therefore laid at the bottom, as a basic foundation for music, a chord of four notes, adding the perfect seventh to the common chord. Bartók's testimony carries great weight, for he was one of the very greatest modern composers. Therefore, for the evolution of music, it is very urgent that we find an equal temperament suitable for the reproduction of Bartók's fundamental chord, which may be specified by the harmonic numbers 4:5:6:7. The "tricesimoprimal" equal temperament discovered by Huygens 3 centuries ago fulfills the requirement.
The aim of an equal temperament is the reproduction of certain intervals, which are chosen because they are considered fundamental, as integer manifolds of one single unit. This means that the problem is, given the measures of the intervals as certain numbers, to find their common divisor. Of course, it is well known that the musical intervals are represented by irrational numbers and that such numbers can never have a common divisor. Thus we must try to find an approximate divisor. The octave (1:2) measures, by definition, O = 1200 cents (the cent is the hundredth part of an equal semitone). The perfect fifth (2:3) has Q = 702 cents. The perfect third has T = 386 cents, and the perfect seventh S = 969 cents. Any divisor common to two numbers is also a divisor of their difference. Now, looking at

O - 3T = 42 = 1 × 42
2T - Q = 70 = 2 × 35
O - S = 231 = 6 × 38.5
T = 386 = 10 × 38.6

one finds that T must have 10 units; O = 31 units; Q = 18 units; and S = 25 units. The result O = 31 units, whence 1 unit = 38.7 cents, fully confirms the discovery of Huygens. It could not have failed to do so.
The question of how to handle the harmonic seventh in music has been raised several times. The use of it has been advocated by several authors, beginning as early as the middle of the 18th century. Among them was Giuseppe Tartini in Padova, who allotted this note a place in the gamut. Accidentally, it happened that in those years the duodecimal temperament started its conquest of Western music. It could reproduce intervals of 900 and 1000 cents, but not 969 cents, as required by the harmonic seventh. The duodecimal temperament therefore inhibited the use of Tartini's seventh. The tricesimoprimal temperament had never received any attention, and therefore Tartini's attempt was doomed to fail, and it also fell into oblivion. The time for recollection might now have come.
From the mathematical side, Leonhard Euler, a contemporary of Tartini - neither known to the other - showed another method for introducting the seventh into the structure of actual music. Euler was also unaware of Huygens and his harmonic cycle of 31.
Giuseppe Tartini left us a means for notation on the stave. He introduced half-flats and one-and-half-flats. The perfect seventh on G is slightly flatter than the discordant F, and we may call it F-minus, or F-half-flat. The perfect seventh on C is slightly flatter than B-flat, and we may call it B-flat-minus or B-one-and-a-half-flat. Tartini's sign for a half-flat is a hook pointing down, and his one-and-a-half-flat is a certain contraction of this hook with an ordinary flat (Fig. 1).

In the tricesimoprimal temperament, F-half-flat is identified with E-sharp. Musicians have a certain liking for connection a relation of a sixth with the letters G and E. Therefore it might be safer to write F-minus instead of E-sharp. The same remark applies to the use of B-flat-minus rather than A-sharp if one meets the harmonic seventh related to C.
Obviously one could benefit by having a notation for a half-sharp too. The perfect seventh below C is a trifle sharper than D. Let us call it D-plus, or D-half-sharp. Again, the perfect seventh below B is a trifle sharper than C-sharp, call it C-sharp-plus, or C-one-and-a-half sharp. The sign for a sharp having two vertical strokes, it might be natural to write one-half sharp with a single vertical stroke, and similarly, one-and-a-half sharp with three vertical strokes. The subharmonic primary chords with four notes would therefore show on the stave as they do in Fig. 2.

A sequence of notes proceeding by fifths of a tone is shown below in Fig. 3 twice, first the available signs of Tartini, second with available sharps. At once the identity, in the tricesimoprimal temperament, of, for example, E-flat and D-sharp-plus or E-double-flat and D-one-half-sharp, is obvious.

I return to the two-dimensional representation of all the notes related by the numbers 2, 3, and 5. If we now recognize the harmonic seventh as a legitimate concord, there is evidently good reason to extend the representation to a three-dimensional one. That will appear as a kind of three-dimensional chessboard. Crystallographers would speak of a crystal lattice with notes instead of atoms. In crystals we meet a certain periodicity of structure and elementary cells are reproduced over and over again, so that not all atoms are different, and the variety is a finite variety only. In a similar way, by establishing identities between various notes, the threefold endless multitude of notes in the harmonic lattice is reduced to a finite set (Fig. 4).

In addition to the identification of notes in the two-dimensional harmonic plane of relationships made by threes and fives, therefore, identifications related to 7 and one or both of the others. We have already come across the identification of 225 = 32 × 52 and 224 = 25 × 7, that is, the identification of A-sharp and B-flat-minus relative to C. Another identification is reasonable, that of 343 = 73 and 341 1/3 = 1024/3 = 210/3. In relation to C, these notes come out as G-double-flat-minus and F. Three such identities, or - speaking geometrically, three identity vectors - delimit a unit cell in the boundless three-dimensional field.
The foregoing sections dealt with the history and the systematics of the problem of equal temperament. Now the practical problem is before us to put the insights into practical use. Of course one wants to be in a position to have all the notes available ready for use at one time. For an organ there is no specific difficulty in providing more pipes. The valves and opened and shut by electropneumatic devices. The problem lies in the construction of the keyboard, and in the tuning of the pipes. The keyboard is shown in Fig. 5. There are two manuals, the second being placed behind the first, joining it at the same level, but rising at an increasing inclination, steeper than the first. The keys are arranged like tiles on a roof, each key midway between a lower pair and a higher pair of keys. The keys in a horizontal row sound notes with distances of whole tones between them. Next to E one finds D and F. In stepping from E to F one finds the next higher horizontal row, which ontains G, A, B, C, D, E... The key for E is just in front of and lower than the key for F. This shows the situation of the steps of one diesis, that is, one-fifth of a tone. Keys for a series of notes such as D, D-plus, D-sharp, E-flat, E-minus, and E lie in a straight tier running away up the keyboard.
The keys have been coloured to facilitate orientation. The lettered notes A, B, C, D, E, F, G have white keys, inconformity with ordinary keyboards. The black keys of the ordinary keyboard appear here in pairs, for example, C and D, D and E, and so forth. Other notes have blue keys. It happens that every white key lies in a straight rising tier between two blues: D between D-minus and D-plus, E between E-minus and E-plus (= F), F between F-minus (= E) and F-plus, and so forth. In all there are 7 × 1 whites, 7 × 2 blues, and 5 × 2 blacks, or 31 keys per octave.
In order to facilitate playing, every note is represented by two keys. Therefore there are 11 horizontal rows. The keys in the lowest row have two replicas, one in the middle row and one in the highest row. Thereby the little finger and the thumb may find the keys they need in one range, the three long fingers may find their wants in a higher range. Pairs of keys in a similar relative position always bear the same interval. Therefore is it not necessary to change the fingering when one is playing various scales or melodies at different pitches. For transposing one simply displaces the hand in a uniform translation. Besides the straight horizontal rows and the straight upward tiers there are skewed tiers. Rising to the right, one finds ascending major semitones: A-minus (= G), A, B, C, D, E (= D-plus), D (= E-minus), E (= F-minus), F-plus (= G), G, G natural. Thus the interval of the fifth C : G in one tier is divided into six steps; in the other tier it is divided into nine steps. The fifth being divided into six parts, it may also be divided into three equal parts. These are superseconds, which have the relationship 7:8. They are elementary steps in the gamelan of Java and Bali. Again, the fifth may be divided into two equal parts. The result is a pair of "thirds" intermediate between minor and major thirds. These intermediate thirds make a very beautiful rendering of the cuckoo's call. In human civilization they are used in music of the Arabs according to the teaching of the famous medieval musician Zalzal. Thus the tricesimoprimal temperament provides means for a synthesis of many musics. The pedal (Fig. 6) shows an arrangement along lines similar to those of the manuals.

Fig. 5

The tuning of the pipes is made by thirds. The deviation from perfect consonance is so small that, owing ot the acoustic coupling by the intermediate air, two pipes - such as A and C - sounded together do not produce beats. Therefore an electric ocillating generator is first tuned against the pipe A to a perfect third. This is done by means of the Lissajous figure shown on the screen of an oscillograph which puts the sound of the pipe against the oscillations of the electric generator. If the generator is oscillating at the correct frequency, the pipe C is sounded and its sound taken to the oscillograph. The pipe is adjusted until the resulting Lissajous figure shows the required number of beats, which might be any number, for example, one in seven seconds, or one in five seconds, as the case may be.
The 31-keyed organ has now been installed in Teylers Museum at Haarlem. It was built by B. Pels en Zoon, Alkmaar, Netherlands. It has a second console with ordinary keyboards. By electrical switchings, these keyboards can be connected to special selections of the complete scale. One of these selections is the one shown previously, with E and G. It provides the notes as used and conceived by the 17th and 18th century composers and it thereby restores the ancient beauty of their creations. This is the console looking back to the classical past.
Recitals of ancient classical music are regularly presented to the public. In addition, the works of modern composers who avail themselves of the new possibilities (Jan van Dijk, Henk Badings, Paul Ch. van Westering, Martinus J. Lürsen, and Arie de Klein) have been executed at the console with the new keyboards. That is the console looking to the future.

Adriaan Daniël Fokker, October 1955