ABOUT 31-TONE EQUAL TEMPERAMENT

by
Paul Rapoport

From: "About 31-tone equal temperament and six songs written in it. An essay with Songs of fruits and vegetables" (score), Corpus Microtonale vol. 23, Diapason Press, Utrecht, 1987

1  INTRODUCTION
Readers who already know the basic theory behind 31-tone equal temperament will find much here that is familiar. Nonetheless, there are several things about the presentation in Sections 2 and 3 which are new. Principal among these are:
(1) the derivation of the scale of 31 degrees; (2) the tabular representation of its enharmonic equivalents; and (3) the names for the intervals in 31-tone equal temperament which are not found in 12-tone equal temperament.
Concerning (1) and (2): the 31-tone scale is often presented on a musical staff from lowest to highest tone within an octave. In this arrangement, the long series of notes, some with new accidental signs, may be overwhelming to the beginner. I believe that after some explanation, Figure 1 and Table II together will make the relationships of the notes (and their names) much clearer.
Concerning (3): the best set of interval names hitherto published of which I am aware is that of Kendall Stallings in Adriaan Fokker's New music with 31 notes (Bonn - Bad Godesberg, 1975). But Stallings' terms do not lend themselves to simple and unambiguous abbreviation. More importantly, in two categories they do not indicate the precise melodic relationship of the new intervals to the more familiar ones. The interval names used in this essay solve both these problems while maintaining the consistency of Stallings' names. They also point the way towards a suitable set of names in some other tuning systems.
Those who know little or nothing about 31-tone equal temperament will, I hope, find the explanation of it simple and the system itself intriguing. Choirs wanting information on how to sing in 31-tone equal temperament will find some in Section 4 of this essay.

2  31-TONE EQUAL TEMPERAMENT: THE RATIONALE
"Why 31?" is the question usually asked, with traces of skepticism. The general answer is this: there are three interrelated reasons to use tuning systems other than the one we are used to, i.e. 12-tone equal temperament. Other systems may provide: (1) harmonies that are more in tune, or less in tune, if desired; (2) new sounds in both melodies and chords, sounds which may resemble those we know or be quite foreign; and (3) different tonal structures. More will be said about all three of these things in due course.
Twelve-tone equal temperament has been with us so long that we may think nothing else is possible. Or if something is, that it must merely subdivide the 12-tone equal-temperament semitone and therefore sound terribly out of tune. This attitude is often revealed in the use of the term `quarter-tones' for any system that is not 12-tone equal temperament. But quarter-tones, which are exactly half the size of the semitone of 12-tone equal temperament, are not the only possibility for a different system. There is no reason why we must accept the twelve pitches we have as a necessary starting point. The pitch spectrum is continuous: we may mark off divisions of the octave anywhere, subject only to our ability to hear and reproduce them. We need not even have octaves in our system, although nearly all tuning systems that have been used or described do have them.
In the discussion which follows, the intervals of various scales will be represented mostly in cents, a logarithmic measure devised by the scientist Alexander John Ellis (1814-1890). By definition, the octave contains 1200 cents, and thus the 12-tone equal-temperament semitone 100 cents, the perfect fifth (seven semitones) 700 cents, etc. These round but exact numbers should not lead us to believe that the intervals of 12-tone equal temperament are all maximally in tune, for they are not, as the following discussion will show.

Music example 1.
Music example 1. The first eight tones of the harmonic series on D.

Music example 1 is one representation of the first eight tones of the harmonic series on D. Depending on the timbre of that sounding D, these harmonics (and some still higher) are heard in varying strengths whenever the D is sung, bowed, blown, etc. In principle, the harmonic series may be used to derive all the intervals of any tuning system, although adjustments to the intervals and pitches chosen may be needed. These adjustments are called temperings, which accounts for the word `temperament' in names of many tunings. `Equal' in `equal temperament' simply means that the octave is divided into a number of parts (e.g. 12, 31) which are all the same size, i.e. contain the same number of cents.
We may now investigate the size of intervals in the various tuning systems. Taking the major third as an example, in 12-tone equal temperament it is four semitones, therefore 400 cents. But if a major third is perfectly in tune, the harmonic series of any fundamental pitch shows us that the frequency ratio of the lower pitch to the higher one is 4:5 (see Music example 1). This is because the major third occurs between pitches 4 and 5 in the harmonic series. The actual pitches may have frequencies such as 293.333 Hz (D) and 366.667 Hz (F): what is important for present purposes is not the absolute frequencies but the ratio between them. In this case, the 4:5 ratio implies that the fifth harmonic of the lower tone (i.e. 5 × 293.333 = 1466.667 Hz) is exactly the same as the fourth harmonic of the upper tone. This contributes to the major third with this particular ratio sounding in tune. When such an interval, represented by a ratio expressed in small integers, is slightly out of tune, that fact is usually quite noticeable, due to the presence of an unpleasant wavering sound known as `beats'.
At this point we need to compare 4:5 (the ratio of the lower frequency to the higher in the pure or `just' major third) with 400 cents. The formula for converting a ratio to cents is:

I = 1200 × (log i)/(log 2)  cents,

in which I is the interval size (in cents), and i the ratio of the higher frequency to the lower. We find, then, that the just major third is 386.314 cents (rounded off to three decimal places, as are other numbers in this essay). The major third of 12-tone equal temperament is thus sharp by 400 - 386.314 = 13.686 cents.

Table I. Just ratios and interval sizes of various consonances, and deviations from these in 12-tone equal temperament, 31-tone equal temperament, meantone tuning, and Pythagorean tuning.

Deviation (cents) from just in:

Interval
Just
ratio
Just size
(cents)
12-tone
equal
temperament
31-tone
equal
temperament
meantone
tuning
Pythagorean
tuning

perfect octave 1:2 1200 0 0 0 0
perfect fifth 2:3 701.955 - 1.955 -5.181 - 5.377 0
major third 4:5 386.314 +13.686 +0.783 0 +21.506
minor third 5:6 315.641 -15.641 -5.964 - 5.377 -21.506
harmonic seventh4:7 968.826 +31.174 -1.084 +38.017 +27.264

The amount of 13.686 cents is between 1/8 and 1/7 of a 12-tone equal-temperament semitone, and represents a difference easily heard in most musical contexts, melodic or harmonic. Differences of merely a few cents are often detectable in chords, depending on the timbre of the instrument(s), dynamics, tempo, range, and other musical considerations. "But wait," some may say, "the major third of 12-tone equal temperament does not sound out of tune at all." That may be true for a piano or for a large performing group like a modern orchestra; but in an orchestra the players are spread out and produce many different timbres; a piano also has acoustically complicated sounds. Thus the orchestra's and piano's approximations to the interval 4:5 sound more acceptable than the same approximations do on, for example, most harpsichords and organs.
Table I presents the four principal consonant intervals of tonality, as well as the `harmonic seventh' (which will be discussed shortly). It shows their just ratios and sizes, and the amount of sharpness or flatness for each of them as represented in four different tuning systems. `Just ratio' refers to the intervals determined by the harmonic series: these give the perfectly in-tune reference points. The plus-sign means sharp, the minus-sign flat. The two tuning systems on the far right will be dealt with after brief discussion of 12- and 31-tone equal temperament.
From Table I it may be seen that the perfect fifth is very close to just in 12-tone but slightly flat in 31-tone equal temperament, the major third is quite sharp in 12-tone but nearly just in 31-tone equal temperament, and the minor third is quite flat in 12-tone but slightly flat in 31-tone equal temperament. A table for the other consonances, namely perfect fourth, minor sixth, and major sixth, could of course be easily constructed because these three intervals are simply harmonic inversions of the middle three intervals in Table I.
The implications of Table I go far beyond the comparison of a few intervals. But before some of those implications are taken up, one further interval must make its appearance: the harmonic seventh, which is the consonance formed by the fourth and seventh tones of the harmonic series (see Music example 1). In 12-tone equal temperament its closest representation, i.e. the minor seventh, is so sharp that we must conclude that the harmonic seventh is simply not present in that tuning system. In 31-tone equal temperament there is an interval which comes very close to the just harmonic seventh. Although nearly all Western music ignores this interval (the ratio 4:7), 31-tone equal temperament makes it usable and thus more than only a theoretical consonance. While the other consonances already discussed are more important because they have lower numbers in their ratios, the harmonic seventh need not be ignored. Of course, any music which uses it extensively will sound different from music in 12-tone equal temperament for all three reasons mentioned in the first paragraph of this section.
In passing, this explains why the sign in front of the C in Music example 1 is necessary: because the seventh harmonic up from D is very far from the minor seventh C, a new sign is needed to indicate it.
By now it may be suspected that 31-tone equal temperament is not so strange. Indeed, it shares much with 12-tone equal temperament. The diatonic progressions in 12-tone equal temperament behave similarly in 31-tone equal temperament, as do many chromatic progressions. But two basic differences are important: the actual sound of the intervals in the two systems even when the notation for the same music in both systems is identical, and the vastly increased chromatic possibilities in 31-tone equal temperament.
The first difference has already been partly discussed. The major thirds in the two systems do not sound the same. Moreover, from the chromatic realm, an ascending or descending scale such as D - D - E - F - F etc. or D - D - C - B - B etc. (or a scale containing both sharps and flats) also sounds different in 31-tone equal temperament. The reason is this: in 12-tone equal temperament, D and E (to choose one pair) are the same; and they divide the distance between D and E exactly in half. But in 31-tone equal temperament, D is lower than E by 38.710 cents. The former is 2/5 of the way up from D to E, the latter 3/5 of the way, the whole tone being divided not into two equal parts but into five. That this must be so will be demonstrated shortly, but at present it is sufficient to note that the chromatic semitone (e.g. D - D) and the diatonic semitone (e.g. D - E) are different intervals in 31-tone equal temperament in actual sound.
There are three basic results of this which should be readily grasped. First, the 12-tone chromatic scale is `uneven' in 31-tone equal temperament; the two different semitones give such a scale a particularly expressive effect which is lacking in 12-tone equal temperament. Second, the leading-tone interval (e.g. D to E) is larger in 31-tone than in 12-tone equal temperament. Third, the specific enharmonic equivalents of 12-tone equal temperament (e.g. D = E, C = B, F = G) are not valid at all in 31-tone equal temperament, which has an entirely different set of enharmonic equivalents. This last point will become clearer when the note and interval names of 31-tone equal temperament are presented in Section 3.
It is now appropriate to consider the relationship of 31-tone equal temperament to the system on its right in Table I: `meantone'. Meantone temperament traces its origins at least as far back as the early sixteenth century. It experienced widespread use as a keyboard tuning at least until the late seventeenth century, and in some areas for certain instruments until well into the nineteenth century. It arose because the prevailing Pythagorean system was inadequate for polyphonic music using thirds as consonances. In the Pythagorean system, the major third is even sharper than in 12-tone equal temperament and thus sounds poor as a consonance or stable interval. This situation arises because the only just consonances in Pythagorean tuning are, by definition, the octave and the perfect fifth, which are the intervals used to generate all the others. Accordingly, the size of the Pythagorean major third is determined by adding four conjoined perfect fifths, e.g. (ascending) D - A - E - B - F, and subtracting two octaves. In cents: 4 × 701.955 - 2 × 1200 = 407.820, which, as Table I shows, is 21.506 cents sharper than the perfectly in-tune just major third (i.e. between 1/5 and 1/4 of a 12-tone equal-temperament semitone sharper).
The amount of 21.506 cents is known as a syntonic comma. Clearly, if we flatten each of the four perfect fifths, for example D up to F, by 1/4 of this comma, i.e. 5.377 cents, we will create a just major third (after subtracting the two octaves). Thus, each new perfect fifth is 701.955 - 5.377 = 696.578 cents. We may now confirm this procedure by adding four of these perfect fifths together and subtracting two octaves: 4 × 696.578 - 2 × 1200 = 386.314 cents, which Table I indicates is the size of the just major third, and which the formula given earlier in this section provides. (The results of this calculation, like all others in this essay, are corrected for any round-off error.)
Thus the meantone tuning system takes a small amount away from the just perfect fifths in order to create just major thirds. As Table I shows, the meantone minor third is also flat by 1/4 comma which compares favorably to the very flat Pythagorean and 12-tone equal-temperament minor thirds. It should also be noted that in meantone tuning, as in 31-tone and 12-tone equal temperament, the major third is divided exactly in half by the major second: the major third comprises four perfect fifths, and the major second two of them. This is not the case if we derive scales only from the first five tones of the harmonic series and use no tempering: the just major third will have two distinct major seconds, the 8:9 and the 9:10. Meantone tuning, in fact, gets its name from its major second, because this major second is the geometric mean between the 9:10 second (182.404 cents) and the 8:9 second (203.910 cents). Its ratio is thus 5/2, its size 193.157 cents.
There is one further important result of all this. In meantone tuning, as in Pythagorean tuning, the series of conjoined perfect fifths does not close; i.e. no matter how many perfect fifths are added, we never arrive back at the note name we started with. In 12-tone equal temperament, of course, adding twelve perfect fifths brings us back to the starting point, if we subtract seven octaves: 12 × 700 - 7 × 1200 = 0 cents. Although the proof of the noncyclic nature of meantone tuning is beyond the scope of this discussion, it will easily be seen that twelve perfect fifths in this system leave us short: 12 × 696.578 - 7 × 1200 = -41.059 cents. So, for example, going from E up to D by twelve perfect fifths leaves us 41.059 cents short of seven octaves. This amount is known as a diesis.
On a keyboard with only twelve tones per octave we must make certain choices. In practice, the most common arrangement from the time of the discovery of meantone tuning provided these twelve tones, with three sharps and two flats:

C-C-D-E-E-F-F-G-G-A-B-B.

Thus E but not D, G but not A, etc. With these choices, the perfect fifth E - B is the usual meantone perfect fifth, but the perfect fifth A - E does not exist. The closest available interval is G - E, which is one diesis too large, in sum: 696.578 + 41.059 = 737.637 cents. This is far too large with respect to the just perfect fifth (701.955 cents) to be used as one. Thus, with these basic pitches, there is no perfect fifth A - E in meantone tuning unless the number of the tones per octave increases, to provide both G and A. But if we include A, we should consider including D, and possibly G; and since we have G, why not D and A?
It will be seen at once that extending the series of perfect fifths in either direction, while desirable in order to provide acceptable perfect fifths in all the keys, could go on without limit, since, as mentioned, we never arrive back where we started: D E, C B, F G, etc. In other words, there are no enharmonic equivalents at all. Clearly this leads in theory to a temperament of a huge or even unlimited number of pitches. While practicable with computers, and to some extent with performing media having unrestricted gradations along the pitch spectrum (such as voice, orchestral strings, trombone), this is impracticable whenever other forces are used which have fixed frets, holes, valves, etc. or fixed lengths of string, pipe, tubing, etc.
But a curious thing happens if we extend meantone tuning as far as a series of 31 perfect fifths. Regardless of their note names, it is clear that 31 meantone fifths are 31 × 696.578 = 21593.931 cents. This is only 6.069 cents less than eighteen octaves. In other words, adding 31 meantone perfect fifths brings us very close to the note name we started with. If we temper each of those 31 perfect fifths by adding 6.069/31 = 0.196 cents back onto each, then we will have a closed cycle of 31 identical perfect fifths, each of which is only 0.196 cents sharper than those of meantone tuning.
Although not in these terms, this fact was recognized as early as the mid-sixteenth century. The first person known to have built a 31-tone instrument was Nicola Vicentino (1511-1572). Even though his instrument, the archicembalo, may not have been tuned to 31-tone equal temperament, it must have been very close to that. Other 31-tone instruments have been constructed in the later centuries. In 1950 the Dutch physicist Adriaan Fokker (1887-1972) had a 31-tone equal-temperament pipe-organ built, which is still in use in Teyler's Museum in Haarlem (Netherlands). In 1970 the archiphone (named after Vicentino's instrument) was built, also in Holland, by Herman van der Horst. It was the first electronic instrument in 31-tone equal temperament. A few years later, Motorola Inc. of Chicago, under the direction of Richard Harasek, built another electronic instrument, the Scalatron. Since it was retunable, it could encompass 31-tone equal temperament as well as many other tunings. It is a late model of the Scalatron which was used in the composition of the six songs.
To return to 31-tone equal temperament: it is obviously an excellent expansion of meantone tuning. In 31-tone equal temperament there are very close approximations to the intervals of meantone tuning, plus a closed circle of perfect fifths. Furthermore, the `equal' part of equal temperament means, of course, that any interval has the same size wherever it is found in an instrument's range. A full modulatory capability is provided, not only to the twelve theoretical keys of meantone tuning, but to nineteen others as well.

3  31-TONE EQUAL TEMPERAMENT: THE NOMENCLATURE
In 31-tone equal temperament, the smallest interval is called a diesis, as it is close to the diesis of meantone tuning in size and also is the interval between any tone and the tone twelve perfect fifths away, less seven octaves. The diesis of 31-tone equal temperament is therefore 1200/31 = 38.710 cents, about 3/8 of a 12-tone equal-temperament semitone. The major third is still four perfect fifths; but as each perfect fifth is 696.774 cents, the major third is 387.097 cents. Disregarding round-off inaccuracies, this is seen to be ten 31-tone equal-temperament dieses. The whole tone, which is half of the major third, is thus equal to 10/2 = 5 dieses. (At this point, we may recall earlier remarks about the whole tone of 31-tone equal temperament being divisible into five equal parts.) Since D is lower than E by a diesis, and appending a sharp to any tone must raise it as much as appending a flat lowers it, we arrive at this configuration, which partly fills in the note names within the major third D - F:

Since D up to D must be two dieses, the same as E to E and F to F (and E to E the same as F to F), we may fill in more of the major third as follows:

Since sharps and flats alter a given pitch by two dieses, double sharps and double flats will do so by four dieses:

The obvious awkwardness of this arrangement is lessened if we invent a kind of D which is half-way between D and D and which will thus be the enharmonic equivalent of E. This is D, i.e. D semi-sharp. Similarly, D will be equivalent to E, i.e. E semi-flat. The same relationships holding true for the tones between E and F, we may now rewrite the major third D to F as follows:

Occasionally we will also need something between a sharp and a double sharp: this will be a sharp-and-a-half, or sesqui-sharp, with the symbol . The sesqui-flat has the symbol . With all these signs employed, our major third would look like this, with enharmonic equivalents appearing vertically, as above:

It may look unfathomable, but a little practice makes it all very clear. In any case, enharmonic equivalents are needed scarcely more often than they are in 12-tone equal temperament. The triple equivalent of E = D = F would have to be dealt with about as often as 12-tone equal temperament's E = D = F: almost never. The same is true for the other triple equivalent shown above.
Despite any initial confusion caused by the symbols and their relationships, it should be obvious that the enharmonic equivalents of 12-tone and 31-tone equal temperament are mutually exclusive. This has very important effects on the different tonal structures alluded to earlier.
The reader should now be able to construct the entire 31-tone scale: six whole tones of five dieses each, plus one diesis to return to the starting pitch (e.g. six whole tones D - E - F - G - A - B - C, plus one more diesis C - D). All the notes may be seen set out in Figure 1. The C major scale is the best place to start. Its ascending whole tones always move one note slightly upwards to the right. Its ascending `semitones' (actually 3/5 tones) always move one note downwards to the right. Once this much is clear, the reader may ascertain that any of the 31 major-mode scales of this system are read the same way.
A small problem arises because Figure 1 does not extend very far in the left-right dimension. Therefore, finding all the notes of most other ascending scales requires jumping eventually from the right side of the figure to the next note on the left in order to continue the scale. The note on the left, of course, is a perfect octave lower than the missing note on the right. For example, the D major scale is D - E - F - G - A - B - C - D. If this is read starting with the D near the top of Figure 1, then the major second higher than B, namely C, must be found an octave lower on the left, at the beginning of the second row of notes in the figure, if one considers the rows to run in whole tones, i.e. slightly upwards to the right. If the same scale is read starting with the D near the bottom of Figure 1, the minor second higher than C, namely D, must be found an octave lower on the left, where the scale started (at the beginning of the eighth row of notes). With practice, one can jump from the missing note on either side to the required one on the opposite side quite easily, because two notes a perfect octave apart in Figure 1 or its horizontal extension are always situated the same distance apart and on the same horizontal axis.
In fact, we may generalize to state that in Figure 1 any given interval has the same configuration, regardless of the actual notes. Because of this consistency and the vertical extent of the figure, it contains a number of duplicates: notes which are a perfect unison `apart', i.e. are identical. It is occasionally convenient or necessary to jump higher or lower in the figure to continue a pattern. The duplicates are always situated two notes directly above plus one sharply upwards to the right, or, in the opposite direction, one note sharply downwards to the left plus two directly beneath.
We may now proceed to name the intervals in 31-tone equal temperament. For the sake of the illustration, they will all be considered to relate to the note D (see Table II). All intervals are assumed to be ascending, although it would be easy enough to read Table II with all descending, as we will do soon. Table II shows the kind of interval from D to any note contained in the three-, four-, or seven-tone group covered by the interval name. For example, going from D to any tone in the major group gives a major interval, and from D to any tone in the semi-augmented group a semi-augmented interval. Thus, from D up to C is a major seventh, and from D up to C is a semi-augmented seventh. The fact that C lies lower than D in the table is immaterial.
The left column of note and interval names is Pythagorean in origin. The sole generating interval is the perfect fifth. The left column is also the same for Pythagorean tuning and 12-tone equal temperament in both notes and intervals, although the enharmonic equivalents for 12-tone equal temperament (e.g. D = E) are not shown. The right column gives the new notes and intervals required in 31-tone equal temperament. It too uses only the perfect fifth as generating interval but also shows enharmonic equivalents of 31-tone equal temperament. By reading across we see, for example, that minor sixth = sesqui-augmented fifth (B = A, augmented sixth = semi-diminished seventh (B = C). Several of the note and interval names given here are very rare, but the table is extended as shown for the sake of clarity. (Even so, the flats in the right column could extend lower than, and the sharps higher than the left-column D, at which the 2 flats and 2 sharps are shown converting to each other.) In any case, by observing the enharmonic equivalents given for 31-tone equal temperament, it should be clear that the notes in both columns may repeat after every 31 note names.

Figure 1.
Figure 1. Schematic representation of the generalized keyboard as used for the Scalatron, to illustrate the structure of 31-tone equal temperament.

Table II. Note names and interval names in 31-tone equal temperament. Notes and intervals on the same line are enharmonic equivalents. The interval names refer to the intervals formed ascending from D to each note.

Interval name Note names Interval name

Doubly augmented seventh C
third F
sixth B - C seventh Neutral
second E - F third
fifth A - B sixth
unison D - E second
fourth G - A fifth Semi-diminished
Augmented seventh C
- D unison
third F - G fourth
sixth B - C seventh
second E - F third
fifth A - B sixth
unison D - E second
fourth G - A fifth Sesqui-diminished
Major seventh C - D unison
third F - G fourth
sixth B - C seventh
second E - F third
Perfect fifth A - B sixth
unison D - E second
- C seventh Sesqui-augmented
fourth G - F third
Minor seventh C - B sixth
third F - E second
sixth B - A fifth
second E - D unison
Diminished fifth A
- G fourth
unison D - C seventh Semi-augmented
fourth G - F third
seventh C - B sixth
third F - E second
sixth B - A fifth
second E - D unison
Doubly diminished fifth A
- G fourth
unison D - C seventh Neutral
fourth G - F third
seventh C - B sixth
third F - E second
sixth B
second E

E.g. F (on the right, near the top and bottom) is the same as both E (left) and G (left).
It may be concluded from Table II that all the familiar intervals of 12-tone equal temperament are present in 31-tone equal temperament and invert harmonically the same way: perfect fifth inverts to perfect fourth, major sixth inverts to minor third, augmented sixth inverts to diminished third, etc. To see this, we need only read the table as containing descending intervals instead of ascending. The interval names would then change positions symmetrically about the horizontal axis running through the arbitrarily chosen base note D. Thus, D - F may be read as a descending minor sixth, because, `minor sixth' is as far below D as `major third' is above D.
In size, the semi-augmented intervals are between perfect or major intervals and augmented intervals:

D - A (ascending): perfect fifth
D - A: semi-augmented fifth
D - A: augmented fifth,

and:

D - E (ascending): major second
D - E: semi-augmented second
D - E: augmented second.

Similarly, in size the semi-diminished intervals are between perfect or minor intervals and diminished intervals. The sesqui-augmented and sesqui-diminished intervals are one diesis larger than augmented and diminished intervals respectively: D up to A is a sesqui-augmented fifth, D down to G is a sesqui-augmented fifth, D down to E is a sesqui-diminished seventh. From this it may be seen that the semi-augmented and semi-diminished intervals invert harmonically to each other: D up to E is a semi-augmented second, but D down to E is a semi-diminished seventh. Similarly, the sesqui-augmented and sesqui-diminished intervals invert to each other. Finally, in size the neutral intervals are between major and minor, and they invert to other neutral intervals: D up to F is a neutral third, and D down to F is a neutral sixth.

Table III. Abbreviations for interval types in 31-tone equal temperament.

Interval typeAbbreviation Interval typeAbbreviation

Doubly augmentedDA
AugmentedASesqui-augmentedSA
MajorMSemi-augmentedsA
PerfectPNeutraln
MinormSemi-diminishedsd
DiminisheddSesqui-diminishedSd
Doubly diminishedDd

Table III provides a convenient set of abbreviations for all the common interval types in 31-tone equal temperament.
It is now necessary to return to the three interrelated reasons for tuning studies first mentioned at the beginning of Section 2. "Harmonies that are more in tune, or less in tune, if desired" have been dealt with; but more may now be added about this in connection with the neutral intervals. In a tonal context full of major and minor intervals, the neutral intervals sound like out-of-focus distortions - a useful quality at times. Yet the neutral third is only 0.979 cents sharper than a pure 9:11 interval, the one formed by the ninth and eleventh harmonics above a fundamental. Needless to say, at 347.408 cents, the 9:11 interval is not found in 12-tone equal temperament, whose minor third is 300 cents and major third 400 cents.
The interdependence of the three reasons may now be illustrated with the neutral intervals. As just mentioned, the neutral third may sound less in tune than intervals of a third which come close to ratios having prime numbers no higher than 5, but more in tune if one is trying to approximate thirds whose ratios include the prime number 11. Because of this, the neutral third, merely as a sound out of context, would have to be classified as `foreign'. It is not even close to anything in 12-tone equal temperament or to the commonly used intervals in Pythagorean or meantone tuning. Moreover, because its size is exactly half way between that of the minor and the major third, the neutral third has the unique function of dividing the perfect fifth exactly in half: C up to E and E up to G are both neutral thirds. One could therefore write a real sequence moving up by neutral thirds: one repetition of one such move would arrive at the dominant. Obviously such a procedure is impossible in 12-tone equal temperament, in which the perfect fifth, being seven semitones, is not divisible exactly in half.
This concludes the outline of some basic principles involved in 31-tone equal temperament and its notation. Much has been left out in this presentation, because further understanding of harmonic relationships which are broader or deeper or simply different than what has been presented here depends on some musical experience with the system. Nonetheless, additional points will emerge in the discussion of the songs which follows.

4  SINGING IN 31-TONE EQUAL TEMPERAMENT
The six Songs of fruits and vegetables were written partly to illustrate certain aspects of 31-tone equal temperament. Of course, 31-tone equal temperament is but one of many equal temperaments, and there are also many unequal temperaments and just tunings. Furthermore, 31-tone equal temperament alone may receive different treatments in composition, some of which are more tonal than others, some of which may use an equality of the third, fifth and seventh harmonics and their close approximations in 31-tone equal temperament to derive many new types of scales. Such an approach was described by Adriaan Fokker, following hints by the 18th-century mathematician Leonhard Euler (1707-1783).
Nonetheless, I chose to write these songs in ways that would illustrate some of the properties of 31-tone equal temperament which relate to traditional tonality, because in this regard 31-tone equal temperament is both similar to 12-tone equal temperament and rich in remarkable but easily grasped differences.
Choirs wishing to sing the songs without access to a 31-tone instrument such as the archiphone or Scalatron will admittedly have greater difficulty. But as with music in any tuning system, the high degree of pitch accuracy required depends somewhat on the harmonic context as well as tempo, dynamics, and other musical factors. Cadences and other relatively stable, structurally important points require a greater degree of tuning precision than, say, fast passages full of chromatic decorations and few or no consonances. In any case, the following points are useful to recall in singing in 31-tone equal temperament:
  (1) For practical purposes, the major third, minor sixth, and semi-diminished seventh are pure (`just'). These are in fact easier to sing than the 12-tone equal-temperament equivalents.
  (2) The perfect fifth and minor third are nearly just. In some situations they will sound just; in others they may be slightly but not obtrusively flat. Their harmonic inversions, the nearly just perfect fourth and major sixth, may sound slightly sharp.
  (3) The chromatic semitone is about 3/4 the size that it is in 12-tone equal temperament; the diatonic semitone is about 7/6 the size that it is in 12-tone equal temperament.
(If perfect fifths, major thirds, minor thirds, and their harmonic inversions are sung pure, the resulting chromatic semitone is only about 7 cents smaller than the 31-tone equal-temperament semitone, and the resulting diatonic semitone is only about 4.5 cents larger than the 31-tone equal-temperament diatonic semitone.)
Therefore, the temptation must be resisted to sing the leading tone sharp and thereby make the diatonic semitone small (as it is in Pythagorean tuning). As shown earlier, the chromatic scale familiar from 12-tone equal temperament is `unequal' in 31-tone equal temperament, because the combination of chromatic semitones and diatonic semitones produces a combination of intervals of 2 and 3 dieses respectively.
  (4) The `tritone' (augmented fourth) in 31-tone equal temperament is only 1.867 cents flatter than the interval 5:7. It thus sounds more stable than the tritone in 12-tone equal temperament, which is 17.488 cents sharper than 5:7. Correspondingly, the diminished fifth, being the harmonic inversion of the augmented fourth, is 1.867 cents sharper in 31-tone equal temperament than 7:10, while in 12-tone equal temperament, the diminished fifth, which is the same as its augmented fourth, is 17.488 cents flatter. In practice, the 31-tone equal-temperament augmented fourth and diminished fifth sound like the just 5:7 and 7:10 respectively.
  (5) In several situations, 31-tone equal temperament has two or three tones a diesis or two apart which sound like close inflections of one scale degree rather than different scale degrees altogether. The easiest case to illustrate is the `dominant-seventh' chord, which, if built on A, may be either A - C - E - G (spelled in ascending thirds in root position) or A - C - E - G. In the former E - G sounds like a nearly just 5:6, and C - G is a diminished fifth. In the latter, A - G sounds like a just 4:7 and C - G is a sesqui-diminished fifth, the enharmonic equivalent of an augmented fourth (which here would be C - F). Although the interval E - G may sound like a flat version of E - G, it is very close to the interval 6:7. The root-position chord A - C - G (with fifth omitted) sounds especially stable, as the slightly flat perfect fifth is not heard.
If choirs need to check their pitches, a tuner accurate to within 10 cents (about 1.5 Hz around middle C) or less is recommended. Relating pitches or intervals to those of a piano is not advised: 31-tone equal temperament is not an offshoot of 12-tone equal temperament, even though in a sense 12-tone equal temperament is contained within 31-tone equal temperament. Retuning a piano to produce twelve of the tones of 31-tone equal temperament is also not recommended.
Despite all these explanations and warnings, choirs may be assured that singing in 31-tone equal temperament is basically not difficult. It may seem so at first, but it is also rewarding in ways which cannot be experienced by singing in 12-tone equal temperament.

Paul Rapoport, 1987