Background 2: What is Just Intonation?

We saw and heard that ordinary 12 tone equal temperament generates harsh beating. If we want to avoid that, we must find a system of musical intervals, where each interval:

• is positioned right at the tip of a V-shape or very close to it (so it generates no beating or very slow beating);
• or is positioned far away from a V-shape (so its beating is too fast to be perceived as such);
• or is dispensable in chords.

One such system, the one investigated here, is 5 limit just intonation. It is best explained by first putting the tones in a matrix as depicted below, borrowed from [HE 1885, p. 463]. But beware: from this point onward you really need some familiarity with note names and musical intervals.

In fact this matrix can be extended with more and more note names to all sides without bounds. Each note name stands for a whole silo of tones/pitches. For instance the middle A stands for pitch 440 Hz, but also for lower octave A's 220 Hz, 110 Hz, 55 Hz etc. and for higher octave A's 880 Hz, 1760 Hz, 3520 Hz etc.

In this matrix, from left to right we have pure fifth intervals, such as C♯-G♯ (top left) and B♭-F (bottom right). We shall assume such a pure fifth to have a pitch ratio of 3/2. From bottom to top we have major third intervals, such as A-C♯ (top left) and F-A (bottom right), which assume a pitch ratio of 5/4.

All other intervals follow from 3/2 and 5/4. For instance, the interval of A-B (a whole tone) is two pure fifths to the right from central A (which reaches B). The pitch ratio of A-B is therefore 3/2 x 3/2 = 9/4. It is customary to bring such pitch ratios within the numerical range of [1...2]. Therefore the final B is lowered by one octave, which makes the pitch ratio 9/4 x 1/2 = 9/8 (which equals 1.125).

But there is another B near central A. From the central A we reach that other B by going to the left two pure fifths (reaching G) and then up one major third. The pitch ratio of that A-B interval (also called a whole tone) is therefore 2/3 x 2/3 x 5/4 = 20/36 = 5/9. Bringing it within the numerical range of [1...2] gives us 10/9 (which equals 1.111...).

The following matrix extends most cells with the pitch ratio of the interval from the central A to the tone in that particular cell.

Because the basic pitch ratios are 3/2 and 5/4, all pitch ratios are composed of the prime numbers 2, 3 and 5. For instance 100/81 (top left) = 5 x 5 x 2 x 2 / 3 x 3 x 3 x 3. In this form of just intonation, the highest prime number is 5, hence the tuning's name: 5 limit just intonation. Some cells have a pitch ratio between brackets, with a factor 7 in it, such as (7/4). Such septimal pitch ratios are very close to the main pitch ratios in those cells. They will be the basis for tempered just intonation.

This is how these intervals sound, ordered by pitch ratio:

Objections

If you are already familiar with just intonation, you may raise the following objections.

1. The matrix gives two pitch ratios for a pure fifth such as A-E: 3/2 and 40/27. The latter is not just intonation. Likewise for pure fourths. The ratio 40/27 shall be used only where it cannot be avoided. The only case where 40/27 is used in a harmonic interval is in rare chords such as C-G-D-A-E: consecutive pure fifths where the end points are supposed to form a major third of 5/4. The chord's dissonance could blur the beating of the single 40/27 pure fifth. (See Sleep.)
2. The matrix gives three pitch ratios for a major third such as A-C♯: 5/4, 81/64 and 100/81. Only the first one is just intonation. Likewise for minor sixths. The ratios 81/64 and 100/81 shall be used only in melodic intervals, and only where they cannot be avoided.
3. The matrix gives two pitch ratios for a minor third such as A-C: 6/5 and 32/27. The latter is not just intonation. Likewise for major sixths. The ratio 6/5 shall be used whereever possible. The ratio 32/27 shall be used only in melodic intervals and in (harmonic) chords which are dissonant anyway.
4. The matrix gives two pitch ratios for an augmented fourth (also known as a tritone) such as A-D♯: 45/32 and 25/18. The latter is not mentioned by Renaissance scholars [Du 2006]. Likewise for diminished fifths. The preference is understandable, because tritones with pitch ratio 45/32 are on the verge of consonance, as that ratio is so close to 7/5. But the 25/18 pitch ratio is in the ascending minor scale [HE 1885, p. 274] and in one of the chords of that scale [HE 1885, p. 338]. Both pitch ratios (45/32 and 25/18) shall be used.
5. The interval A-G is a minor seventh, which is often supposed to have pitch ratio 7/4. In the matrix it is 9/5 or 16/9. One could use A-F♯♯ instead of A-G, resulting in a pitch ratio close to 7/4 [HE 1885, p. 461]. Nonetheless, 9/5 and 16/9 have their own applications in just intonation music.

Tempered Just Intonation

In the matrix above, we see the interval A-F♯♯ with a pitch ratio 225/128 (=1.7578125). That is close to 7/4 (=1.75). The proximity to the tip of the V-shape of 7/4 predicts a moderately slow beating. That is fine if it happens all the time, but on unexpected rare occasions in otherwise beatless music, even slow beats may be distracting.

The same is true for the other grey background cells, which carry alternative septimal pitch ratios (i.e. with a factor 7 in it).

Now suppose we fiddle just a tiny bit with the basic pitch ratios 3/2 and 5/4 that make up the matrix. We could then slow down the beating (of the 225/128 ratio) and at the same time introduce very slow beating in the otherwise beatless intervals, to the point that all of these intervals beat very slowly. Then there is no occasional beating. For the pure fifth 1.498164 (2.12 cents too narrow) is used instead of 3/2 = 1.5. For the major third 1.249026 (1.35 cents too narrow) is used instead of 5/4 = 1.25. All other pitch ratios follow from these in the manner described earlier.

Finlandia Hymn

Here are the first few measures of the theme of the 'Finlandia Hymn' by Jean Sibelius:

The chord at A is the first dissonant in a majority of consonant triads. In just intonation that is a marked opposition:

(just intonation B♭)

If we replace the B♭ by an A♯, the chord at point A becomes a chord of the Italian Sixth [HE 1885, p. 461] with the interval E-A♯ (at point A) having a pitch ratio close to 7/5 that generates a slow beat:

(just intonation A♯)

But a slow beat in a beatless context stands out. That is cured in the tempered just intonation version:

(tempered just intonation A♯)

Admittedly, several sound qualities had to be experimented with to find one in which the fragments above actually and plainly expose the difference between tempered and ordinary just intonation.

Post Renaissance Chords

Tempered just intonation slightly improves some chords that were probably unknown in Renaissance days. Say we have the chord C-E-G♯-C. Now C-E and E-G♯ have pitch ratios 5/4 because they are major thirds. But G♯-C is a diminished fourth with, according to the matrix, pitch ratio 32/25. If we temper the basic 3/2 and 5/4 pitch ratios slightly, we can make G♯-C approach 9/7. Maybe you can hear the difference:

(12 tone equal temperament)

(just intonation)

(tempered just intonation)

Note that this chord would not benefit from exact (untempered) pitch ratios 5/4 and 9/7, because the two major thirds and the diminished fourth would then combine to a pitch ratio of 5/4 x 5/4 x 9/7 = 225/112 = 2.008928571 while that should be an exact octave: 2/1.

Analogously for the chord C-E♭-G♭-A-C, where C-E♭, E♭-G♭ and A-C are minor thirds, but G♭-A is an augmented second with pitch ratio 256/225 which is close to 8/7:

(12 tone equal temperament)

(just intonation)

(tempered just intonation)

Reger and Bach

We are now about to complete the background material. Please compare and judge for yourself whether tempered just intonation can match the sound quality of untempered just intonation:

The Reger fragment we heard earlier (modulation from C major to D♯ major), first in just intonation, then in tempered just intonation:

The Bach fragment we heard earlier (closing measures of Trio Sonata BWV526, second movement), first in just intonation, then in tempered just intonation:

Next: Automatically assigning pitches to notes

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