Raymond Zhao

Approximating Just Intonation with Equal Temperament

Written March 9, 2019.

Abstract

Just intonation (JI) and equal temperament (ET) are two different tuning systems. JI tuning systems use the overtone series to generate intervals to use in musical composition, whereas ET tunings start with a single interval (we only consider the octave in this article) and then divide it into n equal pieces, which we then write as n-ET. For example, 12-ET is the ubiquitous tuning system used in western music, with twelve semitones/steps per octave.

This article first extracts the more consonant 5-limit and 7-limit JI intervals, which can be used for tertiary and septimal harmonies, respectively. It then finds which n-ETs give the best approximation for those sets of intervals, and their approximation errors.

In the appendix, we repeat this process, but with 11/8 and 11/9 added, for anyone who wants to tack on some 11-limit xenharmonic intervals in their compositions.

JI Limits

Considering that JI intervals can all be expressed as reduced fractions, the concept of a limit \(p\), where \(p\) is prime, falls naturally into place. We say an interval is in \(p\)-limit JI if it can be expressed as the fraction \(n/d\) where the highest prime in the prime factorizations of both \(n\) and \(d\) is at most \(p\). Thus, 3-limit JI contains all intervals of the form \(2^a3^b\) for all integers \(a, b\).

In a sense, the primes are independent – prime factorizations are unique modulo the order in which the factors are listed, so every prime adds a new “dimension” in a sense. Thus, 3-limit corresponds to a two-dimensional space with an octave and fifth forming its basis, 5-limit corresponds to a three-dimensional space with an octave, fifth, and major third forming its basis, and so on.

Relating limits back to Western music

It turns out that Pythagorean tuning was formed by 3-limit JI – all the intervals are just produced by stacking 3/2 perfect fifths on top of each other until once again nearly reaching the original note and being off by a Pythagorean comma (the difference between twelve 3/2 fifths and seven 2/1 octaves).

Eventually, meantone temperament, well temperament, and then equal temperament followed along, but there were still only 12 notes, with the consonances approximating the following JI intervals:
unison: 1/1, major second: 9/8 or 10/9, minor third: 6/5, major third: 5/4, perfect fifth: 3/2. (Note: inverting the fraction and then multiplying the numerator by 2’s until it exceeds the denominator gives the inverted interval, so we don’t need to list all of the intervals.) Meanwhile, the tritone and minor second are both more dissonant, and cannot be approximated by fractions with such small numerators and denominators. For the purposes of just getting JI consonances in tune though, we can conclude that 12-ET is a good approximation of 5-limit JI, and based on the intervals given above, is fairly nice for writing tertian harmonies.

But, why 5-limit rather than 7-limit? Because 12-ET does not give a good approximation to 7/4 (harmonic seventh), only offering much better approximations to the 5-limit 16/9 (minor seventh) and 15/8 (major seventh) instead, 12-ET is not a good approximation of 7-limit JI. Yet, 7-limit JI is still worth considering, because barbershop quartets sing the harmonic seventh all the time, and blues musicians often flatten sevenths to approximate septimal harmonies anyway!

Finding better equal-tempered 5-limit and 7-limit JI approximations

From the previous section, we concluded 12-ET is a good approximator to 5-limit JI, or at least, the more consonant intervals of 5-limit JI, because every JI system contains a countably infinite number of intervals.

However, 12-ET’s major third is around 14 cents sharper than the 5/4 major third of 5-limit JI, leading to the former’s major triad sounding slightly more dissonant than that of the latter. In fact, when choirs “lock in” on a major triad in a 12-ET piece, they’re probably singing it with a slightly flattened major third to push the triad into a JI tuning. As a violinist, when playing major third scales, I also end up adjusting the thirds to be in JI, otherwise they just sound more dissonant.

Considering that most people can tell the difference between notes spaced over 5 cents apart (which is 5/1200th of an Octave), I’ll consider an equal-temperated approximation of the 5-limit amazing if it approximates 3/2, 5/4, 6/5, and 9/8 all within 5 cents, each. Likewise, I’ll consider an equal-tempered approximation of the 7-limit amazing if it approximates 3/2, 5/4, 6/5, 9/8, 7/4, 9/7 (septimal major third), 7/6 (septimal minor third), 8/7 (septimal major second), and 7/5 (septimal tritone) all within 5 cents, each. Note that my definition of amazing here is completely subjective – feel free to call it precise or even just passable.

Using a Python script (source here), we get the following results:

12 EDO (3/2: note 7 with error -1.96 cents, 5/4: note 4 with error +13.69 cents, 6/5: note 3 with error -15.64 cents, 9/8: note 2 with error -3.91 cents)
24 EDO (3/2: note 14 with error -1.96 cents, 5/4: note 8 with error +13.69 cents, 6/5: note 6 with error -15.64 cents, 9/8: note 4 with error -3.91 cents)
36 EDO (3/2: note 21 with error -1.96 cents, 5/4: note 12 with error +13.69 cents, 6/5: note 9 with error -15.64 cents, 9/8: note 6 with error -3.91 cents)
41 EDO (3/2: note 24 with error +0.48 cents, 5/4: note 13 with error -5.83 cents, 6/5: note 11 with error +6.31 cents, 9/8: note 7 with error +0.97 cents)
53 EDO (3/2: note 31 with error -0.07 cents, 5/4: note 17 with error -1.41 cents, 6/5: note 14 with error +1.34 cents, 9/8: note 9 with error -0.14 cents)
106 EDO (3/2: note 62 with error -0.07 cents, 5/4: note 34 with error -1.41 cents, 6/5: note 28 with error +1.34 cents, 9/8: note 18 with error -0.14 cents)
159 EDO (3/2: note 93 with error -0.07 cents, 5/4: note 51 with error -1.41 cents, 6/5: note 42 with error +1.34 cents, 9/8: note 27 with error -0.14 cents)

12 EDO (3/2: note 7 with error -1.96 cents, 5/4: note 4 with error +13.69 cents, 6/5: note 3 with error -15.64 cents, 9/8: note 2 with error -3.91 cents, 7/4: note 10 with error +31.17 cents, 9/7: note 4 with error -35.08 cents, 7/6: note 3 with error +33.13 cents, 7/5: note 6 with error +17.49 cents)
24 EDO (3/2: note 14 with error -1.96 cents, 5/4: note 8 with error +13.69 cents, 6/5: note 6 with error -15.64 cents, 9/8: note 4 with error -3.91 cents, 7/4: note 19 with error -18.83 cents, 9/7: note 9 with error +14.92 cents, 7/6: note 5 with error -16.87 cents, 7/5: note 12 with error +17.49 cents)
36 EDO (3/2: note 21 with error -1.96 cents, 5/4: note 12 with error +13.69 cents, 6/5: note 9 with error -15.64 cents, 9/8: note 6 with error -3.91 cents, 7/4: note 29 with error -2.16 cents, 9/7: note 13 with error -1.75 cents, 7/6: note 8 with error -0.20 cents, 7/5: note 17 with error -15.85 cents)
72 EDO (3/2: note 42 with error -1.96 cents, 5/4: note 23 with error -2.98 cents, 6/5: note 19 with error +1.03 cents, 9/8: note 12 with error -3.91 cents, 7/4: note 58 with error -2.16 cents, 9/7: note 26 with error -1.75 cents, 7/6: note 16 with error -0.20 cents, 7/5: note 35 with error +0.82 cents)
99 EDO (3/2: note 58 with error +1.08 cents, 5/4: note 32 with error +1.57 cents, 6/5: note 26 with error -0.49 cents, 9/8: note 17 with error +2.15 cents, 7/4: note 80 with error +0.87 cents, 9/7: note 36 with error +1.28 cents, 7/6: note 22 with error -0.20 cents, 7/5: note 48 with error -0.69 cents)
171 EDO (3/2: note 100 with error -0.20 cents, 5/4: note 55 with error -0.35 cents, 6/5: note 45 with error +0.15 cents, 9/8: note 29 with error -0.40 cents, 7/4: note 138 with error -0.40 cents, 9/7: note 62 with error +0.00 cents, 7/6: note 38 with error -0.20 cents, 7/5: note 83 with error -0.06 cents)

Note that it uses the term EDO (equal divison of the octave) instead of ET, because firstly, the script only considers equal divisions of the octave, and secondly, EDOs are slightly more general than ETs. For the purposes of this article, the difference is negligible.

Analysis and Conclusions

53-ET is the lowest amazing ET for 5-limit JI, and 72-ET is the lowest amazing ET for 7-limit JI, and is also amazing for 5-limit JI. However, these temperaments approximate the just intonation systems so well that they introduce commas (small intervals such as 256/243) back into the music. Thus, compositions in 53-ET and 72-ET must watch out for accidental comma drift, where the entire piece unintentionally starts to slide down by a comma every repetition of a chord progression.

As a compromise, we can instead pick lower temperaments such as 36-ET or 41-ET, which I unfortunately don’t know nearly as much about. However, I have noticed others choosing the much more manageable 19-ET or 31-ET instead, which are both meantone temperaments. In this case, there are no comma drifts, but there is still an uneven alternating step size – anyone who wants to play a 12-note chromatic scale on these temperaments will quickly realize that half steps come in two sizes. For example, some half steps in 19-ET could take two keys, whereas others take just one.

Thus, for anyone willing to explore beyond just 12-ET, but also wanting to keep their consonances in tune, there are 19, 24, 31, 36, 41, 53, 72, 99, and 171-ETs to consider. More specifically, 41-ET and 53-ET are strictly better 5-limit approximations than 12-ET, and 36-ET, 72-ET, 99-ET, and 171-ET are strictly better 7-limit approximations than 12-ET.

Appendix: For 11-limit (more specifically: 11/8 and 11/9) fanatics

11-limit JI is the first JI system with xenharmonic intervals, or intervals completely foreign to Western music. Whereas many 11-limit intervals sound very close to the 7-limit or 5-limit seconds, thirds, fifths, and so on (at least to me), the 11/8 and 11/9 are distinct from all of them.

11/8 is a superfourth, residing between the 4/3 perfect fourth and the 7/5 tritone. I’ve heard it being called the Alphorn fa, so it must be special. More seriously though, it’s a fairly dissonant fourth, but when used as an eleventh (11/4), sounds brighter than 8/3, and not as dissonant. Meanwhile, 11/9 is a neutral third, residing between the 6/5 minor third and 5/4 major third. Thus, we add 11/8 and 11/9 into the Python script and see what shows up:

12 EDO (3/2: note 7 with error -1.96 cents, 5/4: note 4 with error +13.69 cents, 6/5: note 3 with error -15.64 cents, 9/8: note 2 with error -3.91 cents, 7/4: note 10 with error +31.17 cents, 9/7: note 4 with error -35.08 cents, 7/6: note 3 with error +33.13 cents, 7/5: note 6 with error +17.49 cents, 11/8: note 6 with error +48.68 cents, 11/9: note 3 with error -47.41 cents)
24 EDO (3/2: note 14 with error -1.96 cents, 5/4: note 8 with error +13.69 cents, 6/5: note 6 with error -15.64 cents, 9/8: note 4 with error -3.91 cents, 7/4: note 19 with error -18.83 cents, 9/7: note 9 with error +14.92 cents, 7/6: note 5 with error -16.87 cents, 7/5: note 12 with error +17.49 cents, 11/8: note 11 with error -1.32 cents, 11/9: note 7 with error +2.59 cents)
48 EDO (3/2: note 28 with error -1.96 cents, 5/4: note 15 with error -11.31 cents, 6/5: note 13 with error +9.36 cents, 9/8: note 8 with error -3.91 cents, 7/4: note 39 with error +6.17 cents, 9/7: note 17 with error -10.08 cents, 7/6: note 11 with error +8.13 cents, 7/5: note 23 with error -7.51 cents, 11/8: note 22 with error -1.32 cents, 11/9: note 14 with error +2.59 cents)
72 EDO (3/2: note 42 with error -1.96 cents, 5/4: note 23 with error -2.98 cents, 6/5: note 19 with error +1.03 cents, 9/8: note 12 with error -3.91 cents, 7/4: note 58 with error -2.16 cents, 9/7: note 26 with error -1.75 cents, 7/6: note 16 with error -0.20 cents, 7/5: note 35 with error +0.82 cents, 11/8: note 33 with error -1.32 cents, 11/9: note 21 with error +2.59 cents)
144 EDO (3/2: note 84 with error -1.96 cents, 5/4: note 46 with error -2.98 cents, 6/5: note 38 with error +1.03 cents, 9/8: note 24 with error -3.91 cents, 7/4: note 116 with error -2.16 cents, 9/7: note 52 with error -1.75 cents, 7/6: note 32 with error -0.20 cents, 7/5: note 70 with error +0.82 cents, 11/8: note 66 with error -1.32 cents, 11/9: note 42 with error +2.59 cents)
198 EDO (3/2: note 116 with error +1.08 cents, 5/4: note 64 with error +1.57 cents, 6/5: note 52 with error -0.49 cents, 9/8: note 34 with error +2.15 cents, 7/4: note 160 with error +0.87 cents, 9/7: note 72 with error +1.28 cents, 7/6: note 44 with error -0.20 cents, 7/5: note 96 with error -0.69 cents, 11/8: note 91 with error +0.20 cents, 11/9: note 57 with error -1.95 cents)

72-ET thus shows up as the first amazing approximation of these intervals, with 48-ET or even 24-ET as simpler (but not as amazing) alternatives.

However, whereas the pattern for 7-limit JI (strictly better) approximations was 72 to 99 to 171, here it is 72 to 144 to 198, which means that 99-ET and 171-ET are better for people working only with 7-limit intervals, but 144-ET and 198-ET are better for those also adding in 11/8 and 11/9 along with those seventh intervals. Due to how small the errors are by 72-ET though, I’d say 72-ET is amazing enough for both 7-limit and 11-limit (or more specifically 7-limit plus 11/8 and 11/9) JI.