COMPARISON OF REGULAR TEMPERAMENTS

to just intervals

HOW TUNING CAN BE STUBBORN

An underlying and recurring problem in music is establishing compatibility between tuning systems used in ensemble performances.

In choral music, the purest chords are provided by just intonation, providing the simplest mathematical relationship between notes. Pitches must be adjusted note by note, to correspond to the lead voice. Such constant adjustment is commonplace for a singer. It is also possible on some instruments, by exact placement of a finger along an unfretted viol string or movement of a trombone slide.

However, the majority of musical instruments are difficult to retune, including woodwinds with permanent holes, guitars with glued frets, organs with solid pipes, and pianos with 88 taut and interrelated strings. The value of these instruments depends on their ability to modulate between keys without retuning. Therefore, their tuning reflects a compromise between purity and versatility. In Western music, compromise usually results in 12-tone equal temperament: adding together pentatonic (black key) and heptatonic (white key) scales, so that semi-tones and chromatic intervals are made to coincide.

Although 12-tone music can provide a reasonably accurate expression of music from Bach to Bacharach, it cannot capture the subtlety of music designed to fit divergent theoretical contexts: African neutral intervals, enharmonic scales of ancient Greece, augmented intervals of Hungary, 17-note Arabic tuning, 22-note Indian tuning, quarter-tone compositions of Charles Ives, African-American blue notes, and Indonesian 5-tone slendro and 7-tone pelog scales (to name a few). Moreover, 12TET instruments cannot even produce a true 4:5:6:7 barbershop seventh chord, an American tradition.

Many non-Western systems are also mutually incompatible. Composers who want to combine scales from various countries risk writing music that cannot be played anywhere, by anyone.

HOW TO VARY THE DIET OF MUSICAL INSTRUMENTS

So, what can be done? These are several options:

Perform the music in tune, with a gamelan orchestra, sitar, or oud. People from different cultures will be able to play little of each other's music. Analysis will suffer because of the large number of variables imposed between the systems. Cross-cultural sharing will be limited.
Retune instruments of one culture to play music from another. This is expensive and usually results in the instrument becoming useless in its original tuning and difficult to play in the new tuning. Therefore, it's unlikely to ever be more than a novelty.
Play the music out-of-tune. Thus it sounds not only strange, but also unpleasant. Composers and fans are understandably offended to have their beloved music corrupted this way. Debasement of foreign aesthetic values will make understanding and appreciation of them difficult.
Develop electronic instruments that can retune themselves at the push of a button. For the short-term at least, this will rob us of the rich timbre of acoustic instruments, especially woodwinds. It's also certain to limit portability, unless a method is devised to plug in an amplifier at a campfire.
Put many notes on instruments, to anticipate every scale. Thus, instruments would become so huge and complicated as to be impractical to manufacture and play.
Put movable pitch controls on instruments, to precise control by the performer. This would also complicate design, manufacture, preparation, and performance of instruments. It would be cheaper and easier to use several instruments.
Tune instruments to approach various popular tuning systems equally well. There are many advocates of this practice. They differ on decisions of which notes to include. The fewer notes needed to do the job, the more efficient the tuning. (Commonly, this approach is known as microtonality, a somewhat misleading term. Tuning is unlikely to require a thousandfold division, let alone a million.)

HOW TUNING OF INSTRUMENTS APPROACHES JUST INTONATION

It is said that the basis of music is the human voice. While this statement ignores older phenomena such as bird calls, it is largely true. Children learn to sing before they learn to play pianos and violins. Around the world, most song seems to be based on just intonation. (There may be exceptions.) Therefore, the first test of versatility is whether a tuning system can reasonably substitute for just intonation, by imitating commonly-used just intervals.

The chart below provides a comparison of several regular tuning systems to just intervals. (It is a work in progress, so any corrections should be sent to aloe@rev.net.) See also a comparison of weighted harmonic errors.

"Meantone", or "regular", indicates that each system repeats a constant interval of a fourth or fifth to produce all notes of the scale. (Just intonation uses ratios of whole numbers.) Alternatively, each scale can be defined in terms of the whole-tone, which is the distance between the fourth and fifth.

The regular tuning systems shown on the chart are of several types:

Equal-Temperament (shown in white):

Equal-temperament scales use fractions (ratios) of base-2 logarithms to divide the scale in equal proportions. Any fraction (ratio) of a base-2 logarithm will close at a number of steps equal to the denominator of the fraction, creating an equal-temperament scale. These are some of the equal-temperament systems compared on the chart:

5-tone, Uganda
7-tone, Thailand
12-tone, Europe
53-tone, Ottoman

Incremental:

These tuning systems employ a repeated fixed interval to add additional notes to the scale. Normally, this results in intervals of two different sizes.

Rational meantone (shown in orange):

These tuning systems assume one interval, with all others having a fractional relationship to it. Two of the meantones on the chart attempt to produce just thirds (either minor or major).

Pythagorean (shown in yellow):: A special instance of regular tuning uses fractions based on powers of 3 to derive all notes of the scale. (Coincidentally, Pythagorean tuning qualifies as a 3-limit just intonation system.) If extended, Pythagorean tuning can produce a close approximation of a 53-note Ottoman equal temperament scale, within 3.62 cents (2^0.003012538) of closure.

Constant (shown in gold):

These scales have been devised using irrational numbers. Both the absolute numbers and their base-2 logarithms are irrational (unable to be expressed as a fraction of integers). Many such scales are theoretically possible. How the intervals are derived is less important than whether they have aesthetic value. Two constant scales, both based on common mathematical constants, appear on the chart:

Pi-based Lucy Tuning:: Developed by Charles E. H. Lucy from the work of John Harrison, this interesting tuning employs a whole tone ratio of 2^{Pi^-2}. (Pi is the ratio between the circumference and diameter of a circle.) Closure to equal temperament is closely approached by Lucy scales of 19, 69, and 88 tones.

Phi-based golden meantone:: Developed by Thorvald Kornerup and advocated by Jacques Dudon, this tuning is based on Phi, the golden mean, equal to 1/2+5^1/2/2 or 1.61803398875. (Phi is a unique constant. It is the ratio of two quantities, so that the smaller is to the greater as the greater is to the sum of the greater and the smaller. It is one larger than its own reciprocal. It is the mean of quotients of adjacent elements in a Fibonacci series. Phi is the ratio of width to height of many Greek temples, because ancient architects considered it pleasant to the eye. Whether it is as pleasing to the ear is left to listeners' taste.) The whole tone ratio is 2^0.160357456, or 1.117564002. Closure to equal temperament is roughly approached along a Fibonacci series of {5, 7, 12, 19, 31, 50, 81, . . .}. (See comparative data.)

Tunings are listed in order of size of wide interval (whole-tone or fifth), from narrowest to widest.

Measurements are given in cents (a standard having an unfortunate basis in 12-tone equal temperament). They are the difference between frequencies of notes in the given scale and overtones formed by fractions (ratios).

Equal temperaments are truncated when a finite number is reached.

Entries that would make the scale irregular are left blank.

Circle of Fifths

Steps 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Accidental Perfect Maj Maj Maj Maj Aug Aug Aug Aug Aug Aug Aug x Neutral Neutral
Note 5th 2nd 6th 3rd 7th 4th 1st 5th 2nd 6th 3rd 7th 4th 2nd 6th

Numerator 3 9 5 5 15 7 25 14 7 7 64 35 16 12 18
Denominator 2 8 3 4 8 5 24 9 6 4 49 18 11 11 11

Cents 702 204 884 386 1088 583 71 765 267 969 462 1151 649 151 853

Number of notes in scale to 5th Ratio of tone to diatonic semitone
7 Equal 7 4 1.00 -16.2¢ -32.5¢ -27.2¢
1/2 Comma Meantone 1.27 -10.8¢ -21.5¢ -10.8¢ -21.5¢ -32.3¢ -35.3¢ -32.3¢ -35.3¢ -46.1¢
26 Equal 26 15 1.33 -9.65¢ -19.3¢ -7.4¢ -17.1¢ -26.7¢ -28.7¢ -24.5¢ -26.5¢ -36.1¢
45 Equal 45 26 1.40 -8.6¢ -17.2¢ -4.4¢ -13.0¢ -21.6¢ -22.5¢ -17.3¢ -18.2¢ -26.9¢
19 Equal 19 11 1.50 -7.2¢ -14.4¢ -0.1¢ -7.4¢ -14.6¢ -14.1¢ -7.5¢ -7.0¢ -14.2¢
1/3 Comma Meantone 1.50 -7.2¢ -14.3¢ 0.0¢ -7.2¢ -14.3¢ -13.8¢ -7.2¢ -6.6¢ -13.8¢ -21.0¢ -19.7¢ -13.8¢ -16.5¢ -30.8¢
88 Equal 88 51 1.56 -6.5¢ -13.0¢ +2.0¢ -4.5¢ -11.0¢ -9.8¢ -2.5¢ -1.3¢ -7.8¢ -14.3¢ -12.3¢ -5.8¢ -7.8¢ -14.3¢ -20.8¢
Pi-based 1.56 -6.5¢ -12.9¢ +2.1¢ -4.3¢ -10.8¢ -9.6¢ -2.2¢ -1.0 -7.4¢ -13.9¢ -11.9¢ -13.7¢ -7.3¢ -28.1¢ -20.2¢
Meta Meantone 1.57 -6.3¢ -12.6¢ +2.5¢ -3.8¢ -10.1¢ -8.7¢ -1.3¢ +0.1¢ -6.2¢ -12.5¢ -10.4¢ -3.7¢ -5.5¢ -11.8¢ -18.1¢
69 Equal 69 40 1.57 -6.3¢ -12.6¢ +2.6¢ -3.7¢ -10.0¢ -8.6¢ -1.1¢ +0.3¢ -6.0¢ -12.3¢ -10.2¢ -3.4¢ -5.2¢ -11.5¢ -17.8¢
2/7-Comma Meantone 1.58 -6.1¢ -12.2¢ +3.1¢ -3.1¢ -9.2¢ -7.7¢ 0.0¢ +1.6¢ -4.6¢ -10.7¢ -8.4¢ -1.5¢ -3.1¢ -9.3¢ -15.4¢
5/18-Comma Meantone 1.60 -6.0¢ -11.9¢ +3.6¢ -2.4¢ -8.4¢ -6.6¢ +1.2¢ +2.9¢ -3.0¢ -9.0¢ -6.6¢ +0.5¢ -0.9¢ -6.9¢ -12.9¢
50 Equal 50 29 1.60 -6.0¢ -11.9¢ +3.6¢ -2.3¢ -8.3¢ -6.5¢ +1.3¢ +3.1¢ -2.9¢ -8.9¢ -6.3¢ +0.8¢ -0.7¢ -6.6¢ -12.6¢
Golden 1.62 -5.7¢ -11.5¢ +4.3¢ -1.5¢ -7.2¢ -5.2¢ +2.8¢ +4.8¢ -0.9¢ -6.7¢ -4.0¢ +3.3¢ +2.1¢ -3.6¢ -9.4¢
81 Equal 81 47 1.63 -5.7¢ -11.3¢ +4.5¢ -1.1¢ -6.8¢ -4.7¢ +3.4¢ +5.5¢ -0.2¢ -5.9¢ -3.1¢ +4.3¢ +3.2¢ -2.5¢ -8.1¢
1/4 Comma Meantone 1.65 -5.4¢ -10.8¢ +5.4¢ 0.0¢ -5.4¢ -3.0¢ +5.4¢ +7.7¢ +2.3¢ -3.0¢ 0.0¢ +7.7¢ +6.8¢ -1.5¢ -3.9¢
31 Equal 31 18 1.67 -5.2¢ -10.4¢ +6.0¢ +0.8¢ -4.4¢ -1.9¢ +6.7¢ +9.3¢ +4.1¢ -1.1¢ +2.17¢ +10.1¢ +9.4¢ +4.2¢ -0.0¢
43 Equal 43 25 1.75 -4.3¢ -8.6¢ +8.7¢ +4.4¢ +0.1¢ +3.5¢ +13.0¢ +16.5¢ +12.2¢ +7.9¢ +12.1¢ +20.9¢ +21.1¢ +16.8¢ +12.5¢
55 Equal 55 32 1.80 -3.8¢ -7.5¢ +10.2¢ +6.4¢ +2.6¢ +6.6¢ +16.6¢ +20.5¢ +16.8¢ +13.0¢ +17.7¢ +27.0¢ +27.7¢ +23.9¢ +20.1¢
67 Equal 67 39 1.83 -3.4¢ -6.9¢ +11.2¢ +7.7¢ +4.3¢ +8.5¢ +18.9¢ +23.1¢ +19.7¢ +16.2¢ +21.2¢ +30.9¢ +31.9¢ +28.5¢ +25.0¢
79 Equal 79 46 1.86 -3.2¢ -6.4¢ +11.8¢ +8.6¢ +5.4¢ +9.9¢ +20.5¢ +25.0¢ +21.7¢ +18.5¢ +23.7¢ +33.6¢ +34.9¢ +31.6¢ +28.4¢
91 Equal 91 53 1.88 -3.1¢ -6.1¢ +12.3¢ +9.3¢ +6.2¢ +10.9¢ +21.6¢ +26.3¢ +23.2¢ +20.2¢ +25.6¢ +35.6¢ +37.0¢ +34.0¢ +30.9¢
12 Equal 12 7 2.00 -2.0¢ -3.9¢ +15.6¢ +13.7¢ +11.7¢ +17.5¢
53 Equal 53 31 2.25 -0.1¢ -0.1¢ +21.3¢ +21.2¢ +21.2¢ +28.8¢
Pythagorean 2.26 0.0¢ 0.0¢ +21.5¢ +21.5¢ +21.5¢ +29.2¢
41 Equal 41 24 2.33 +0.5¢ +1.0¢ +23.0¢ +23.4¢ +23.9¢ +32.1¢
29 Equal 29 17 2.50 +1.5¢ +3.0¢ +26.0¢ +27.5¢ +29.0¢ +38.2¢
17 Equal 17 10 3.00 +3.9¢ +7.9¢ +33.3¢ +37.2¢ +41.1¢ +52.8¢
22 Equal 22 13 4.00 +7.1¢ +14.3¢ +42.9¢ +50.0¢ +57.2¢ +72.0¢
5 Equal 5 3 +18.0¢ +36.1¢

Accidental Perfect Maj Maj Maj Maj Aug Aug Aug Aug Aug Aug Aug x Neutral Neutral
Note 5th 2nd 6th 3rd 7th 4th 1st 5th 2nd 6th 3rd 7th 4th 2nd 6th

Steps 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Circle of Fourths

Steps 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Accidental Perfect min min min min dim dim dim dim dim dim dim bb Neutral Neutral
Note 4th 7th 3rd 6th 2nd 5th 1st 4th 7th 3rd 6th 2nd 5th 7th 3rd

Numerator 4 16 6 8 16 10 48 9 12 8 49 36 11 11 11
Denominator 3 9 5 5 15 7 25 7 7 7 32 35 8 6 9

Cents 498 996 316 814 112 617 1129 435 933 231 738 49 551 1049 347

Number of notes in scale to 4th Ratio of tone to diatonic semitone
7 Equal 7 3 1.00 +16.2¢ +32.5¢ +27.2¢
1/2 Comma Meantone 1.27 +10.8¢ +21.5¢ +10.8¢ +21.5¢ +32.3¢ +35.3¢ +32.3¢ +35.3¢ +46.1¢
26 Equal 26 11 1.30 +9.65¢ +19.3¢ +7.4¢ +17.1¢ +26.7¢ +28.7¢ +24.5¢ +26.5¢ +36.1¢
45 Equal 45 19 1.40 +8.6¢ +17.2¢ +4.4¢ +13.0¢ +21.6¢ +22.5¢ +17.3¢ +18.2¢ +26.9¢
19 Equal 19 8 1.50 +7.2¢ +14.4¢ +0.1¢ +7.4¢ +14.6¢ +14.1¢ +7.5¢ +7.0¢ +14.2¢
1/3 Comma Meantone 1.50 +7.2¢ +14.3¢ 0.0¢ +7.2¢ +14.3¢ +13.8¢ +7.2¢ +6.6¢ +13.8¢ +21.0¢ +19.7¢ +13.8¢ +16.5¢ +30.8¢
88 Equal 88 37 1.56 +6.5¢ +13.0¢ -2.0¢ +4.5¢ +11.0¢ +9.8¢ +2.5¢ +1.3¢ +7.8¢ +14.3¢ +12.3¢ +5.8¢ +7.8¢ +14.3¢ +20.8¢
Pi-based 1.56 +6.5¢ +12.9¢ -2.1¢ +4.3¢ +10.8¢ +9.6¢ +2.2¢ +1.0¢ +7.4¢ +13.9¢ +11.9¢ +5.3¢ +7.3¢ +13.7¢ +20.2¢
Meta Meantone 1.57 +6.3¢ +12.6¢ -2.5¢ +3.8¢ +10.1¢ +8.7¢ +1.3¢ -0.1¢ +6.2¢ +12.5¢ +10.4¢ +3.7¢ +5.5¢ +11.8¢ +18.1¢
69 Equal 69 29 1.57 +6.3¢ +12.6¢ -2.6¢ +3.7¢ +10.0¢ +8.6¢ +1.1¢ -0.3¢ +6.0¢ +12.3¢ +10.2¢ +3.4¢ +5.2¢ +11.5¢ +17.8¢
2/7-Comma Meantone 1.58 +6.1¢ +12.2¢ -3.1¢ +3.1¢ +9.2¢ +7.7¢ 0.0¢ -1.6¢ +4.6¢ +10.7¢ +8.4¢ +1.5¢ +3.1¢ +9.3¢ +15.4¢
5/18-Comma Meantone 1.60 +6.0¢ +11.9¢ -3.6¢ +2.4¢ +8.4¢ +6.6¢ -1.2¢ -2.9¢ +3.0¢ +9.0¢ +6.6¢ -0.5¢ +0.9¢ +6.9¢ +12.9¢
50 Equal 50 21 1.60 +6.0¢ +11.9¢ -3.6¢ +2.3¢ +8.3¢ +6.5¢ -1.3¢ -3.1¢ +2.9¢ +8.8¢ +6.3¢ -0.8¢ -0.7¢ +6.6¢ +12.6¢
Golden 1.62 +5.7¢ +11.5¢ -4.3¢ +1.5¢ +7.2¢ +5.2¢ -2.8¢ -4.8¢ +0.9¢ +6.7¢ -4.0¢ -3.3¢ -2.1¢ +3.6¢ +9.4¢
81 Equal 81 34 1.63 +5.7¢ +11.3¢ -4.5¢ +1.1¢ +6.8¢ +4.7¢ -3.4¢ -5.5¢ +0.2¢ +5.9¢ +3.1¢ -4.3¢ -3.2¢ +2.5¢ +8.1¢
1/4 Comma Meantone 1.65 +5.4¢ +10.8¢ -5.4¢ 0.0¢ +5.4¢ +3.0¢ -5.4¢ -7.7¢ -2.3¢ +3.0¢ 0.0¢ -7.7¢ -6.8¢ +1.5¢ +3.9¢
31 Equal 31 13 1.67 +5.2¢ +10.4¢ -6.0¢ -0.8¢ +4.4¢ +1.9¢ -6.7¢ -9.3¢ -4.1¢ +1.1¢ -2.17¢ -10.1¢ -9.4¢ -4.2¢ +1.0¢
43 Equal 43 18 1.75 +4.3¢ +8.6¢ -8.7¢ -4.4¢ -0.1¢ -3.5¢ -13.0¢ -16.5¢ -12.2¢ -7.9¢ -12.1¢ -20.9¢ -21.1¢ -16.8¢ -12.5¢
55 Equal 55 23 1.80 +3.8¢ +7.5¢ -10.2¢ -6.4¢ -2.6¢ -6.6¢ -16.6¢ -20.5¢ -16.8¢ -13.0¢ -17.7¢ -27.0¢ -27.7¢ -23.9¢ -20.1¢
67 Equal 67 28 1.83 +3.4¢ +6.9¢ -11.2¢ -7.7¢ -4.3¢ -8.5¢ -18.9¢ -23.1¢ -19.7¢ -16.2¢ -21.2¢ -30.9¢ -31.9¢ -28.5¢ -25.0¢
79 Equal 79 33 1.86 +3.2¢ +6.4¢ -11.8¢ -8.6¢ -5.4¢ -9.9¢ -20.5¢ -25.0¢ -21.7¢ -18.5¢ -23.7¢ -33.6¢ -34.9¢ -31.6¢ -28.4¢
91 Equal 91 38 1.88 +3.1¢ +6.1¢ -12.3¢ -9.3¢ -6.2¢ -10.9¢ -21.6¢ -26.3¢ -23.2¢ -20.2¢ -25.6¢ -35.6¢ -37.0¢ -34.0¢ -30.9¢
12 Equal 12 5 2.00 +2.0¢ +3.9¢ -15.6¢ -13.7¢ -11.7¢
53 Equal 53 22 2.25 +0.1¢ +0.1¢ -21.3¢ -21.2¢ -21.2¢
Pythagorean 2.26 0.0¢ 0.0¢ -21.5¢ -21.5¢ -21.5¢
41 Equal 41 17 2.33 -0.5¢ -1.0¢ -23.0¢ -23.4¢ -23.9¢
29 Equal 29 12 2.50 -1.5¢ -3.0¢ -26.0¢ -27.5¢ -29.0¢
17 Equal 17 7 3.00 -3.9¢ -7.9¢ -33.3¢ -37.2¢ -41.1¢
22 Equal 22 9 4.00 -7.1¢ -14.3¢ -42.9¢ -50.0¢ -57.2¢
5 Equal 53 23 -18.0¢ -36.1¢

Accidental Perfect min min min min dim dim dim dim dim dim dim bb Neutral Neutral
Note 4th 7th 3rd 6th 2nd 5th 1st 4th 7th 3rd 6th 2nd 5th 7th 3rd

Steps 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

WHY THIS PAGE ENDS HERE

Providing compatibility between instrumental tunings is a complex problem.

In technologically-limited societies, tuning theory often diverges from practice. The actual theoretical basis of the tuning remains a matter of dipute, if not a complete mystery.

Until the facts are known and the need is demonstrated, the compatibility of instrumental tuning systems will not be addressed further on this page.

Send suggestions or comments to Ear O'Corn at aloe@rev.net.

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Last revised: 20 May 2009

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