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.

So, what can be done?

HOW TO VARY THE DIET OF MUSICAL INSTRUMENTS

These are several options:

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 xenharmonium@gmail.com.) 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 (20.003012538) 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 2Pi-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+51/2/2 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 20.160357456, 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
Steps123456789101112131415
AccidentalPerfectMajMajMajMajAugAugAugAugAugAugAugxNeutralNeutral
Note5th2nd6th3rd7th4th1st5th2nd6th3rd7th4th2nd6th
Numerator39551572514776435161218
Denominator283485249644918111111
Cents7022048843861088583717652679694621151649151853
Number of notesin scaleto 5thRatio of tone to diatonic semitone
7 Equal741.00-16.2¢-32.5¢-27.2¢
1/2 Comma Meantone1.27-10.8¢-21.5¢-10.8¢-21.5¢-32.3¢-35.3¢-32.3¢-35.3¢-46.1¢
26 Equal26151.33-9.65¢-19.3¢-7.4¢-17.1¢-26.7¢-28.7¢-24.5¢-26.5¢-36.1¢
45 Equal45261.40-8.6¢-17.2¢-4.4¢-13.0¢-21.6¢-22.5¢-17.3¢-18.2¢-26.9¢
19 Equal19111.50-7.2¢-14.4¢-0.1¢-7.4¢-14.6¢-14.1¢-7.5¢-7.0¢-14.2¢
1/3 Comma Meantone1.50-7.2¢-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 Equal88511.56-6.5¢-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-based1.56-6.5¢-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 Meantone1.57-6.3¢-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 Equal69401.57-6.3¢-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 Meantone1.58-6.1¢-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 Meantone1.60-6.0¢-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 Equal50291.60-6.0¢-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¢
Golden1.62-5.7¢-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 Equal81471.63-5.7¢-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 Meantone1.65-5.4¢-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 Equal31181.67-5.2¢-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 Equal43251.75-4.3¢-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 Equal55321.80-3.8¢-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 Equal67391.83-3.4¢-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 Equal79461.86-3.2¢-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 Equal91531.88-3.1¢-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 Equal1272.00-2.0¢-3.9¢+15.6¢+13.7¢+11.7¢+17.5¢
53 Equal53312.25-0.1¢-0.1¢+21.3¢+21.2¢+21.2¢+28.8¢
Pythagorean2.260.0¢0.0¢+21.5¢+21.5¢+21.5¢+29.2¢
41 Equal41242.33+0.5¢+1.0¢+23.0¢+23.4¢+23.9¢+32.1¢
29 Equal29172.50+1.5¢+3.0¢+26.0¢+27.5¢+29.0¢+38.2¢
17 Equal17103.00+3.9¢+7.9¢+33.3¢+37.2¢+41.1¢+52.8¢
22 Equal22134.00+7.1¢+14.3¢+42.9¢+50.0¢+57.2¢+72.0¢
5 Equal53+18.0¢+36.1¢
AccidentalPerfectMajMajMajMajAugAugAugAugAugAugAugxNeutralNeutral
Note5th2nd6th3rd7th4th1st5th2nd6th3rd7th4th2nd6th
Steps123456789101112131415
Circle of Fourths
Steps123456789101112131415
AccidentalPerfectminminminmindimdimdimdimdimdimdimbbNeutralNeutral
Note4th7th3rd6th2nd5th1st4th7th3rd6th2nd5th7th3rd
Numerator4166816104891284936111111
Denominator3955157257773235869
Cents4989963168141126171129435933231738495511049347
Number of notesin scaleto 4thRatio of tone to diatonic semitone
7 Equal731.00+16.2¢+32.5¢+27.2¢
1/2 Comma Meantone1.27+10.8¢+21.5¢+10.8¢+21.5¢+32.3¢+35.3¢+32.3¢+35.3¢+46.1¢
26 Equal26111.30+9.65¢+19.3¢+7.4¢+17.1¢+26.7¢+28.7¢+24.5¢+26.5¢+36.1¢
45 Equal45191.40+8.6¢+17.2¢+4.4¢+13.0¢+21.6¢+22.5¢+17.3¢+18.2¢+26.9¢
19 Equal1981.50+7.2¢+14.4¢+0.1¢+7.4¢+14.6¢+14.1¢+7.5¢+7.0¢+14.2¢
1/3 Comma Meantone1.50+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 Equal88371.56+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-based1.56+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 Meantone1.57+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 Equal69291.57+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 Meantone1.58+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 Meantone1.60+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 Equal50211.60+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¢
Golden1.62+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 Equal81341.63+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 Meantone1.65+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 Equal31131.67+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 Equal43181.75+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 Equal55231.80+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 Equal67281.83+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 Equal79331.86+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 Equal91381.88+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 Equal1252.00+2.0¢+3.9¢-15.6¢-13.7¢-11.7¢
53 Equal53222.25+0.1¢+0.1¢-21.3¢-21.2¢-21.2¢
Pythagorean2.260.0¢0.0¢-21.5¢-21.5¢-21.5¢
41 Equal41172.33-0.5¢-1.0¢-23.0¢-23.4¢-23.9¢
29 Equal29122.50-1.5¢-3.0¢-26.0¢-27.5¢-29.0¢
17 Equal1773.00-3.9¢-7.9¢-33.3¢-37.2¢-41.1¢
22 Equal2294.00-7.1¢-14.3¢-42.9¢-50.0¢-57.2¢
5 Equal5323-18.0¢-36.1¢
AccidentalPerfectminminminmindimdimdimdimdimdimdimbbNeutralNeutral
Note4th7th3rd6th2nd5th1st4th7th3rd6th2nd5th7th3rd
Steps123456789101112131415

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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 xenharmonium@gmail.com.

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Last revised: 28 August 2016

Last edited: 31 May 2019

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