IVOR DARREG

November, 1981

Relative Importance of some Equal Temperaments

This is a first draft. To my knowledge no-one has charted the equal temperaments in quite this manner before; if someone has, please speak up! Opinions have reinforced by practical experience with many of them, and various persons' tables and comparisons. The big Zero at the beginning is to stres the importance of noises and noise-with-tones in practical music-as-heard, as well as the random tunings and unpitched sounds used in actual musical performances. It is also a reminder of how many instruments int he Real WOrld are out-of-tune but nevertheless often heard.

Major criteria have been: suitability for melody; suitability for harmony; distinctive mood and difference from 12-tone; emotional impact of compositions played in certain temperaments; quality of fifths and major thirds in a given system.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

108 109 110 111 112 113...

Beyond 113, the following should be mentioned: 118, 120, 144, 171, 200--higher numbers are mainly theoretical. 301, 559, and 612 occur in theoretical treasises.

Just intonation is of course the most important system of all, and may be carried out to various numbners of tones. Comparing it here would be insulting. For 1/4 comma meantone, see 31--there is no practical difference so far as this chart is concerned. For 1/3 comma meantone, see 19. For 1/5 comma meantone, see 43.

Carrillo's array of systems is as follows: 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96, but he composed in: 12 18 24 48 96 for the most part. Haba's array of system is: 12 18 24 30 36 72. He composed in 24 and 36 for the most part. Yassers theoretical array of systems is as follows, a Fibonacci Series: 2 5 7 12 19 31 50 81 131 212...all this beyond 12 being put into a distant future.

This chart is not about unequal temperaments, but that does not mean any disresepct for them! Do not read more into it than was put there.

Many of the above equal temperaments can be stretched or shrunk by basing them on the equal division of another interval than the octave: for instance the 49th root of 3 instead of the 31st root of 2, the 19th root of 3 instead of the 12th root of 2.