DANIELOU'S "UNIVERSAL SCALE" COMPARED TO 72 TONE EQUAL TEMPERAMENT
David Canright has prepared the following chart, comparing Danielou's "Universal Scale" to it's approximations in 72-tET.
This scale can be found in: Danielou, Alain, Music and the Power of Sound, The Influence of Tuning and Interval on Consciousness, Inner Traditions, Rochester, Vermont, USA 1995, p42.
| Note | Indian solfege | sruti | ratio, | cents error, | note, | key |
|---|---|---|---|---|---|---|
| 1 | Sa | 0 | 1/1 | +0.0¢ | 0 | 0:0 |
| 2 | 81/80 | +4.8¢ | 1 | 1:0 | ||
| 46/45 | +4.7¢ | 2 | 2:0 | |||
| 3 | 128/125 | +7.7¢ | 2 | 2:0 | ||
| 31/30 | +6.8¢ | 3 | 3:0 | |||
| 4 | 1 | 25/24 | +4.0¢ | 4 | 4:0 | |
| 256/243 | +6.9¢ | 5 | 5:0 | |||
| 5 | 135/128 | -7.8¢ | 6 | 0:1 | ||
| 6 | 2 | 16/15 | -4.9¢ | 7 | 1:1 | |
| 2187/2048 | -3.0¢ | 7 | 1:1 | |||
| 7 | 27/25 | -0.1¢ | 8 | 2:1 | ||
| 135/124 | -2.9¢ | 9 | 3:1 | |||
| 8 | 800/729 | -5.8¢ | 10 | 4:1 | ||
| 11/10 | -1.7¢ | 10 | 4:1 | |||
| 9 | 3 | 10/9 | -0.9¢ | 11 | 5:1 | |
| 10 | Re | 4 | 9/8 | +3.9¢ | 12 | 0:2 |
| 11 | 256/225 | +6.8¢ | 13 | 1:2 | ||
| 8/7 | -2.2¢ | 14 | 2:2 | |||
| 2187/1900 | -6.5¢ | 15 | 3:2 | |||
| 12 | 15/13 | -2.3¢ | 15 | 3:2 | ||
| 93/80 | -6.0¢ | 16 | 4:2 | |||
| 13 | 75/64 | +7.9¢ | 16 | 4:2 | ||
| 14 | 5 | 32/27 | -5.9¢ | 18 | 0:3 | |
| 15 | 6 | 6/5 | -1.0¢ | 19 | 1:3 | |
| 75/62 | -3.8¢ | 20 | 2:3 | |||
| 243/200 | +3.8¢ | 20 | 2:3 | |||
| 16 | 8000/6561 | -6.7¢ | 21 | 3:3 | ||
| 17 | 100/81 | -1.9¢ | 22 | 4:3 | ||
| 18 | Ga | 7 | 5/4 | +3.0¢ | 23 | 5:3 |
| 19 | 8 | 81/64 | +7.8¢ | 24 | 0:4 | |
| 19/15 | -7.4¢ | 25 | 1:4 | |||
| 20 | 32/25 | -6.0¢ | 26 | 2:4 | ||
| 31/24 | -6.9¢ | 27 | 3:4 | |||
| 21 | 125/96 | +7.0¢ | 27 | 3:4 | ||
| 22 | 320/243 | -6.8¢ | 29 | 5:4 | ||
| 23 | Ma | 9 | 4/3 | -2.0¢ | 30 | 0:5 |
| 24 | 10 | 27/20 | +2.9¢ | 31 | 1:5 | |
| 25 | 512/375 | +5.8¢ | 32 | 2:5 | ||
| 2187/1600 | +7.7¢ | 32 | 2:5 | |||
| 62/45 | +4.8¢ | 33 | 3:5 | |||
| 26 | 25/18 | +2.1¢ | 34 | 4:5 | ||
| 7/5 | -0.8¢ | 35 | 5:5 | |||
| 27 | 11 | 45/32 | +6.9¢ | 35 | 5:5 | |
| 28 | 12 | 64/45 | -6.9¢ | 37 | 1:6 | |
| 29 | 36/25 | -2.1¢ | 38 | 2:6 | ||
| 45/31 | -4.8¢ | 39 | 3:6 | |||
| 19/13 | +7.0¢ | 39 | 3:6 | |||
| 30 | 375/256 | -5.8¢ | 40 | 4:6 | ||
| 31 | 40/27 | -2.9¢ | 41 | 5:6 |
| Note | Indian solfege | sruti | ratio | cents error, | note | key |
|---|---|---|---|---|---|---|
| 32 | Pa | 13 | 3/2 | +2.0¢ | 42 | 0:7 |
| 33 | 243/160 | +6.8¢ | 43 | 1:7 | ||
| 34 | 192/125 | -7.0¢ | 45 | 3:7 | ||
| 19683/12800 | -5.0¢ | 45 | 3:7 | |||
| 31/20 | -7.9¢ | 46 | 4:7 | |||
| 35 | 25/16 | +6.0¢ | 46 | 4:7 | ||
| 36 | 14 | 128/81 | -7.8¢ | 48 | 0:8 | |
| 19/12 | -4.4¢ | 48 | 0:8 | |||
| 37 | 15 | 8/5 | -3.0¢ | 49 | 1:8 | |
| 50/31 | -5.7¢ | 50 | 2:8 | |||
| 38 | 81/50 | +1.9¢ | 50 | 2:8 | ||
| 39 | 400/243 | -3.8¢ | 52 | 4:8 | ||
| 40 | Dha | 16 | 5/3 | +1.0¢ | 53 | 5:8 |
| 41 | 17 | 27/16 | +5.9¢ | 54 | 0:9 | |
| 42 | 128/75 | -7.9¢ | 56 | 2:9 | ||
| 12/7 | -0.2¢ | 56 | 2:9 | |||
| 31/18 | +7.8¢ | 56 | 2:9 | |||
| 43 | 125/72 | +5.0¢ | 57 | 3:9 | ||
| 7/4 | +2.2¢ | 58 | 4:9 | |||
| 44 | 225/128 | -6.8¢ | 59 | 5:9 | ||
| 45 | 18 | 16/9 | -3.9¢ | 60 | 0:10 | |
| 46 | 19 | 9/5 | +0.9¢ | 61 | 1:10 | |
| 29/16 | -3.8¢ | 62 | 2:10 | |||
| 729/400 | +5.8¢ | 62 | 2:10 | |||
| 47 | 4000/2187 | -4.7¢ | 63 | 3:10 | ||
| 48 | 50/27 | +0.1¢ | 64 | 4:10 | ||
| 13/7 | +5.0¢ | 64 | 4:10 | |||
| 49 | Ni | 20 | 15/8 | +4.9¢ | 65 | 5:10 |
| 21 | 256/135 | +7.8¢ | 66 | 0:11 | ||
| 50 | 243/128 | -6.9¢ | 67 | 1:11 | ||
| 51 | 48/25 | -4.0¢ | 68 | 2:11 | ||
| 60/31 | -6.8¢ | 69 | 3:11 | |||
| 31/16 | -5.0¢ | 69 | 3:11 | |||
| 52 | 125/64 | -7.7¢ | 70 | 4:11 | ||
| 53 | 160/81 | -4.8¢ | 71 | 5:11 | ||
| 1 | Sa | 22 | 2/1 | +0.0¢ | 72 | 0:12 |
Each entry includes the ratio, cents error relative to nearest tempered (so, for example, 3/2 is +2.0c relative to note #42), tempered note # (0-71), keyboard # (0-5), key # (0-11), where keyboards 1 -5 are~ assumed progressively [17¢ globally] sharper relative to keyboard 0 [which are all 12-tET].
Tues., 1 May 2001 from "Canright, David"
DAVID CANRIGHT PERSONAL WEBPAGE
IMPROVISATION IN CANRIGHTS 13-LIMIT 12 TONE SCALE
Maurice Courant allows himself to write: "Needless to say, the sixty degrees in the octave are scarcely perceptible and are difficult to realize, a slight difference of temperature bringing a significant variation in the sound compared with the interval of two successive degrees. Such a scale can never be accurate." Courant, Maurice. "Chine et Corée, essai historique sur la musique classique des chinois." In Encylopédie de la musique et dictionnaire du Conservatoire. Paris: Delagrave, 1922., in Daniélou, Alain, Music and The Power of Sound, Inner Traditions, Rochester, VT, 1995, p55]
. . . We should not forget that the problem is not to play intervals of one comma in succession but to play intervals with an accuracy of one comma. A difference of one comma in a fifth or an octave is not only perceptible but extremely disagreeable even to an untrained ear. The same difference in a third or in a major second (it is then the difference between the major and the minor tone) completely changes the color of the note and its expression. One can even say, as a rule, that such differences are the very basis of vocal and melodic expression, . . . . [Daniélou, Alain, Music and The Power of Sound, Inner Traditions, Rochester, VT, 1995, p55]