60edo: Difference between revisions
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== Theory == | == Theory == | ||
Since 60 = 5×12, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo|-19 12}}. In the [[5-limit]], it tempers out both the [[magic comma]], 3125/3072, and the [[amity comma]], 1600000/1594323, and supplies the optimal patent val for 5-limit [[magic]], tempering out 3125/3072. In the [[7-limit]] it tempers out [[875/864]], [[245/243]], [[225/224]] and [[10976/10935]], and supports [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] scores lower in [[badness]] than the [[patent val]], and makes for an excellent tritonic tuning. It tempers out [[121/120]] and [[441/440]], whereas the patent val tempers out [[100/99]], [[385/384]] and [[540/539]]. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for [[13-limit]] tritonic. As a no-fives val, it is also excellent for the 2.3.7.11.13 [[Chromatic_pairs#Bleu|bleu temperament]]. | Since 60 = 5×12, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo|-19 12}}. In the [[5-limit]], it tempers out both the [[magic comma]], 3125/3072, and the [[amity comma]], 1600000/1594323, and supplies the optimal patent val for 5-limit [[magic]], tempering out 3125/3072. In the [[7-limit]] it tempers out [[875/864]], [[245/243]], [[225/224]] and [[10976/10935]], and supports [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] scores lower in [[badness]] than the [[patent val]], and makes for an excellent tritonic tuning. It tempers out [[121/120]] and [[441/440]], whereas the patent val tempers out [[100/99]], [[385/384]] and [[540/539]]. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for [[13-limit]] tritonic. As a no-fives val, it is also excellent for the 2.3.7.11.13 [[Chromatic_pairs#Bleu|bleu temperament]]. | ||
{{harmonics in equal|60}} | |||
== Intervals == | == Intervals == | ||
{| class="wikitable center-all right-2 left-3 left-4" | {| class="wikitable center-all right-2 left-3 left-4" | ||
Revision as of 22:57, 20 January 2022
The 60 equal division divides the octave into 60 parts of exactly 20 cents each.
Theory
Since 60 = 5×12, 60edo belongs to the family of edos which contain 12edo, and like the other small edos of this kind, it tempers out the Pythagorean comma, 531441/524288 = [-19 12⟩. In the 5-limit, it tempers out both the magic comma, 3125/3072, and the amity comma, 1600000/1594323, and supplies the optimal patent val for 5-limit magic, tempering out 3125/3072. In the 7-limit it tempers out 875/864, 245/243, 225/224 and 10976/10935, and supports magic, compton and tritonic temperaments. In the 11-limit, the 60e val scores lower in badness than the patent val, and makes for an excellent tritonic tuning. It tempers out 121/120 and 441/440, whereas the patent val tempers out 100/99, 385/384 and 540/539. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for 13-limit tritonic. As a no-fives val, it is also excellent for the 2.3.7.11.13 bleu temperament.
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.96 | -6.31 | -8.83 | -3.91 | +8.68 | -0.53 | -8.27 | -4.96 | +2.49 | +9.22 | -8.27 |
| Relative (%) | -9.8 | -31.6 | -44.1 | -19.6 | +43.4 | -2.6 | -41.3 | -24.8 | +12.4 | +46.1 | -41.4 | |
| Steps (reduced) |
95 (35) |
139 (19) |
168 (48) |
190 (10) |
208 (28) |
222 (42) |
234 (54) |
245 (5) |
255 (15) |
264 (24) |
271 (31) | |
Intervals
| Degrees | Cents | Approximate Ratios in the 2.3.5.13.17 subgroup |
Additional Ratios of 7 and 11 (tending flat, i.e. 60e val) |
|---|---|---|---|
| 0 | 0 | 1/1 | |
| 1 | 20 | 81/80 | 49/48 |
| 2 | 40 | 50/49, 64/63, 33/32 | |
| 3 | 60 | 25/24 | 28/27, 36/35 |
| 4 | 80 | ||
| 5 | 100 | 17/16, 18/17 | |
| 6 | 120 | 16/15 | 15/14, 14/13 |
| 7 | 140 | 13/12 | |
| 8 | 160 | 12/11, 11/10 | |
| 9 | 180 | 10/9 | |
| 10 | 200 | 9/8 | |
| 11 | 220 | 17/15 | |
| 12 | 240 | 15/13 | 8/7 |
| 13 | 260 | 7/6 | |
| 14 | 280 | 20/17 | 13/11, 33/28 |
| 15 | 300 | 32/27 | |
| 16 | 320 | 6/5 | |
| 17 | 340 | 39/32 | 11/9, 17/14 |
| 18 | 360 | 16/13 | 27/22, 21/17 |
| 19 | 380 | 5/4 | |
| 20 | 400 | 81/64 | |
| 21 | 420 | 33/26, 14/11 | |
| 22 | 440 | 9/7, 22/17 | |
| 23 | 460 | 13/10, 17/13 | 21/16 |
| 24 | 480 | ||
| 25 | 500 | 4/3 | |
| 26 | 520 | 27/20 | |
| 27 | 540 | 11/8, 15/11 | |
| 28 | 560 | 18/13 | |
| 29 | 580 | 7/5 | |
| 30 | 600 | 17/12, 24/17 | |
| 31 | 620 | 10/7 | |
| 32 | 640 | 13/9 | |
| 33 | 660 | 16/11, 22/15 | |
| 34 | 680 | 40/27 | |
| 35 | 700 | 3/2 | |
| 36 | 720 | ||
| 37 | 740 | 20/13, 26/17 | 32/21 |
| 38 | 760 | 14/9, 17/11 | |
| 39 | 780 | 52/33, 11/7 | |
| 40 | 800 | 160/81 | |
| 41 | 820 | 8/5 | |
| 42 | 840 | 13/8 | 44/27, 34/21 |
| 43 | 860 | 64/39 | 18/11, 28/17 |
| 44 | 880 | 5/3 | |
| 45 | 900 | 27/16 | |
| 46 | 920 | 17/10 | 22/13, 56/33 |
| 47 | 940 | 12/7 | |
| 48 | 960 | 26/15 | 7/4 |
| 49 | 980 | 30/17 | |
| 50 | 1000 | 16/9 | |
| 51 | 1020 | 9/5 | |
| 52 | 1040 | 11/6, 20/11 | |
| 53 | 1060 | ||
| 54 | 1080 | 15/8 | 28/15, 13/7 |
| 55 | 1100 | 17/9, 32/17 | |
| 56 | 1120 | ||
| 57 | 1140 | 48/25 | 27/14, 35/18 |
| 58 | 1160 | 49/25, 63/32, 64/33 | |
| 59 | 1180 | 160/81 | 96/49 |
| 60 | 1200 | 2/1 |
Compositions
Rojqoq (So-Called Peace) play by William Sethares
Black Salt - White Pepper play
all by Robin Perry
Dingshi and
Gene's Jitterbug (ogg) Gene's Jitterbug (mp3) Score
by Graham Breed
Images
Robin Perry guitar