User:BudjarnLambeth/272ed23
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| ← 271ed23 | 272ed23 | 273ed23 → |
272 equal divisions of the 23rd harmonic (abbreviated 272ed23) is a nonoctave tuning system that divides the interval of 23/1 into 272 equal parts of about 20 ¢ each. Each step represents a frequency ratio of 231/272, or the 272nd root of 23.
272ed23 is primarily intended to be used as 60edo but with slightly compressed octaves.
Theory
Compared to pure-octaves 60edo, 272ed23 features a relatively large improvement to 7/1 and 11/1, at the cost of moderate worsening of 2/1, 3/1 and 5/1.
It also causes the vals to flip for 5/1, 7/1, 13/1 and 17/1.
These characteristics, particularly the flipped 5/1 and 13/1, are preferred over pure-octaves 60edo for catnip temperament specifically. They change catnip’s warts from 60cf to 272dg (later letters in the alphabet are better).
Given this, 272ed23 can be thought of as a 60edo clone tailor-made for catnip.
60edo is already on the optimal ET sequence for catnip, so 272ed23 is one of the most optimal catnip tunings concievable.
Prime harmonics
In the 47-limit, 272ed3 has less than 40% relative error on primes 2, 3, 5, 7, 11, 17, 23, 29, 31, 37, 41, 43 and 47.
This makes it a solid tuning for the entire no-13, no-19 47-limit.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.59 | -6.05 | +7.65 | +3.89 | -0.28 | +9.86 | +4.44 | -8.51 | +0.00 | -2.17 | +2.12 |
| Relative (%) | -13.0 | -30.3 | +38.3 | +19.5 | -1.4 | +49.4 | +22.2 | -42.6 | +0.0 | -10.8 | +10.6 | |
| Steps (reduced) |
60 (60) |
95 (95) |
140 (140) |
169 (169) |
208 (208) |
223 (223) |
246 (246) |
255 (255) |
272 (0) |
292 (20) |
298 (26) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.84 | -2.94 | -5.57 | +0.09 | -8.33 | +5.57 | +7.73 | +4.96 | +4.35 | -3.83 | -0.88 |
| Relative (%) | -24.2 | -14.7 | -27.9 | +0.5 | -41.8 | +27.9 | +38.7 | +24.8 | +21.8 | -19.2 | -4.4 | |
| Steps (reduced) |
313 (41) |
322 (50) |
326 (54) |
334 (62) |
344 (72) |
354 (82) |
357 (85) |
365 (93) |
370 (98) |
372 (100) |
379 (107) | |
60edo for comparison
In the 47-limit, 60edo has less than 40% relative error on primes 2, 3, 5, 13, 17, 19, 31, 47.
This makes it a solid tuning for the no-7, no-11 19-limit (or dual-7, dual-11).
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | -6.31 | -8.83 | +8.68 | -0.53 | -4.96 | +2.49 | -8.27 | -9.58 | -5.04 |
| Relative (%) | +0.0 | -9.8 | -31.6 | -44.1 | +43.4 | -2.6 | -24.8 | +12.4 | -41.4 | -47.9 | -25.2 | |
| Steps (reduced) |
60 (0) |
95 (35) |
139 (19) |
168 (48) |
208 (28) |
222 (42) |
245 (5) |
255 (15) |
271 (31) |
291 (51) |
297 (57) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.66 | -9.06 | +8.48 | -5.51 | +6.50 | +0.83 | +3.12 | +0.69 | +0.30 | -7.79 | -4.54 |
| Relative (%) | +43.3 | -45.3 | +42.4 | -27.5 | +32.5 | +4.1 | +15.6 | +3.5 | +1.5 | -38.9 | -22.7 | |
| Steps (reduced) |
313 (13) |
321 (21) |
326 (26) |
333 (33) |
344 (44) |
353 (53) |
356 (56) |
364 (4) |
369 (9) |
371 (11) |
378 (18) | |
Subsets and supersets
272ed23 is quite composite, with subset ed23s 1, 2, 4, 8, 16, 17, 34, 68, 136.
Unlike pure-octaves 60edo, 272ed23 does not have high consistency at all. 60edo is both consistent and distinctly consistent up to the 9-integer limit, while 272ed23 is consistent and distinctly consistently only up to the 4-integer limit.
Notation
272ed23 can be notated using ups and downs notation using Helmholtz-Ellis accidentals as if it were 60edo:
| Semitones | 0 | 1⁄5 | 2⁄5 | 3⁄5 | 4⁄5 | 1 | 11⁄5 | 12⁄5 | 13⁄5 | 14⁄5 | 2 | 21⁄5 | 22⁄5 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol |  |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
| Flat symbol |  |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with three arrows.
Intervals
| Degrees | Cents | Approximate ratios in the 11-limit |
Additional ratios in the no-13, no-19 47-limit |
Additional ratios in the full 47-limit |
|---|---|---|---|---|
| 0 | 0 | 1/1 | ||
| 1 | 19.96 | 55/54, 56/55, 64/63, 81/80 | 51/50, 63/62, 69/68, 70/69, 75/74, 82/81, 85/84, 88/87 | 52/51, 53/52, 57/56, 58/57, 65/64, 66/65, 76/75, 77/76, 78/77 |
| 2 | 39.91 | 33/32, 36/35 | 34/33, 35/34, 37/36, 41/40, 42/41, 43/42, 44/43 | 38/37, 39/38, 40/39 |
| 3 | 59.87 | 25/24, 28/27 | 24/23, 29/28, 30/29, 88/85 | 26/25, 27/26 |
| 4 | 79.83 | 21/20, 22/21 | 23/22, 45/43 | 19/18, 20/19 |
| 5 | 99.78 | 16/5, 35/33 | 17/16, 18/17 | |
| 6 | 119.74 | 15/14 | 29/27, 44/41, 74/69 | 14/13 |
| 7 | 139.70 | 25/23, 51/47 | 13/12, 38/35 | |
| 8 | 159.66 | 11/10, 12/11 | 23/21, 34/31 | 57/52 |
| 9 | 179.61 | 10/9 | 31/28, 41/37, 51/46 | 21/19 |
| 10 | 199.57 | 9/8, 28/25, 55/49 | 37/33, 46/41 | 19/17 |
| 11 | 219.53 | 8/7 (patent) | 17/15, 42/37 | |
| 12 | 239.48 | 8/7 (catnip) | 23/20, 31/27, 54/47, 85/74 | 15/13 |
| 13 | 259.44 | 7/6 | 29/25, 36/31, 43/37 | 22/19 |
| 14 | 279.4 | 20/17, 27/23, 47/40 | 13/11 | |
| 15 | 299.35 | 25/21 | 44/37 | 19/16 |
| 16 | 319.31 | 6/5, 77/64 | 29/24, 35/28, 41/34 | 23/19, 47/39, 65/54 |
| 17 | 339.27 | 11/9 | 17/14, 28/23, 45/37 | |
| 18 | 359.22 | 27/22 | 43/35 | 16/13, 91/74 |
| 19 | 379.18 | 5/4, 56/45 | 21/17, 31/25, 36/29, 41/33, 46/37, 51/41 | 26/21 |
| 20 | 399.14 | 63/50 | 29/23, 34/27 | 19/15, 24/19 |
| 21 | 419.09 | 14/11 | 23/18, 37/29, 51/40 | 65/51 |
| 22 | 439.05 | 9/7 | 22/17, 31/24, 40/31, 58/45 | 49/38 |
| 23 | 459.01 | 176/135 | 30/23, 43/33 | 13/10, 17/13 |
| 24 | 478.97 | 21/16, 33/25 | 29/22, 62/47 | 25/19, 91/69 |
| 25 | 498.92 | 4/3, 75/56 | 43/32, 47/35, 63/47, 99/74 | 39/29, 51/38, 87/65, 91/68 |
| 26 | 518.88 | 27/20 | 23/17, 31/23, 58/43, 85/63 | 19/14 |
| 27 | 538.84 | 15/11 | 41/30, 56/41, 86/63 | 26/19 |
| 28 | 558.79 | 11/8 | 29/21, 40/29, 69/50 | 18/13 |
| 29 | 578.75 | 7/5, 25/18, 88/63 | 32/23, 46/33, 60/43, 81/58 | 39/28, 95/68 |
| 30 | 598.71 | 17/12, 24/17, 41/29 | 65/46 | |
| 31 | 618.66 | 10/7, 63/44 | 23/16, 33/23, 43/30 | 93/65 |
| 32 | 638.62 | 81/56 | 29/20, 42/29, 68/47 | 13/9, 55/38, 94/65 |
| 33 | 658.58 | 16/11, 22/15 | 41/28, 60/41 | 19/13 |
| 34 | 678.53 | 25/17, 31/21, 34/32, 37/25 | 28/19 | |
| 35 | 698.49 | 3/2, 49/33 | 55/37, 64/43, 70/47, 82/55, 94/63 | 52/35, 58/39, 76/51, 85/57 |
| 36 | 718.45 | 32/21, 50/33 | 29/19, 35/23, 41/27, 44/29, 47/31 | 38/25 |
| 37 | 738.40 | 49/32 | 23/15, 72/47 | 20/13, 26/17 |
| 38 | 748.36 | 14/9 | 17/11, 31/20, 48/31 | |
| 39 | 778.32 | 11/7, 25/16 | 36/23, 47/30, 69/44 | |
| 40 | 798.28 | 27/17, 46/29 | 19/12, 65/41 | |
| 41 | 818.23 | 8/5, 45/28, 77/48 | 29/18, 37/23, 69/43 | 21/13 |
| 42 | 838.19 | 34/21, 47/29, 60/37 | 13/8, 21/13 | |
| 43 | 858.15 | 18/11 | 23/14, 41/25 | 64/39 |
| 44 | 878.10 | 5/3, 33/20 | 28/17, 48/29, 58/35, 68/41, 93/56 | 38/23, 43/26, 63/38, 78/47 |
| 45 | 898.06 | 27/16, 42/25 | 37/22, 47/28 | 32/19 |
| 46 | 918.02 | 56/33 | 17/10 | 22/13, 39/23 |
| 47 | 937.97 | 12/7, 55/32 | 31/18, 43/25 | 19/11, 98/57 |
| 48 | 958.93 | 7/4 (catnip) | 19/11, 40/23 | 26/15, 33/19 |
| 49 | 977.89 | 7/4 (patent), 44/25 | 30/17, 37/21, 51/29 | 23/13, 95/54 |
| 50 | 997.84 | 16/9, 25/14 | 41/23 | 57/32 |
| 51 | 1017.8 | 9/5 | 29/16, 56/31, 74/41, 92/51 | 38/21, 47/26, 65/36 |
| 52 | 1037.76 | 20/11 | 31/17, 51/28 | |
| 53 | 1057.72 | 11/6 | 94/51 | 24/13, 35/18 |
| 54 | 1077.67 | 28/15 | 41/22, 54/29 | 13/7, 95/51 |
| 55 | 1097.63 | 15/8, 66/35 | 17/9, 32/17 | 49/26 |
| 56 | 1117.59 | 21/11, 40/21 | 82/43 | 19/10 |
| 57 | 1137.54 | 27/14 | 23/12, 29/15, 56/29 | 25/3 |
| 58 | 1157.50 | 35/18 | 31/16, 33/17, 41/21, 80/41 | 37/19, 39/20 |
| 59 | 1177.46 | 49/25, 55/28, 63/32 | 47/24, 57/29, 69/35 | 51/26, 65/33, 75/38, 77/39 |
| 60 | 1200 | 2/1, 99/50 |
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 12 | 12\272ed23 (2\272ed23) |
239.48 (39.91) |
8/7 (36/35) |
Catnip (272dg) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
(This table is incomplete.)
Scales
- 60edo scales
These are useable in 272ed23, simply apply an octave compression of 0.99784 in Scale Workshop.
Nearby equal-step tunings
There are a few other useful equal-step tunings which occur close to 272ed23 in step size:
- 207ed11, 168ed7
The tunings 207ed11 and 168ed7 are almost identical. Each is 60edo but with slightly stretched octaves.
Each causes relatively large improvement to 5/1, 7/1 and 11/1 at the cost of moderate worsening of 2/1 and 3/1.
Each also causes the vals to flip for 11/1 and 13/1.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.28 | +3.24 | +1.29 | +0.36 | +0.00 | -8.44 | +8.44 | -3.63 | +6.54 | +6.33 | -8.85 |
| Relative (%) | +16.4 | +16.2 | +6.4 | +1.8 | +0.0 | -42.1 | +42.1 | -18.1 | +32.6 | +31.6 | -44.1 | |
| Steps (reduced) |
60 (60) |
95 (95) |
139 (139) |
168 (168) |
207 (0) |
221 (14) |
245 (38) |
254 (47) |
271 (64) |
291 (84) |
296 (89) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.15 | +3.04 | +0.99 | +0.00 | -0.44 | -8.92 | +7.92 | -4.17 | +5.96 | +5.71 | -9.49 |
| Relative (%) | +15.7 | +15.1 | +4.9 | +0.0 | -2.2 | -44.5 | +39.5 | -20.8 | +29.7 | +28.5 | -47.3 | |
| Steps (reduced) |
60 (60) |
95 (95) |
139 (139) |
168 (0) |
207 (39) |
221 (53) |
245 (77) |
254 (86) |
271 (103) |
291 (123) |
296 (128) | |
- 301zpi
The tuning 301zpi, the 301st zeta peak index, is 60edo but with slightly stretched octaves.
It causes relatively large improvement to 3/1, 5/1, 7/1, 11/1 and 17/1 at the cost of relatively small worsening of 2/1 and relatively large worsening of 13/1.
It also causes the val for 11/1 to flip from 208 steps to 207 steps.
301zpi is both consistent and distinctly consistent up to the 10-integer-limit, which is unusually high for a two digit edo or three digit zpi.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.60 | +0.58 | -2.61 | -4.35 | -5.80 | +5.39 | +1.57 | +9.28 | -1.05 | -1.82 | +2.88 |
| Relative (%) | +8.0 | +2.9 | -13.0 | -21.7 | -29.0 | +26.9 | +7.9 | +46.3 | -5.3 | -9.1 | +14.4 | |
| Step | 60 | 95 | 139 | 168 | 207 | 222 | 245 | 255 | 271 | 291 | 297 | |
- 60edo
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | -6.31 | -8.83 | +8.68 | -0.53 | -4.96 | +2.49 | -8.27 | -9.58 | -5.04 |
| Relative (%) | +0.0 | -9.8 | -31.6 | -44.1 | +43.4 | -2.6 | -24.8 | +12.4 | -41.4 | -47.9 | -25.2 | |
| Steps (reduced) |
60 (0) |
95 (35) |
139 (19) |
168 (48) |
208 (28) |
222 (42) |
245 (5) |
255 (15) |
271 (31) |
291 (51) |
297 (57) | |
- 255ed19
The tuning 255ed19 is 60edo but with slightly compressed octaves.
It causes a relatively large improvement to 11/1, at the cost of relatively small worsening of every smaller prime.
It also causes the val for 7/1 to flip from 168 steps to 169.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.59 | -2.88 | -7.67 | +9.53 | +6.65 | -2.69 | -7.34 | +0.00 | +9.07 | +7.57 | -7.93 |
| Relative (%) | -2.9 | -14.4 | -38.4 | +47.7 | +33.3 | -13.5 | -36.7 | +0.0 | +45.4 | +37.9 | -39.7 | |
| Steps (reduced) |
60 (60) |
95 (95) |
139 (139) |
169 (169) |
208 (208) |
222 (222) |
245 (245) |
255 (0) |
272 (17) |
292 (37) |
297 (42) | |
- 208ed11
The tuning 208ed11 is 60edo but with slightly compressed octaves.
It causes a relatively large improvement to 7/1 and 11/1, at the cost of moderate worsening of 2/1, 3/1 and 5/1.
It also causes the vals to flip for 5/1, 7/1 and 17/1.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.50 | -5.92 | +7.84 | +4.12 | +0.00 | -9.79 | +4.78 | -8.16 | +0.37 | -1.77 | +2.53 |
| Relative (%) | -12.5 | -29.7 | +39.3 | +20.6 | +0.0 | -49.1 | +23.9 | -40.9 | +1.9 | -8.8 | +12.7 | |
| Steps (reduced) |
60 (60) |
95 (95) |
140 (140) |
169 (169) |
208 (0) |
222 (14) |
246 (38) |
255 (47) |
272 (64) |
292 (84) |
298 (90) | |
- 272ed23
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.59 | -6.05 | +7.65 | +3.89 | -0.28 | +9.86 | +4.44 | -8.51 | +0.00 | -2.17 | +2.12 |
| Relative (%) | -13.0 | -30.3 | +38.3 | +19.5 | -1.4 | +49.4 | +22.2 | -42.6 | +0.0 | -10.8 | +10.6 | |
| Steps (reduced) |
60 (60) |
95 (95) |
140 (140) |
169 (169) |
208 (208) |
223 (223) |
246 (246) |
255 (255) |
272 (0) |
292 (20) |
298 (26) | |
Instruments
Due to 60edo's highly composite nature, 272ed23 has an unusually high number of ways it can be subdivided. This means it has multiple good skip-fretting systems which can be used to create stringed instruments with playable fret spacings that still span the full gamut. Probably the best of these is tuning a compressed-octaves 20edo guitar to major thirds, emulating the pure-octaves one by Robin Perry in the image below. This is very closely related to the Kite Guitar.
