253edo: Difference between revisions
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== Theory == | == Theory == | ||
253edo is [[consistent]] to the [[17-odd-limit]], approximating the fifth by 148\253 (0.021284 cents sharper than the just 3/2), and the prime harmonics from 5 to 17 are all slightly flat. It tempers out [[32805/32768]] in the 5-limit; [[2401/2400]] in the 7-limit; [[385/384]], 1375/1372 and [[4000/3993]] in the 11-limit; [[325/324]], [[1575/1573]] and [[2200/2197]] in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides a good tuning for | 253edo is [[consistent]] to the [[17-odd-limit]], approximating the fifth by 148\253 (0.021284 cents sharper than the just 3/2), and the prime harmonics from 5 to 17 are all slightly flat. It tempers out [[32805/32768]] in the 5-limit; [[2401/2400]] in the 7-limit; [[385/384]], 1375/1372 and [[4000/3993]] in the 11-limit; [[325/324]], [[1575/1573]] and [[2200/2197]] in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides the [[optimal patent val]] for the [[tertiaschis]] temperament, and a good tuning for the [[sesquiquartififths]] temperament in the higher limits. | ||
253 = 11 × 23, and has subset edos [[11edo]] and [[23edo]]. | 253 = 11 × 23, and has subset edos [[11edo]] and [[23edo]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|253}} | {{Harmonics in equal|253|columns=11}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal 8ve <br> | ! rowspan="2" | Optimal 8ve <br>Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 73: | Line 73: | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per Octave | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |||
| 1 | |||
| 35\253 | |||
| 166.01 | |||
| 11/10 | |||
| [[Tertiaschis]] | |||
|- | |- | ||
| 1 | | 1 | ||
Revision as of 21:20, 31 August 2022
| ← 252edo | 253edo | 254edo → |
(semiconvergent)
Theory
253edo is consistent to the 17-odd-limit, approximating the fifth by 148\253 (0.021284 cents sharper than the just 3/2), and the prime harmonics from 5 to 17 are all slightly flat. It tempers out 32805/32768 in the 5-limit; 2401/2400 in the 7-limit; 385/384, 1375/1372 and 4000/3993 in the 11-limit; 325/324, 1575/1573 and 2200/2197 in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides the optimal patent val for the tertiaschis temperament, and a good tuning for the sesquiquartififths temperament in the higher limits.
253 = 11 × 23, and has subset edos 11edo and 23edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.02 | -2.12 | -1.24 | -1.12 | -1.00 | -0.61 | +1.30 | -2.19 | -0.33 | -1.95 |
| Relative (%) | +0.0 | +0.4 | -44.8 | -26.1 | -23.6 | -21.1 | -12.8 | +27.4 | -46.1 | -6.9 | -41.2 | |
| Steps (reduced) |
253 (0) |
401 (148) |
587 (81) |
710 (204) |
875 (116) |
936 (177) |
1034 (22) |
1075 (63) |
1144 (132) |
1229 (217) |
1253 (241) | |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [401 -253⟩ | [⟨253 401]] | -0.007 | 0.007 | 0.14 |
| 2.3.5 | 32805/32768, [-4 -37 27⟩ | [⟨253 401 587]] | +0.300 | 0.435 | 9.16 |
| 2.3.5.7 | 2401/2400, 32805/32768, 390625/387072 | [⟨253 401 587 710]] | +0.335 | 0.381 | 8.03 |
| 2.3.5.7.11 | 385/384, 1375/1372, 4000/3993, 19712/19683 | [⟨253 401 587 710 875]] | +0.333 | 0.341 | 7.19 |
| 2.3.5.7.11.13 | 325/324, 385/384, 1375/1372, 1575/1573, 2200/2197 | [⟨253 401 587 710 875 936]] | +0.323 | 0.312 | 6.58 |
| 2.3.5.7.11.13.17 | 325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197 | [⟨253 401 587 710 875 936 1034]] | +0.298 | 0.295 | 6.22 |
Rank-2 temperaments
| Periods per Octave |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 35\253 | 166.01 | 11/10 | Tertiaschis |
| 1 | 37\253 | 175.49 | 448/405 | Sesquiquartififths |
| 1 | 105\253 | 498.02 | 4/3 | Helmholtz |
| 1 | 123\253 | 583.40 | 7/5 | Cotritone |
Scales
- 63 32 63 63 32: Pentic
- 43 43 19 43 43 43 19: Helmholtz[7]
- 41 41 24 41 41 41 24: Meantone[7]
- 35 35 35 35 35 35 35 8: Porcupine[8]
- 33 33 33 11 33 33 33 33 11: "The Hendecapliqued superdiatonic of the Icositriphony"
- 31 31 31 18 31 31 31 31 18: Mavila[9]
- 26 26 15 26 26 26 15 26 26 26 15: Sensi[11]
- 20 20 20 11 20 20 20 20 11 20 20 20 20 11: Ketradektriatoh scale