460edo: Difference between revisions
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The '''460 equal divisions of the octave''' divides the octave into 460 equal parts of 2. | {{Infobox ET | ||
| Prime factorization = 2<sup>2</sup> × 5 × 23 | |||
| Step size = 2.60870¢ | |||
| Fifth = 269\460 (701.74¢) | |||
| Semitones = 43:35 (112.17¢ : 91.30¢) | |||
| Consistency = 21 | |||
}} | |||
The '''460 equal divisions of the octave''' ('''460edo'''), or the '''460(-tone) equal temperament''' ('''460tet''', '''460et''') when viewed from a [[regular temperament]] perspective, divides the octave into 460 equal parts of about 2.61 [[cent]]s each. | |||
== Theory == | |||
460edo is a very strong 19-limit system and is uniquely [[consistent]] to the [[21-odd-limit]], with harmonics of 3 to 19 all tuned flat. It tempers out the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and 65536/65625 in the 7-limit, so that it [[support]]s [[pontiac]]. In the 11-limit it tempers of 43923/43904, [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]]; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], 2058/2057, 2431/2430, [[2601/2600]] and 4914/4913; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank five temperament tempering out 833/832 and 1001/1000. | 460edo is a very strong 19-limit system and is uniquely [[consistent]] to the [[21-odd-limit]], with harmonics of 3 to 19 all tuned flat. It tempers out the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and 65536/65625 in the 7-limit, so that it [[support]]s [[pontiac]]. In the 11-limit it tempers of 43923/43904, [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]]; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], 2058/2057, 2431/2430, [[2601/2600]] and 4914/4913; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank five temperament tempering out 833/832 and 1001/1000. | ||
460 factors into 2<sup>2</sup> × 5 × 23, and has subset edos 2, 4, 5, 10, 20, 23, 46, 92, 115, and 230. | |||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 19:13, 30 January 2022
| ← 459edo | 460edo | 461edo → |
The 460 equal divisions of the octave (460edo), or the 460(-tone) equal temperament (460tet, 460et) when viewed from a regular temperament perspective, divides the octave into 460 equal parts of about 2.61 cents each.
Theory
460edo is a very strong 19-limit system and is uniquely consistent to the 21-odd-limit, with harmonics of 3 to 19 all tuned flat. It tempers out the schisma, 32805/32768, in the 5-limit and 4375/4374 and 65536/65625 in the 7-limit, so that it supports pontiac. In the 11-limit it tempers of 43923/43904, 3025/3024 and 9801/9800; in the 13-limit 1001/1000, 4225/4224 and 10648/10647; in the 17-limit 833/832, 1089/1088, 1225/1224, 1701/1700, 2058/2057, 2431/2430, 2601/2600 and 4914/4913; and in the 19-limit 1331/1330, 1445/1444, 1521/1520, 1540/1539, 1729/1728, 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the optimal patent val for various temperaments such as the rank five temperament tempering out 833/832 and 1001/1000.
460 factors into 22 × 5 × 23, and has subset edos 2, 4, 5, 10, 20, 23, 46, 92, 115, and 230.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.22 | -0.23 | -1.00 | -0.88 | -0.53 | -0.61 | -0.12 | +0.42 | +0.86 | +0.18 |
| Relative (%) | +0.0 | -8.3 | -8.7 | -38.3 | -33.9 | -20.2 | -23.3 | -4.7 | +16.2 | +32.9 | +7.0 | |
| Steps (reduced) |
460 (0) |
729 (269) |
1068 (148) |
1291 (371) |
1591 (211) |
1702 (322) |
1880 (40) |
1954 (114) |
2081 (241) |
2235 (395) |
2279 (439) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-729 460⟩ | [⟨460 729]] | +0.0681 | 0.0681 | 2.61 |
| 2.3.5 | 32805/32768, [6 68 -49⟩ | [⟨460 729 1068]] | +0.0780 | 0.0573 | 2.20 |
| 2.3.5.7 | 4375/4374, 32805/32768, [-4 -2 -9 10⟩ | [⟨460 729 1068 1291]] | +0.1475 | 0.1303 | 4.99 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 32805/32768, 184877/184320 | [⟨460 729 1068 1291 1591]] | +0.1691 | 0.1243 | 4.76 |
| 2.3.5.7.11.13 | 1001/1000, 3025/3024, 4225/4224, 4375/4374, 26411/26364 | [⟨460 729 1068 1291 1591 1702]] | +0.1647 | 0.1139 | 4.36 |
| 2.3.5.7.11.13.17 | 833/832, 1001/1000, 1089/1088, 1225/1224, 1701/1700, 4225/4224 | [⟨460 729 1068 1291 1591 1702 1880]] | +0.1624 | 0.1056 | 4.05 |
| 2.3.5.7.11.13.17.19 | 833/832, 1001/1000, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1617/1615 | [⟨460 729 1068 1291 1591 1702 1880 1954]] | +0.1457 | 0.1082 | 4.15 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 121\460 | 315.65 | 6/5 | Egads |
| 1 | 191\460 | 498.26 | 4/3 | Helmholtz / pontiac |
| 10 | 121\460 (17\460) |
315.65 (44.35) |
6/5 (40/39) |
Deca |