359edo: Difference between revisions
Jump to navigation
Jump to search
+infobox; improve intro; update prime error table |
+RTT table and rank-2 temperaments |
||
| Line 21: | Line 21: | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|359|columns=11}} | {{Harmonics in equal|359|columns=11}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list|Comma List]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | |||
! colspan="2" | Tuning Error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -569 359 }} | |||
| [{{val| 359 569 }}] | |||
| +0.0016 | |||
| 0.0016 | |||
| 0.05 | |||
|- | |||
| 2.3.5 | |||
| 393216/390625, {{monzo| -69 45 -1 }} | |||
| [{{val| 359 569 834 }}] | |||
| -0.2042 | |||
| 0.2910 | |||
| 8.71 | |||
|- | |||
| 2.3.5.7 | |||
| 2401/2400, 3136/3125, {{monzo| -18 24 -5 -3 }} | |||
| [{{val| 359 569 834 1008 }}] | |||
| -0.2007 | |||
| 0.2521 | |||
| 7.54 | |||
|- | |||
| 2.3.5.7.11 | |||
| 2401/2400, 3136/3125, 8019/8000, 42592/42525 | |||
| [{{val| 359 569 834 1008 1242 }}] | |||
| -0.1729 | |||
| 0.2322 | |||
| 6.95 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125 | |||
| [{{val| 359 569 834 1008 1242 1328 }}] (359f) | |||
| -0.2257 | |||
| 0.2426 | |||
| 7.26 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per Octave | |||
! Generator<br>(Reduced) | |||
! Cents<br>(Reduced) | |||
! Associated<br>Ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 58\359 | |||
| 193.87 | |||
| 28/25 | |||
| [[Hemiwürschmidt]] | |||
|- | |||
| 1 | |||
| 116\359 | |||
| 387.74 | |||
| 5/4 | |||
| [[Würschmidt]] (5-limit) | |||
|- | |||
| 1 | |||
| 149\359 | |||
| 498.05 | |||
| 4/3 | |||
| [[Counterschismic]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||
[[Category:Hera]] | [[Category:Hera]] | ||
Revision as of 17:28, 30 August 2022
| ← 358edo | 359edo | 360edo → |
(semiconvergent)
Theory
359edo contains a very close approximation of the pure 3/2 fifth of 701.955 cents, with the 210\359 step of 701.94986 cents. It provides the optimal patent val for the 11-limit hera temperament.
359edo supports a type of exaggerated Hornbostel mode, with an approximation of the blown fifth that he described of the pan flutes of some regions of South America; the Pythagorean fifth (701.955¢) minus the Pythagorean comma (23.46¢) = 678.495¢; in 359edo this is the step 203\359 of 678.55153¢.
Pythagorean diatonic scale: 61 61 27 61 61 61 27
Exaggerated Hornbostel superdiatonic scale: 47 47 47 15 47 47 47 47 15 (fails in the position of Phi and the square root of Pi [+1\359 step of each one][clarification needed]).
359edo is the 72nd prime edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.01 | +1.43 | +0.53 | +0.21 | -1.53 | -1.33 | -0.02 | +0.14 | -0.05 | +1.48 |
| Relative (%) | +0.0 | -0.2 | +42.8 | +16.0 | +6.4 | -45.8 | -39.9 | -0.6 | +4.1 | -1.5 | +44.4 | |
| Steps (reduced) |
359 (0) |
569 (210) |
834 (116) |
1008 (290) |
1242 (165) |
1328 (251) |
1467 (31) |
1525 (89) |
1624 (188) |
1744 (308) |
1779 (343) | |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-569 359⟩ | [⟨359 569]] | +0.0016 | 0.0016 | 0.05 |
| 2.3.5 | 393216/390625, [-69 45 -1⟩ | [⟨359 569 834]] | -0.2042 | 0.2910 | 8.71 |
| 2.3.5.7 | 2401/2400, 3136/3125, [-18 24 -5 -3⟩ | [⟨359 569 834 1008]] | -0.2007 | 0.2521 | 7.54 |
| 2.3.5.7.11 | 2401/2400, 3136/3125, 8019/8000, 42592/42525 | [⟨359 569 834 1008 1242]] | -0.1729 | 0.2322 | 6.95 |
| 2.3.5.7.11.13 | 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125 | [⟨359 569 834 1008 1242 1328]] (359f) | -0.2257 | 0.2426 | 7.26 |
Rank-2 temperaments
| Periods per Octave |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 58\359 | 193.87 | 28/25 | Hemiwürschmidt |
| 1 | 116\359 | 387.74 | 5/4 | Würschmidt (5-limit) |
| 1 | 149\359 | 498.05 | 4/3 | Counterschismic |