359edo: Difference between revisions
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Cleanup; +commas; +subset and supersets |
Cleanup; clarify the title row of the rank-2 temp table; request a citation |
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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. In the 5-limit it tempers out the [[würschmidt comma]] and the [[counterschisma]]; in the 7-limit [[2401/2400]] and [[3136/3125]], supporting [[hemiwürschmidt]]; in the 11-limit, [[8019/8000]], providing the [[optimal patent val]] for 11-limit [[hera]]. | 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. In the 5-limit it tempers out the [[würschmidt comma]] and the [[counterschisma]]; in the 7-limit [[2401/2400]] and [[3136/3125]], supporting [[hemiwürschmidt]]; in the 11-limit, [[8019/8000]], providing the [[optimal patent val]] for 11-limit [[hera]]. | ||
359edo [[support]]s 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¢. | 359edo [[support]]s 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{{citation needed}}; 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 | Pythagorean diatonic scale: 61 61 27 61 61 61 27 | ||
| Line 30: | Line 30: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -569 359 }} | | {{monzo| -569 359 }} | ||
| | | {{mapping| 359 569 }} | ||
| +0.0016 | | +0.0016 | ||
| 0.0016 | | 0.0016 | ||
| Line 37: | Line 37: | ||
| 2.3.5 | | 2.3.5 | ||
| 393216/390625, {{monzo| -69 45 -1 }} | | 393216/390625, {{monzo| -69 45 -1 }} | ||
| | | {{mapping| 359 569 834 }} | ||
| -0.2042 | | -0.2042 | ||
| 0.2910 | | 0.2910 | ||
| Line 44: | Line 44: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 3136/3125, {{monzo| -18 24 -5 -3 }} | | 2401/2400, 3136/3125, {{monzo| -18 24 -5 -3 }} | ||
| | | {{mapping| 359 569 834 1008 }} | ||
| -0.2007 | | -0.2007 | ||
| 0.2521 | | 0.2521 | ||
| Line 51: | Line 51: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 3136/3125, 8019/8000, 42592/42525 | | 2401/2400, 3136/3125, 8019/8000, 42592/42525 | ||
| | | {{mapping| 359 569 834 1008 1242 }} | ||
| -0.1729 | | -0.1729 | ||
| 0.2322 | | 0.2322 | ||
| Line 58: | Line 58: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125 | | 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125 | ||
| | | {{mapping| 359 569 834 1008 1242 1328 }} (359f) | ||
| -0.2257 | | -0.2257 | ||
| 0.2426 | | 0.2426 | ||
| Line 67: | Line 67: | ||
{| 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 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 91: | Line 91: | ||
| [[Counterschismic]] | | [[Counterschismic]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Hera]] | [[Category:Hera]] | ||
Revision as of 15:41, 9 November 2023
| ← 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. In the 5-limit it tempers out the würschmidt comma and the counterschisma; in the 7-limit 2401/2400 and 3136/3125, supporting hemiwürschmidt; in the 11-limit, 8019/8000, providing the optimal patent val for 11-limit hera.
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[citation needed]; 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]).
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) | |
Subsets and supersets
359edo is the 72nd prime edo. 718edo, which doubles it, provides a good correction to the harmonics 5, 13, 17, and 31.
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 8ve |
Generator* | Cents* | 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 |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct