359edo: Difference between revisions
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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{{citation needed}}; the | 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 678.495{{c}} [[262144/177147|Pythagorean diminished sixth]]; in 359edo this is reached using 203 steps, or 678.55153{{c}}. | ||
Pythagorean diatonic scale: 61 61 27 61 61 61 27 | Pythagorean diatonic scale: 61 61 27 61 61 61 27 | ||
| Line 39: | Line 39: | ||
| 393216/390625, {{monzo| -69 45 -1 }} | | 393216/390625, {{monzo| -69 45 -1 }} | ||
| {{mapping| 359 569 834 }} | | {{mapping| 359 569 834 }} | ||
| | | −0.2042 | ||
| 0.2910 | | 0.2910 | ||
| 8.71 | | 8.71 | ||
| Line 46: | Line 46: | ||
| 2401/2400, 3136/3125, {{monzo| -18 24 -5 -3 }} | | 2401/2400, 3136/3125, {{monzo| -18 24 -5 -3 }} | ||
| {{mapping| 359 569 834 1008 }} | | {{mapping| 359 569 834 1008 }} | ||
| | | −0.2007 | ||
| 0.2521 | | 0.2521 | ||
| 7.54 | | 7.54 | ||
| Line 53: | Line 53: | ||
| 2401/2400, 3136/3125, 8019/8000, 42592/42525 | | 2401/2400, 3136/3125, 8019/8000, 42592/42525 | ||
| {{mapping| 359 569 834 1008 1242 }} | | {{mapping| 359 569 834 1008 1242 }} | ||
| | | −0.1729 | ||
| 0.2322 | | 0.2322 | ||
| 6.95 | | 6.95 | ||
| Line 60: | Line 60: | ||
| 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) | | {{mapping| 359 569 834 1008 1242 1328 }} (359f) | ||
| | | −0.2257 | ||
| 0.2426 | | 0.2426 | ||
| 7.26 | | 7.26 | ||
| Line 93: | Line 93: | ||
| [[Counterschismic]] | | [[Counterschismic]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||
; [[Francium]] | ; [[Francium]] | ||
* "This Madness Won't Stop!" from ''End Of Sartorius Membranes'' (2024) | * "This Madness Won't Stop!" from ''End Of Sartorius Membranes'' (2024) – [https://open.spotify.com/track/50O9nTxeMafR8AyBtsPSKa Spotify] | [https://francium223.bandcamp.com/track/this-madness-wont-stop Bandcamp] | [https://www.youtube.com/watch?v=UJyIKzgLVQU YouTube] | ||
[[Category:Hera]] | [[Category:Hera]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||
Revision as of 19:35, 15 January 2025
| ← 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 678.495 ¢ Pythagorean diminished sixth; in 359edo this is reached using 203 steps, or 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 distinct