631edo: Difference between revisions

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== Theory ==
== Theory ==
631edo is [[consistent]] to the [[9-odd-limit]], with all of the odd harmonics having a flat tendency. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[4375/4374]], [[41503/41472]], [[32805/32768]] and [[12005/11979]] in the [[11-limit]]; [[1575/1573]], 4375/4374, [[4459/4455]], [[4225/4224]] and 83349/83200 in the [[13-limit]].
631edo is [[consistent]] to the [[9-odd-limit]], with all of the odd harmonics having a flat tendency. Using the [[patent val]], it [[tempers out]] [[4375/4374]], [[41503/41472]], [[32805/32768]] and [[12005/11979]] in the [[11-limit]]; [[1575/1573]], 4375/4374, [[4459/4455]], [[4225/4224]], and 83349/83200 in the [[13-limit]].


=== Odd harmonics ===
=== Odd harmonics ===
Line 26: Line 26:
| {{monzo|-1000 631}}
| {{monzo|-1000 631}}
| {{mapping|631 1000}}
| {{mapping|631 1000}}
| 0.0668
| +0.0668
| 0.0668
| 0.0668
| 3.51
| 3.51
Line 33: Line 33:
| 32805/32768, {{monzo|-50 -71 70}}
| 32805/32768, {{monzo|-50 -71 70}}
| {{mapping|631 1000 1465}}
| {{mapping|631 1000 1465}}
| 0.0818
| +0.0818
| 0.0585
| 0.0585
| 3.08
| 3.08
Line 40: Line 40:
| 4375/4374, 32805/32768, 678223072849/675000000000
| 4375/4374, 32805/32768, 678223072849/675000000000
| {{mapping|631 1000 1465 1771}}
| {{mapping|631 1000 1465 1771}}
| 0.1361
| +0.1361
| 0.1067
| 0.1067
| 5.61
| 5.61
Line 47: Line 47:
| 4375/4374, 41503/41472, 32805/32768, 12005/11979
| 4375/4374, 41503/41472, 32805/32768, 12005/11979
| {{mapping|631 1000 1465 1771 2183}}
| {{mapping|631 1000 1465 1771 2183}}
| 0.0980
| +0.0980
| 0.1221
| 0.1221
| 6.42
| 6.42
Line 54: Line 54:
| 1575/1573, 4375/4374, 4459/4455, 4225/4224, 83349/83200
| 1575/1573, 4375/4374, 4459/4455, 4225/4224, 83349/83200
| {{mapping|631 1000 1465 1771 2183 2335}}
| {{mapping|631 1000 1465 1771 2183 2335}}
| 0.0797
| +0.0797
| 0.1187
| 0.1187
| 6.24
| 6.24
Line 61: Line 61:
| 1225/1224, 1701/1700, 833/832, 1575/1573, 4459/4455, 4225/4224
| 1225/1224, 1701/1700, 833/832, 1575/1573, 4459/4455, 4225/4224
| {{mapping|631 1000 1465 1771 2183 2335 2579}}
| {{mapping|631 1000 1465 1771 2183 2335 2579}}
| 0.0809
| +0.0809
| 0.1099
| 0.1099
| 5.78
| 5.78
Line 82: Line 82:
| [[Helmholtz]] / [[Pontiac]]
| [[Helmholtz]] / [[Pontiac]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<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]]
* "DUMPSTER FIRE" from ''CAPSLOCK'' (2024) &ndash; [https://open.spotify.com/track/7noSmEcwOfZpnFsiQ6vYNv Spotify] | [https://francium223.bandcamp.com/track/dumpster-fire Bandcamp] | [https://www.youtube.com/watch?v=PWP2o71Kqes YouTube]
* "DUMPSTER FIRE" from ''CAPSLOCK'' (2024) – [https://open.spotify.com/track/7noSmEcwOfZpnFsiQ6vYNv Spotify] | [https://francium223.bandcamp.com/track/dumpster-fire Bandcamp] | [https://www.youtube.com/watch?v=PWP2o71Kqes YouTube]


[[Category:Listen]]
[[Category:Listen]]

Revision as of 19:25, 15 January 2025

← 630edo 631edo 632edo →
Prime factorization 631 (prime)
Step size 1.90174 ¢ 
Fifth 369\631 (701.743 ¢)
Semitones (A1:m2) 59:48 (112.2 ¢ : 91.28 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

631edo is consistent to the 9-odd-limit, with all of the odd harmonics having a flat tendency. Using the patent val, it tempers out 4375/4374, 41503/41472, 32805/32768 and 12005/11979 in the 11-limit; 1575/1573, 4375/4374, 4459/4455, 4225/4224, and 83349/83200 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 631edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.212 -0.260 -0.839 -0.423 +0.188 +0.043 -0.472 -0.360 -0.841 +0.851 -0.699
Relative (%) -11.1 -13.7 -44.1 -22.3 +9.9 +2.3 -24.8 -18.9 -44.2 +44.8 -36.8
Steps
(reduced) 		1000
(369) 		1465
(203) 		1771
(509) 		2000
(107) 		2183
(290) 		2335
(442) 		2465
(572) 		2579
(55) 		2680
(156) 		2772
(248) 		2854
(330)

Subsets and supersets

631edo is the 115th prime EDO.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1000 631⟩ [⟨631 1000]] +0.0668 0.0668 3.51
2.3.5 32805/32768, [-50 -71 70⟩ [⟨631 1000 1465]] +0.0818 0.0585 3.08
2.3.5.7 4375/4374, 32805/32768, 678223072849/675000000000 [⟨631 1000 1465 1771]] +0.1361 0.1067 5.61
2.3.5.7.11 4375/4374, 41503/41472, 32805/32768, 12005/11979 [⟨631 1000 1465 1771 2183]] +0.0980 0.1221 6.42
2.3.5.7.11.13 1575/1573, 4375/4374, 4459/4455, 4225/4224, 83349/83200 [⟨631 1000 1465 1771 2183 2335]] +0.0797 0.1187 6.24
2.3.5.7.11.13.17 1225/1224, 1701/1700, 833/832, 1575/1573, 4459/4455, 4225/4224 [⟨631 1000 1465 1771 2183 2335 2579]] +0.0809 0.1099 5.78

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 262\631 498.257 4/3 Helmholtz / Pontiac

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium
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