653edo: Difference between revisions

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== Theory ==
== Theory ==
653edo is [[distinctly consistent]] to the [[21-odd-limit]], [[tempering out]] {{monzo| 39 -29 3 }} ([[tricot comma]]) and {{monzo| -20 -24 25 }} ([[counterhanson comma]]) in the 5-limit; [[2401/2400]], [[65625/65536]], and {{monzo| 7 -27 13 2 }} in the 7-limit; [[3025/3024]], [[41503/41472]], 496125/495616, and 1953125/1948617 in the 11-limit; [[2080/2079]], 4459/4455, [[6656/6655]], [[10985/10976]], and 170625/170368 in the 13-limit; [[1225/1224]], [[2058/2057]], [[2431/2430]], [[2500/2499]], [[4914/4913]], and 11271/11264 in the 17-limit; [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 3136/3135, 4200/4199, and 4394/4389 in the 19-limit.
653edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] {{monzo| 39 -29 3 }} ([[alphatricot comma]]) and {{monzo| -20 -24 25 }} ([[counterhanson comma]]) in the [[5-limit]]; [[2401/2400]], [[65625/65536]], and {{monzo| 7 -27 13 2 }} in the [[7-limit]]; [[3025/3024]], [[41503/41472]], 496125/495616, and 1953125/1948617 in the [[11-limit]]; [[2080/2079]], [[4459/4455]], [[6656/6655]], [[10985/10976]], and 170625/170368 in the [[13-limit]]; [[1225/1224]], [[2058/2057]], [[2431/2430]], [[2500/2499]], [[4914/4913]], and 11271/11264 in the [[17-limit]]; [[1445/1444]], [[1521/1520]], [[1540/1539]], [[1729/1728]], [[3136/3135]], [[4200/4199]], and 4394/4389 in the [[19-limit]].


=== Prime harmonics ===
=== Prime harmonics ===
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 1035 -653 }}
| {{Monzo| 1035 -653 }}
| {{mapping| 653 1035 }}
| {{Mapping| 653 1035 }}
| −0.0113
| −0.0113
| 0.0113
| 0.0113
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 39 -29 3 }}, {{monzo| -20 -24 25 }}
| {{Monzo| 39 -29 3 }}, {{monzo| -20 -24 25 }}
| {{mapping| 653 1035 1516 }}
| {{Mapping| 653 1035 1516 }}
| +0.0503
| +0.0503
| 0.0875
| 0.0875
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 65625/65536, {{monzo| 7 -27 13 2 }}
| 2401/2400, 65625/65536, {{monzo| 7 -27 13 2 }}
| {{mapping| 653 1035 1516 1833 }}
| {{Mapping| 653 1035 1516 1833 }}
| +0.0709
| +0.0709
| 0.0838
| 0.0838
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 65625/65536, 1953125/1948617
| 2401/2400, 3025/3024, 65625/65536, 1953125/1948617
| {{mapping| 653 1035 1516 1833 2259 }}
| {{Mapping| 653 1035 1516 1833 2259 }}
| +0.0576
| +0.0576
| 0.0795
| 0.0795
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 2401/2400, 3025/3024, 10985/10976, 65625/65536
| 2080/2079, 2401/2400, 3025/3024, 10985/10976, 65625/65536
| {{mapping| 653 1035 1516 1833 2259 2416 }}
| {{Mapping| 653 1035 1516 1833 2259 2416 }}
| +0.0801
| +0.0801
| 0.0882
| 0.0882
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4914/4913, 10985/10976
| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4914/4913, 10985/10976
| {{mapping| 653 1035 1516 1833 2259 2416 2669 }}
| {{Mapping| 653 1035 1516 1833 2259 2416 2669 }}
| +0.0759
| +0.0759
| 0.0823
| 0.0823
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 1225/1224, 1445/1444, 1521/1520, 1540/1539, 2058/2057, 2080/2079, 2401/2400
| 1225/1224, 1445/1444, 1521/1520, 1540/1539, 2058/2057, 2080/2079, 2401/2400
| {{mapping| 653 1035 1516 1833 2259 2416 2669 2774 }}
| {{Mapping| 653 1035 1516 1833 2259 2416 2669 2774 }}
| +0.0608
| +0.0608
| 0.0867
| 0.0867
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 566.00
| 566.00
| 81920/59049
| 81920/59049
| [[Tricot]]
| [[Alphatricot]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if 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]]
* "Seamless Toggle-Style" from ''Take Advantage'' (2024) – [https://open.spotify.com/track/6nGhaXwzE85erYFtxGB9Dt Spotify] | [https://francium223.bandcamp.com/track/seamless-toggle-style Bandcamp] | [https://www.youtube.com/watch?v=TQP9W0vIvqw YouTube]
* "Seamless Toggle-Style" from ''Take Advantage'' (2024) – [https://open.spotify.com/track/6nGhaXwzE85erYFtxGB9Dt Spotify] | [https://francium223.bandcamp.com/track/seamless-toggle-style Bandcamp] | [https://www.youtube.com/watch?v=TQP9W0vIvqw YouTube]

Revision as of 14:04, 16 March 2025

← 652edo 653edo 654edo →
Prime factorization 653 (prime)
Step size 1.83767 ¢ 
Fifth 382\653 (701.991 ¢)
Semitones (A1:m2) 62:49 (113.9 ¢ : 90.05 ¢)
Consistency limit 21
Distinct consistency limit 21

653 equal divisions of the octave (abbreviated 653edo or 653ed2), also called 653-tone equal temperament (653tet) or 653 equal temperament (653et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 653 equal parts of about 1.84 ¢ each. Each step represents a frequency ratio of 21/653, or the 653rd root of 2.

Theory

653edo is distinctly consistent to the 21-odd-limit. As an equal temperament, it tempers out [39 -29 3⟩ (alphatricot comma) and [-20 -24 25⟩ (counterhanson comma) in the 5-limit; 2401/2400, 65625/65536, and [7 -27 13 2⟩ in the 7-limit; 3025/3024, 41503/41472, 496125/495616, and 1953125/1948617 in the 11-limit; 2080/2079, 4459/4455, 6656/6655, 10985/10976, and 170625/170368 in the 13-limit; 1225/1224, 2058/2057, 2431/2430, 2500/2499, 4914/4913, and 11271/11264 in the 17-limit; 1445/1444, 1521/1520, 1540/1539, 1729/1728, 3136/3135, 4200/4199, and 4394/4389 in the 19-limit.

Prime harmonics

Approximation of prime harmonics in 653edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.036 -0.403 -0.373 -0.016 -0.711 -0.208 +0.190 +0.210 -0.481 -0.166
Relative (%) +0.0 +1.9 -21.9 -20.3 -0.9 -38.7 -11.3 +10.3 +11.4 -26.2 -9.0
Steps
(reduced) 		653
(0) 		1035
(382) 		1516
(210) 		1833
(527) 		2259
(300) 		2416
(457) 		2669
(57) 		2774
(162) 		2954
(342) 		3172
(560) 		3235
(623)
Approximation of prime harmonics in 653edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73
Error Absolute (¢) +0.417 -0.885 -0.645 -0.269 -0.610 -0.672 +0.420 -0.287 +0.395 +0.082
Relative (%) +22.7 -48.1 -35.1 -14.7 -33.2 -36.6 +22.9 -15.6 +21.5 +4.5
Steps
(reduced) 		3402
(137) 		3498
(233) 		3543
(278) 		3627
(362) 		3740
(475) 		3841
(576) 		3873
(608) 		3961
(43) 		4016
(98) 		4042
(124)

Subsets and supersets

653edo is the 119th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [1035 -653⟩ [⟨653 1035]] −0.0113 0.0113 0.61
2.3.5 [39 -29 3⟩, [-20 -24 25⟩ [⟨653 1035 1516]] +0.0503 0.0875 4.76
2.3.5.7 2401/2400, 65625/65536, [7 -27 13 2⟩ [⟨653 1035 1516 1833]] +0.0709 0.0838 4.56
2.3.5.7.11 2401/2400, 3025/3024, 65625/65536, 1953125/1948617 [⟨653 1035 1516 1833 2259]] +0.0576 0.0795 4.33
2.3.5.7.11.13 2080/2079, 2401/2400, 3025/3024, 10985/10976, 65625/65536 [⟨653 1035 1516 1833 2259 2416]] +0.0801 0.0882 4.80
2.3.5.7.11.13.17 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4914/4913, 10985/10976 [⟨653 1035 1516 1833 2259 2416 2669]] +0.0759 0.0823 4.48
2.3.5.7.11.13.17.19 1225/1224, 1445/1444, 1521/1520, 1540/1539, 2058/2057, 2080/2079, 2401/2400 [⟨653 1035 1516 1833 2259 2416 2669 2774]] +0.0608 0.0867 4.72

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 21\653 38.59 45/44 Hemitert
1 42\653 77.18 256/245 Tertiaseptal
1 172/653 316.08 6/5 Counterhanson
1 308/653 566.00 81920/59049 Alphatricot

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

Music

Francium
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