125edo: Difference between revisions

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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| -198 125 }}
| {{Monzo| -198 125 }}
| [{{val| 125 198 }}]
| {{Mapping| 125 198 }}
| +0.364
| +0.364
| 0.364
| 0.364
Line 32: Line 32:
| 2.3.5
| 2.3.5
| 15625/15552, 17433922005/17179869184
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| {{Mapping| 125 198 290 }}
| +0.575
| +0.575
| 0.421
| 0.421
Line 39: Line 39:
| 2.3.5.7
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| {{Mapping| 125 198 290 351 }}
| +0.362
| +0.362
| 0.519
| 0.519
Line 46: Line 46:
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| +0.528
| 0.570
| 0.570
Line 53: Line 53:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| +0.680
| 0.622
| 0.622
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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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|-
|-
| 5
| 5
| 26\125<br />(1\125)
| 26\125<br>(1\125)
| 249.6<br />(9.6)
| 249.6<br>(9.6)
| 81/70<br />(176/175)
| 81/70<br>(176/175)
| [[Hemipental]]
| [[Hemiquintile]]
|-
|-
| 5
| 5
| 52\125<br />(2\125)
| 52\125<br>(2\125)
| 499.2<br />(19.2)
| 499.2<br>(19.2)
| 4/3<br />(81/80)
| 4/3<br>(81/80)
| [[Pental (temperament)|Pental]]
| [[Quintile]]
|}
|}
<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


[[Category:Catakleismic]]
[[Category:Catakleismic]]

Revision as of 14:28, 19 February 2025

← 124edo 125edo 126edo →
Prime factorization 53
Step size 9.6 ¢ 
Fifth 73\125 (700.8 ¢)
Semitones (A1:m2) 11:10 (105.6 ¢ : 96 ¢)
Consistency limit 9
Distinct consistency limit 9

125 equal divisions of the octave (abbreviated 125edo or 125ed2), also called 125-tone equal temperament (125tet) or 125 equal temperament (125et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 125 equal parts of exactly 9.6 ¢ each. Each step represents a frequency ratio of 21/125, or the 125th root of 2.

Theory

The equal temperament tempers out 15625/15552 in the 5-limit; 225/224 and 4375/4374 in the 7-limit; 385/384 and 540/539 in the 11-limit. It defines the optimal patent val for 7- and 11-limit slender temperament. In the 13-limit the 125f val ⟨125 198 290 351 432 462] does a better job, where it tempers out 169/168, 325/324, 351/350, 625/624 and 676/675, providing a good tuning for catakleismic. Among well-known intervals, the approximation of 10/9, as 19 steps, is notable for being a strong convergent, within 0.004 cents.

Prime harmonics

Approximation of prime harmonics in 125edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.16 -2.31 +0.77 -4.12 +4.27 +0.64 +0.09 -4.27 -2.38 -2.64
Relative (%) +0.0 -12.0 -24.1 +8.1 -42.9 +44.5 +6.7 +0.9 -44.5 -24.8 -27.5
Steps
(reduced) 		125
(0) 		198
(73) 		290
(40) 		351
(101) 		432
(57) 		463
(88) 		511
(11) 		531
(31) 		565
(65) 		607
(107) 		619
(119)

Subsets and supersets

Since 125 factors into 53, 125edo contains 5edo and 25edo as its subsets. Being the cube closest to division of the octave by the Germanic long hundred, 125edo has a unit step which is the cubic (fine) relative cent of 1edo. Using every 9th step, or 1ed86.4c still encapsulates many of its best-tuned harmonics.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-198 125⟩ [⟨125 198]] +0.364 0.364 3.80
2.3.5 15625/15552, 17433922005/17179869184 [⟨125 198 290]] +0.575 0.421 4.39
2.3.5.7 225/224, 4375/4374, 589824/588245 [⟨125 198 290 351]] +0.362 0.519 5.40
2.3.5.7.11 225/224, 385/384, 1331/1323, 4375/4374 [⟨125 198 290 351 432]] +0.528 0.570 5.94
2.3.5.7.11.13 169/168, 225/224, 325/324, 385/384, 1331/1323 [⟨125 198 290 351 432 462]] (125f) +0.680 0.622 6.47

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 4\125 38.4 49/48 Slender
1 12\125 115.2 77/72 Semigamera
1 19\125 182.4 10/9 Mitonic
1 24\125 230.4 8/7 Gamera
1 33\125 316.8 6/5 Catakleismic
1 52\125 499.2 4/3 Gracecordial
1 61\125 585.6 7/5 Merman
5 26\125
(1\125) 		249.6
(9.6) 		81/70
(176/175) 		Hemiquintile
5 52\125
(2\125) 		499.2
(19.2) 		4/3
(81/80) 		Quintile

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

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