135edo: Difference between revisions

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Expand the theory with 2.3.7.11 interpretation. For full 13-limit, talk about 135c and 135f instead since they make more sense
Partial reversal -- last edit removed the note on the edo's notability in the 2.3.7.11 subgroup
 
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The '''135 equal divisions of the octave''' ('''135edo'''), or the '''135(-tone) equal temperament''' ('''135tet''', '''135et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 135 parts of about 8.89 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the 5th and the 13th harmonics.  
135edo is [[consistent]] to the [[7-odd-limit]], but with large relative error for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo|270et]], 135et makes most sense to use as a [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]] [[regular temperament|temperament]], where it is characterized by [[tempering out]] the [[garischisma]], the [[septiennealimma]], the [[symbiotic comma]], the [[argyria]], the [[chrysia]], and the [[olympia]]. On top of this, it also has fairly good approximations to primes [[17/1|17]], [[29/1|29]], and [[31/1|31]].  


Using the 135f [[val]] {{val| 135 214 313 379 467 '''499''' }}, which tends flat, 135et tempers out 32805/32768 ([[schisma]]) and 30517578125/29386561536 (quintriyo comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], 2200/2187, 12005/11979 and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.  
If we consider the full 13-limit, the flat-tending {{val| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.  


Using the 135c val {{val| 135 214 '''314''' 379 467 500 }}, which tends sharp, it tempers out 1594323/1562500 and 50331648/48828125 in the 5-limit; [[126/125]], [[10976/10935]], and 589824/588245 in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and 16384/16335 in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and 6656/6655 in the 13-limit.  
Using the 135f val, it tempers out 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (pentadecal comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], [[2200/2187]], [[12005/11979]] and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.  


As every other step of the full 13-limit monster – [[270edo|270et]], 135et probably makes more sense as a 2.3.7.11 [[subgroup]] temperament, where it tempers out the [[garischisma]] and the [[symbiotic comma]].  
Using the 135c val, it tempers out 1594323/1562500 ([[unicorn comma]]) and 50331648/48828125 ([[magus comma]]) in the 5-limit; [[126/125]], [[10976/10935]], and [[589824/588245]] in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and [[16384/16335]] in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and [[6656/6655]] in the 13-limit.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|135}}
{{Harmonics in equal|135|columns=11}}
{{Harmonics in equal|135|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 135edo (continued)}}


[[Category:Equal divisions of the octave]]
=== Subsets and supersets ===
[[Category:Quartismic]]
Since 135 factors into primes as {{nowrap| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, notably provides extremely good corrections for the approximation to harmonics 5, 13, and 19.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 214 -135 }}
| {{Mapping| 135 214 }}
| −0.0843
| 0.0843
| 0.95
|-
| 2.3.7
| 33554432/33480783, 40353607/40310784
| {{Mapping| 135 214 379 }}
| −0.0637
| 0.0747
| 0.84
|-
| 2.3.7.11
| 19712/19683, 41503/41472, 43923/43904
| {{Mapping| 135 214 379 467 }}
| −0.0328
| 0.0840
| 0.94
|-
| 2.3.7.11.17
| 1089/1088, 2058/2057, 5832/5831, 19712/19683
| {{Mapping| 135 214 379 467 552 }}
| −0.1100
| 0.1716
| 1.93
|}
 
== Instruments ==
* [[Lumatone mapping for 135edo]]

Latest revision as of 20:16, 4 May 2026

← 134edo 135edo 136edo →
Prime factorization 33 × 5
Step size 8.88889 ¢ 
Fifth 79\135 (702.222 ¢)
Semitones (A1:m2) 13:10 (115.6 ¢ : 88.89 ¢)
Consistency limit 7
Distinct consistency limit 7

135 equal divisions of the octave (abbreviated 135edo or 135ed2), also called 135-tone equal temperament (135tet) or 135 equal temperament (135et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 135 equal parts of about 8.89 ¢ each. Each step represents a frequency ratio of 21/135, or the 135th root of 2.

Theory

135edo is consistent to the 7-odd-limit, but with large relative error for the 5th and 13th harmonics. As every other step of the full 13-limit monster – 270et, 135et makes most sense to use as a 2.3.7.11-subgroup temperament, where it is characterized by tempering out the garischisma, the septiennealimma, the symbiotic comma, the argyria, the chrysia, and the olympia. On top of this, it also has fairly good approximations to primes 17, 29, and 31.

If we consider the full 13-limit, the flat-tending ⟨135 214 313 379 467 499] (135f) and the sharp-tending ⟨135 214 314 379 467 500] (135c) are reasonable choices.

Using the 135f val, it tempers out 32805/32768 (schisma) and [-11 -15 15⟩ (pentadecal comma) in the 5-limit; 225/224, 3125/3087, and 28824005/28697814 in the 7-limit, 385/384, 540/539, 2200/2187, 12005/11979 and the quartisma in the 11-limit; 169/168 and 364/363 in the 13-limit.

Using the 135c val, it tempers out 1594323/1562500 (unicorn comma) and 50331648/48828125 (magus comma) in the 5-limit; 126/125, 10976/10935, and 589824/588245 in the 7-limit; 176/175, 441/440, 14641/14580 and 16384/16335 in the 11-limit; 196/195, 351/350, 352/351, 676/675, and 6656/6655 in the 13-limit.

Prime harmonics

Approximation of prime harmonics in 135edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.27 -4.09 +0.06 -0.21 +3.92 +1.71 -4.18 +2.84 +1.53 +1.63
Relative (%) +0.0 +3.0 -46.0 +0.7 -2.3 +44.1 +19.3 -47.0 +31.9 +17.3 +18.3
Steps
(reduced) 		135
(0) 		214
(79) 		313
(43) 		379
(109) 		467
(62) 		500
(95) 		552
(12) 		573
(33) 		611
(71) 		656
(116) 		669
(129)
Approximation of prime harmonics in 135edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -2.46 -2.40 +4.04 +1.16 -2.39 -1.39 +3.12 +0.69 -1.92 +3.32 -0.09
Relative (%) -27.6 -27.0 +45.4 +13.1 -26.9 -15.7 +35.0 +7.8 -21.6 +37.4 -1.0
Steps
(reduced) 		703
(28) 		723
(48) 		733
(58) 		750
(75) 		773
(98) 		794
(119) 		801
(126) 		819
(9) 		830
(20) 		836
(26) 		851
(41)

Subsets and supersets

Since 135 factors into primes as 33 × 5, 135edo has subset edos 3, 5, 9, 15, 27, and 45. 270edo, which doubles it, notably provides extremely good corrections for the approximation to harmonics 5, 13, and 19.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [214 -135⟩ [⟨135 214]] −0.0843 0.0843 0.95
2.3.7 33554432/33480783, 40353607/40310784 [⟨135 214 379]] −0.0637 0.0747 0.84
2.3.7.11 19712/19683, 41503/41472, 43923/43904 [⟨135 214 379 467]] −0.0328 0.0840 0.94
2.3.7.11.17 1089/1088, 2058/2057, 5832/5831, 19712/19683 [⟨135 214 379 467 552]] −0.1100 0.1716 1.93

Instruments

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