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{{Infobox ET}}
{{Infobox ET}}
The '''612 equal divisions of the octave''' ('''612edo'''), or the '''612(-tone) equal temperament''' ('''612tet''', '''612et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 612 parts of about 1.96 [[cent]]s each, a size close to [[32805/32768]], the schisma.
{{ED intro}}


== Theory ==
== Theory ==
612edo is a very strong [[5-limit]] system, a fact noted by [[R. H. M. Bosanquet]] and [[James Murray Barbour]]. The equal temperament [[tempering out|tempers out]] the {{monzo| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), {{monzo| -53 10 16 }} (kwazy comma), {{monzo| 54 -37 2 }} ([[monzisma]]), {{monzo| -107 47 14 }} (fortune comma), and {{monzo| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].
612edo is a very strong [[5-limit]] system, a fact noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}{{citation needed}} and {{w|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out|tempers out]] the {{monzo| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), {{monzo| -53 10 16 }} ([[kwazy comma]]), {{monzo| 54 -37 2 }} ([[monzisma]]), {{monzo| -107 47 14 }} (fortune comma), and {{monzo| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].


The 612edo has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].
The 612edo step has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|612|columns=11}}
{{Harmonics in equal|612}}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 23: Line 24:
| 2.3.5
| 2.3.5
| {{monzo| 1 -27 18 }}, {{monzo| -53 10 16 }}
| {{monzo| 1 -27 18 }}, {{monzo| -53 10 16 }}
| [{{val| 612 970 1421 }}]
| {{Mapping| 612 970 1421 }}
| +0.0044
| +0.0044
| 0.0089
| 0.0089
Line 30: Line 31:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| -53 10 16 }}
| 2401/2400, 4375/4374, {{monzo| -53 10 16 }}
| [{{val| 612 970 1421 1718 }}]
| {{Mapping| 612 970 1421 1718 }}
| +0.0210
| +0.0210
| 0.0297
| 0.0297
Line 37: Line 38:
| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4375/4374, {{monzo| 21 -6 -7 -2 3 }}
| 2401/2400, 3025/3024, 4375/4374, {{monzo| 21 -6 -7 -2 3 }}
| [{{val| 612 970 1421 1718 2117 }}]
| {{Mapping| 612 970 1421 1718 2117 }}
| +0.0363
| +0.0363
| 0.0406
| 0.0406
Line 44: Line 45:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2200/2197, 2401/2400, 3025/3024, 4096/4095, 4375/4374
| 2200/2197, 2401/2400, 3025/3024, 4096/4095, 4375/4374
| [{{val| 612 970 1421 1718 2117 2265 }}]
| {{Mapping| 612 970 1421 1718 2117 2265 }}
| +0.0010
| +0.0010
| 0.0871
| 0.0871
Line 51: Line 52:
| 2.3.5.7.11.13.19
| 2.3.5.7.11.13.19
| 1331/1330, 1540/1539, 2200/2197, 2376/2375, 2926/2925, 4096/4095
| 1331/1330, 1540/1539, 2200/2197, 2376/2375, 2926/2925, 4096/4095
| [{{val| 612 970 1421 1718 2117 2265 2600 }}]
| {{Mapping| 612 970 1421 1718 2117 2265 2600 }}
| -0.0168
| −0.0168
| 0.0917
| 0.0917
| 4.68
| 4.68
|}
|}
* 612et has a lower relative error than any previous equal temperaments in the 5-limit. Not until [[1171edo|1171]] do we find a better equal temperament in terms of either absolute error or relative error.  
* 612et has a lower relative error than any previous equal temperaments in the 5-limit. Not until [[1171edo|1171]] do we find a better equal temperament in terms of either absolute error or relative error.  
* Besides, it has a lower absolute error in the 7- and 11-limit than any previous equal temperaments, and is only bettered by [[935edo|935]] and [[836edo|836]], respectively.  
* It also has a lower absolute error in the 7- and 11-limit than any previous equal temperaments, and is only bettered by [[935edo|935]] and [[836edo|836]], respectively.  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+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
|-
|-
Line 84: Line 86:
| 162.75
| 162.75
| 1125/1024
| 1125/1024
| [[Kwazy]]
| [[Crazy]]
|-
|-
| 4
| 4
Line 122: Line 124:
| [[Hemiennealimmal]] (11-limit)
| [[Hemiennealimmal]] (11-limit)
|}
|}
<nowiki>*</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 ==
; [[Eliora]]
; [[Eliora]]
* [https://www.youtube.com/watch?v=_DrkrgkiaAY ''Theme and Variations in Hemiennealimmal''] (2023)
* [https://www.youtube.com/watch?v=_DrkrgkiaAY ''Theme and Variations in Hemiennealimmal''] (2023)
== Notes ==
<references />


[[Category:Ennealimmal]]
[[Category:Ennealimmal]]
[[Category:Hemiennealimmal]]
[[Category:Hemiennealimmal]]
[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 12:55, 31 July 2025

← 611edo 612edo 613edo →
Prime factorization 22 × 32 × 17
Step size 1.96078 ¢ 
Fifth 358\612 (701.961 ¢) (→ 179\306)
Semitones (A1:m2) 58:46 (113.7 ¢ : 90.2 ¢)
Consistency limit 11
Distinct consistency limit 11

612 equal divisions of the octave (abbreviated 612edo or 612ed2), also called 612-tone equal temperament (612tet) or 612 equal temperament (612et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 612 equal parts of about 1.96 ¢ each. Each step represents a frequency ratio of 21/612, or the 612th root of 2.

Theory

612edo is a very strong 5-limit system, a fact noted by Isaac Newton[1], R. H. M. Bosanquet[citation needed] and James Murray Barbour[citation needed]. As an equal temperament, it tempers out the [485 -306⟩ (sasktel comma) in the 3-limit, and in the 5-limit [1 -27 18⟩ (ennealimma), [-52 -17 34⟩ (septendecima), [-53 10 16⟩ (kwazy comma), [54 -37 2⟩ (monzisma), [-107 47 14⟩ (fortune comma), and [161 -84 -12⟩ (atom). In the 7-limit it tempers out 2401/2400 and 4375/4374, so that it supports the ennealimmal temperament, and in fact provides the optimal patent val for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out 3025/3024 and 9801/9800, so that 612 supports the hemiennealimmal temperament. In the 13-limit, it tempers 2200/2197 and 4096/4095.

The 612edo step has been proposed as the logarithmic interval size measure skisma (or sk), since one step is nearly the same size as the schisma (32805/32768), 1/12 of a Pythagorean comma or 1/11 of a syntonic comma. Since 612 is divisible by 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under Table of 612edo intervals.

Prime harmonics

Approximation of prime harmonics in 612edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.006 -0.039 -0.198 -0.338 +0.649 +0.927 +0.526 -0.823 -0.165 +0.062
Relative (%) +0.0 +0.3 -2.0 -10.1 -17.2 +33.1 +47.3 +26.8 -42.0 -8.4 +3.2
Steps
(reduced) 		612
(0) 		970
(358) 		1421
(197) 		1718
(494) 		2117
(281) 		2265
(429) 		2502
(54) 		2600
(152) 		2768
(320) 		2973
(525) 		3032
(584)

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [1 -27 18⟩, [-53 10 16⟩ [⟨612 970 1421]] +0.0044 0.0089 0.46
2.3.5.7 2401/2400, 4375/4374, [-53 10 16⟩ [⟨612 970 1421 1718]] +0.0210 0.0297 1.52
2.3.5.7.11 2401/2400, 3025/3024, 4375/4374, [21 -6 -7 -2 3⟩ [⟨612 970 1421 1718 2117]] +0.0363 0.0406 2.07
2.3.5.7.11.13 2200/2197, 2401/2400, 3025/3024, 4096/4095, 4375/4374 [⟨612 970 1421 1718 2117 2265]] +0.0010 0.0871 4.44
2.3.5.7.11.13.19 1331/1330, 1540/1539, 2200/2197, 2376/2375, 2926/2925, 4096/4095 [⟨612 970 1421 1718 2117 2265 2600]] −0.0168 0.0917 4.68
  • 612et has a lower relative error than any previous equal temperaments in the 5-limit. Not until 1171 do we find a better equal temperament in terms of either absolute error or relative error.
  • It also has a lower absolute error in the 7- and 11-limit than any previous equal temperaments, and is only bettered by 935 and 836, respectively.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 113\612 221.57 8388608/7381125 Fortune
1 127\612 249.02 [-26 18 -1⟩ Monzismic
2 83\612 162.75 1125/1024 Crazy
4 194\612
(41\612) 		380.39
(80.39) 		81/65
(22/21) 		Quasithird
9 133\612
(25\612) 		315.69
(49.02) 		6/5
(36/35) 		Ennealimmal
12 124\612
(22\612) 		243.137
(43.14) 		3145728/2734375
(?) 		Magnesium
12 254\612
(1\612) 		498.04
(1.96) 		4/3
(32805/32768) 		Atomic
17 127\612
(17\612) 		249.02
(33.33) 		[-23 5 9 -2⟩
(100352/98415) 		Chlorine
18 127\612
(9\612) 		249.02
(17.65) 		231/200
(99/98) 		Hemiennealimmal (11-limit)

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

Music

Eliora

Notes

  1. ↑ Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". Empirical Musicology Review 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.
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