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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>2</sup> × 3<sup>2</sup> × 17
{{ED intro}}
| Step size = 1.96078¢
| Fifth = 358\612 (701.961¢) (→ [[306edo|179\306]])
| Semitones = 58:46 (113.725¢ : 90.196¢)
| Consistency = 11
}}
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.


== Theory ==
== Theory ==
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]].


612edo is a very strong [[5-limit]] system, a fact noted by [[Bosanquet]] and [[Barbour]]. It tempers out the sasktel comma, {{monzo| 485 -306 }}, in the 3-limit, and in the 5-limit {{monzo| -52 -17 34 }}, the [[septendecima]], {{monzo| 1 -27 18 }}, the [[ennealimma]], {{monzo| -53 10 16 }}, the kwazy comma, {{monzo| 54 -37 2 }}, the [[monzisma]], {{monzo| -107 47 14 }}, the fortune comma, and {{monzo| 161 -84 -12 }}, the [[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 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]].
 
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, 68 and 72 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|612}}
{{Harmonics in equal|612}}
[[Category:612edo]]
[[Category:Equal divisions of the octave]]
[[Category:Ennealimmal]]
[[Category:Hemiennealimmal]]


== 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]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 35: 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
|
| 0.0089
|
| 0.46
|-
|-
| 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
|
| 0.0297
|
| 1.52
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4375/4374, 2791309312/2790703125
| 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
|
| 0.0406
|
| 2.07
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2200/2197, 2401/2400, 3025/3024, 4096/4095, 24192/24167
| 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
|
| 0.0871
|
| 4.44
|
|-
|-
| 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
|
| 0.0917
|
| 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.
* 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 ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 1
| 113\612
| 221.57
| 8388608/7381125
| [[Fortune]]
|-
| 1
| 127\612
| 249.02
| {{monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 2
| 83\612
| 162.75
| 1125/1024
| [[Crazy]]
|-
| 4
| 194\612<br>(41\612)
| 380.39<br>(80.39)
| 81/65<br>(22/21)
| [[Quasithird]]
|-
| 9
| 133\612<br>(25\612)
| 315.69<br>(49.02)
| 6/5<br>(36/35)
| [[Ennealimmal]]
|-
| 12
| 124\612<br>(22\612)
| 243.137<br>(43.14)
| 3145728/2734375<br>(?)
| [[Magnesium]]
|-
| 12
| 254\612<br>(1\612)
| 498.04<br>(1.96)
| 4/3<br>(32805/32768)
| [[Atomic]]
|-
| 17
| 127\612<br>(17\612)
| 249.02<br>(33.33)
| {{monzo| -23 5 9 -2 }}<br>(100352/98415)
| [[Chlorine]]
|-
| 18
| 127\612<br>(9\612)
| 249.02<br>(17.65)
| 231/200<br>(99/98)
| [[Hemiennealimmal]] (11-limit)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=_DrkrgkiaAY ''Theme and Variations in Hemiennealimmal''] (2023)
== Notes ==
<references />
[[Category:Ennealimmal]]
[[Category:Hemiennealimmal]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/612edo"