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{{Infobox ET}} | |||
{{ED intro}} | |||
== 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]]. | |||
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 | |||
=== Prime harmonics === | |||
{{Harmonics in equal|612}} | |||
== 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]] (%) | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 1 -27 18 }}, {{monzo| -53 10 16 }} | |||
| {{Mapping| 612 970 1421 }} | |||
| +0.0044 | |||
| 0.0089 | |||
| 0.46 | |||
|- | |||
| 2.3.5.7 | |||
| 2401/2400, 4375/4374, {{monzo| -53 10 16 }} | |||
| {{Mapping| 612 970 1421 1718 }} | |||
| +0.0210 | |||
| 0.0297 | |||
| 1.52 | |||
|- | |||
| 2.3.5.7.11 | |||
| 2401/2400, 3025/3024, 4375/4374, {{monzo| 21 -6 -7 -2 3 }} | |||
| {{Mapping| 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 | |||
| {{Mapping| 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 | |||
| {{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]] | |||
Latest revision as of 12:55, 31 July 2025
| ← 611edo | 612edo | 613edo → |
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
| 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
| 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