612edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|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
| 2.3.5
Line 52: Line 44:
| 1331/1330, 1540/1539, 2200/2197, 2376/2375, 2926/2925, 4096/4095
| 1331/1330, 1540/1539, 2200/2197, 2376/2375, 2926/2925, 4096/4095
| {{mapping| 612 970 1421 1718 2117 2265 2600 }}
| {{mapping| 612 970 1421 1718 2117 2265 2600 }}
| -0.0168
| &minus;0.0168
| 0.0917
| 0.0917
| 4.68
| 4.68
|}
{{comma basis end}}
* 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"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 87: Line 73:
|-
|-
| 4
| 4
| 194\612<br>(41\612)
| 194\612<br />(41\612)
| 380.39<br>(80.39)
| 380.39<br />(80.39)
| 81/65<br>(22/21)
| 81/65<br />(22/21)
| [[Quasithird]]
| [[Quasithird]]
|-
|-
| 9
| 9
| 133\612<br>(25\612)
| 133\612<br />(25\612)
| 315.69<br>(49.02)
| 315.69<br />(49.02)
| 6/5<br>(36/35)
| 6/5<br />(36/35)
| [[Ennealimmal]]
| [[Ennealimmal]]
|-
|-
| 12
| 12
| 124\612<br>(22\612)
| 124\612<br />(22\612)
| 243.137<br>(43.14)
| 243.137<br />(43.14)
| 3145728/2734375<br>(?)
| 3145728/2734375<br />(?)
| [[Magnesium]]
| [[Magnesium]]
|-
|-
| 12
| 12
| 254\612<br>(1\612)
| 254\612<br />(1\612)
| 498.04<br>(1.96)
| 498.04<br />(1.96)
| 4/3<br>(32805/32768)
| 4/3<br />(32805/32768)
| [[Atomic]]
| [[Atomic]]
|-
|-
| 17
| 17
| 127\612<br>(17\612)
| 127\612<br />(17\612)
| 249.02<br>(33.33)
| 249.02<br />(33.33)
| {{monzo| -23 5 9 -2 }}<br>(100352/98415)
| {{monzo| -23 5 9 -2 }}<br />(100352/98415)
| [[Chlorine]]
| [[Chlorine]]
|-
|-
| 18
| 18
| 127\612<br>(9\612)
| 127\612<br />(9\612)
| 249.02<br>(17.65)
| 249.02<br />(17.65)
| 231/200<br>(99/98)
| 231/200<br />(99/98)
| [[Hemiennealimmal]] (11-limit)
| [[Hemiennealimmal]] (11-limit)
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Music ==
== Music ==

Revision as of 05:25, 16 November 2024

← 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

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 equal division of the octave into 612 parts of about 1.96 cents each, a size close to 32805/32768, the schisma.

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]. The equal temperament 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 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

Template:Comma basis begin |- | 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 Template:Comma basis end

  • 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

Template:Rank-2 begin |- | 1 | 113\612 | 221.57 | 8388608/7381125 | Fortune |- | 1 | 127\612 | 249.02 | [-26 18 -1⟩ | Monzismic |- | 2 | 83\612 | 162.75 | 1125/1024 | Kwazy |- | 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) Template:Rank-2 end Template:Orf

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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