742edo: Difference between revisions

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742et has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any previous equal temperaments. It is only bettered by [[1178edo|1178et]].  
742et has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any previous equal temperaments. It is only bettered by [[1178edo|1178et]].  
{| 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|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal
! rowspan="2" | Optimal<br>8ve stretch (¢)
8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3.5
| 2.3.5
|6115295232/6103515625, {{monzo|-61 59 -14}}
| {{monzo| 23 6 -14 }}, {{monzo| -84 53 }}
|[{{val|742 1176 1723}}]
| [{{val| 742 1176 1723 }}]
|<nowiki>-0.015690</nowiki>
| -0.0157
|0.055470
| 0.0555
|3.43
| 3.43
|-
|-
|2.3.5.7
| 2.3.5.7
|2401/2400, 14348907/14336000, 6115295232/6103515625
| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}
|[{{val|742 1176 1723 2083}}]
| [{{val| 742 1176 1723 2083 }}]
| -0.003508
| -0.0035
|0.052468
| 0.0525
|3.24
| 3.24
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|9801/9800, 2401/2400, 172032/171875, 1240029/1239040
| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040
|[{{val|742 1176 1723 2083 2567}}]
| [{{val| 742 1176 1723 2083 2567 }}]
| -0.012319
| -0.0123
|0.050128
| 0.0501
|3.10
| 3.10
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|4096/4095, 9801/9800, 2401/2400, 67392/67375, 59535/59488
| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325
|[{{val|742 1176 1723 2083 2567 2746}}]
| [{{val| 742 1176 1723 2083 2567 2746 }}]
| -0.030205
| -0.0302
|0.060773
| 0.0608
|3.76
| 3.76
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|1701/1700, 2058/2057, 2601/2600, 8624/8619, 12376/12375, 14400/14399
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655
|[{{val|742 1176 1723 2083 2567 2746 3033}}]
| [{{val| 742 1176 1723 2083 2567 2746 3033 }}]
|<nowiki>-0.031687</nowiki>
| -0.0317
|0.056382
| 0.0564
|3.49
| 3.49
|}
|}
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
Line 78: Line 78:
| [[Fortune]]
| [[Fortune]]
|-
|-
|1
| 1
|439\742
| 303\742
|709.973
| 490.026
|675/448
| 896/675
|[[Surmarvelpyth]]
| [[Surmarvelpyth]]
|-
|-
| 2
| 2

Revision as of 16:40, 17 September 2022

← 741edo 742edo 743edo →
Prime factorization 2 × 7 × 53
Step size 1.61725 ¢ 
Fifth 434\742 (701.887 ¢) (→ 31\53)
Semitones (A1:m2) 70:56 (113.2 ¢ : 90.57 ¢)
Consistency limit 21
Distinct consistency limit 21

Template:EDO intro

Theory

742edo is a very strong 19-limit system and a zeta peak tuning, and is uniquely consistent in the 21-odd-limit. It tempers out the vishnuzma and the fortune comma in the 5-limit, supporting vishnu and fortune; 2401/2400 in the 7-limit, 9801/9800 in the 11-limit, 4096/4095, 6656/6655, 10648/10647 in the 13-limit, 1701/1700, 2058/2057, 2601/2600, 4914/4913, 5832/5831 in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.

742 = 2 × 7 × 53, so it notably contains 53edo and 14edo. It supports silicon temperament (224 & 742) with period 14 in the 13-limit, and iodine temperament (159 & 742) with period 53 in the 17-limit.

Prime harmonics

Approximation of prime harmonics in 742edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.068 +0.209 -0.093 +0.165 +0.443 +0.166 +0.061 -0.781 +0.611 -0.022
Relative (%) +0.0 -4.2 +12.9 -5.7 +10.2 +27.4 +10.3 +3.8 -48.3 +37.8 -1.4
Steps
(reduced) 		742
(0) 		1176
(434) 		1723
(239) 		2083
(599) 		2567
(341) 		2746
(520) 		3033
(65) 		3152
(184) 		3356
(388) 		3605
(637) 		3676
(708)

Regular temperament properties

742et has a lower 19-limit relative error than any previous equal temperaments. It is only bettered by 1178et.

Subgroup Comma List Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [23 6 -14⟩, [-84 53⟩ [⟨742 1176 1723]] -0.0157 0.0555 3.43
2.3.5.7 2401/2400, 14348907/14336000, [23 6 -14⟩ [⟨742 1176 1723 2083]] -0.0035 0.0525 3.24
2.3.5.7.11 2401/2400, 9801/9800, 172032/171875, 1240029/1239040 [⟨742 1176 1723 2083 2567]] -0.0123 0.0501 3.10
2.3.5.7.11.13 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325 [⟨742 1176 1723 2083 2567 2746]] -0.0302 0.0608 3.76
2.3.5.7.11.13.17 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655 [⟨742 1176 1723 2083 2567 2746 3033]] -0.0317 0.0564 3.49

Rank-2 temperaments

Periods
per Octave 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 137\742 221.563 8388608/7381125 Fortune
1 303\742 490.026 896/675 Surmarvelpyth
2 44\742 71.159 25/24 Vishnu
14 434\742
(10\742) 		701.886
(16.173) 		3/2
(105/104) 		Silicon
53 239\742
(1\742) 		386.523
(1.617) 		5/4
(32805/32768) 		Mercator
53 565\742
(5\742) 		913.746
(8.086) 		441/260
(196/195) 		Iodine
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