742edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 7 × 53
{{ED intro}}
| Step size = 1.61725¢
| Fifth = 434\742 (701.89¢) (→ [[53edo|31\53]])
| Semitones = 70:56 (113.21¢ : 90.57¢)
| Consistency = 21
}}
{{EDO intro|742}}


== Theory ==
== Theory ==
742edo is a very strong 19-limit system and a [[The Riemann zeta function and tuning #Zeta EDO lists|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.
742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency|distinctly consistent]] in the [[21-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[vishnuzma]] and the fortune comma in the 5-limit, [[support]]ing [[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 [[Fractional-octave temperaments#Silicon|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 ===
{{Harmonics in equal|742|columns=11}}
{{Harmonics in equal|742|columns=11|start=12|collapsed=1|title=Approximation of prime harmonics in 742edo (continued)}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Harmonics in equal|742}}
Since 742 factors into 2 × 7 × 53, 742edo has subset edos {{EDOs| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] ({{nowrap|224 & 518}}) with 14 periods per octave in the 13-limit, and [[iodine]] ({{nowrap|159& 583f}}) with 53 periods per octave in the 17-limit.


== Regular temperament properties ==
== Regular temperament properties ==
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" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve stretch (¢)
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! 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| 23 6 -14 }}, {{monzo| -84 53 }}
| {{Mapping| 742 1176 1723 }}
| −0.0157
| 0.0555
| 3.43
|-
| 2.3.5.7
| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}
| {{Mapping| 742 1176 1723 2083 }}
| −0.0035
| 0.0525
| 3.24
|-
| 2.3.5.7.11
| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040
| {{Mapping| 742 1176 1723 2083 2567 }}
| −0.0123
| 0.0501
| 3.10
|-
|-
|2.3.5
| 2.3.5.7.11.13
|6115295232/6103515625, {{monzo|-61 59 -14}}
| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325
|[{{val|742 1176 1723}}]
| {{Mapping| 742 1176 1723 2083 2567 2746 }}
|?
| −0.0302
|?
| 0.0608
|?
| 3.76
|-
|-
|2.3.5.7
| 2.3.5.7.11.13.17
|2401/2400, 14348907/14336000, 6115295232/6103515625
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655
|[{{val|742 1176 1723 2083}}]
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 }}
|
| −0.0317
|
| 0.0564
|
| 3.49
|-
|-
|2.3.5.7.11
| 2.3.5.7.11.13.17.19
|9801/9800, 2401/2400, 172032/171875, 1240029/1239040
| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211
|[{{val|742 1176 1723 2083 2567}}]
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 3152 }}
|
| −0.0295
|
| 0.0531
|
| 3.28
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13.17.19.23
|4096/4095, 9801/9800, 2401/2400, 67392/67375, 59535/59488
| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211
|[{{val|742 1176 1723 2083 2567 2746}}]
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)
|
| −0.0468
|
| 0.0699
|
| 4.32
|}
|}
* 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 [[935edo|935]] in terms of absolute error, and by [[1178edo|1178]] in terms of relative error.
* 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats [[581edo|581]] and is bettered by [[764edo|764]]; in the 23-limit it beats [[718edo|718]] and is bettered by [[814edo|814]].
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per Octave
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Generator<br>(Reduced)
|-
! Cents<br>(Reduced)
! Periods<br>per 8ve
! Associated<br>Ratio
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
Line 70: Line 91:
| 8388608/7381125
| 8388608/7381125
| [[Fortune]]
| [[Fortune]]
|-
| 1
| 243\742
| 392.992
| 2744/2187
| [[Emmthird]] (7-limit)
|-
| 1
| 303\742
| 490.026
| 896/675
| [[Surmarvelpyth]]
|-
|-
| 2
| 2
Line 95: Line 128:
| [[Iodine]]
| [[Iodine]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
== Scales ==
[[Category:Zeta]]
* Silicon[28]: 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43

Latest revision as of 22:24, 25 April 2026

← 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

742 equal divisions of the octave (abbreviated 742edo or 742ed2), also called 742-tone equal temperament (742tet) or 742 equal temperament (742et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 742 equal parts of about 1.62 ¢ each. Each step represents a frequency ratio of 21/742, or the 742nd root of 2.

Theory

742edo is a very strong 19-limit system and a zeta peak edo, and is distinctly consistent in the 21-odd-limit. As an equal temperament, 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.

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)
Approximation of prime harmonics in 742edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -0.670 -0.491 -0.466 +0.801 -0.189 +0.128 +0.635 -0.062 -0.182 +0.243 -0.655
Relative (%) -41.4 -30.4 -28.8 +49.5 -11.7 +7.9 +39.3 -3.8 -11.2 +15.0 -40.5
Steps
(reduced) 		3865
(155) 		3975
(265) 		4026
(316) 		4122
(412) 		4250
(540) 		4365
(655) 		4401
(691) 		4501
(49) 		4563
(111) 		4593
(141) 		4677
(225)

Subsets and supersets

Since 742 factors into 2 × 7 × 53, 742edo has subset edos 2, 7, 14, 53, 106, and 371, of which 7edo, 14edo and 53edo are very notable. It supports silicon (224 & 518) with 14 periods per octave in the 13-limit, and iodine (159& 583f) with 53 periods per octave in the 17-limit.

Regular temperament properties

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
2.3.5.7.11.13.17.19 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211 [⟨742 1176 1723 2083 2567 2746 3033 3152]] −0.0295 0.0531 3.28
2.3.5.7.11.13.17.19.23 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211 [⟨742 1176 1723 2083 2567 2746 3033 3152 3357]] (742i) −0.0468 0.0699 4.32
  • 742et has a lower 19-limit relative error than any previous equal temperaments. It is only bettered by 935 in terms of absolute error, and by 1178 in terms of relative error.
  • 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats 581 and is bettered by 764; in the 23-limit it beats 718 and is bettered by 814.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 137\742 221.563 8388608/7381125 Fortune
1 243\742 392.992 2744/2187 Emmthird (7-limit)
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

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

Scales

  • Silicon[28]: 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43
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