742edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro}}
{{ED intro}}


== Theory ==
== Theory ==
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=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|742}}
{{Harmonics in equal|742|columns=11}}
{{Harmonics in equal|742|columns=11|start=12|collapsed=1|title=Approximation of prime harmonics in 742edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| -84 53 }}
| {{Monzo| 23 6 -14 }}, {{monzo| -84 53 }}
| {{mapping| 742 1176 1723 }}
| {{Mapping| 742 1176 1723 }}
| −0.0157
| −0.0157
| 0.0555
| 0.0555
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}
| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}
| {{mapping| 742 1176 1723 2083 }}
| {{Mapping| 742 1176 1723 2083 }}
| −0.0035
| −0.0035
| 0.0525
| 0.0525
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040
| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040
| {{mapping| 742 1176 1723 2083 2567 }}
| {{Mapping| 742 1176 1723 2083 2567 }}
| −0.0123
| −0.0123
| 0.0501
| 0.0501
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325
| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325
| {{mapping| 742 1176 1723 2083 2567 2746 }}
| {{Mapping| 742 1176 1723 2083 2567 2746 }}
| −0.0302
| −0.0302
| 0.0608
| 0.0608
Line 53: Line 54:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655
| {{mapping| 742 1176 1723 2083 2567 2746 3033 }}
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 }}
| −0.0317
| −0.0317
| 0.0564
| 0.0564
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211
| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211
| {{mapping| 742 1176 1723 2083 2567 2746 3033 3152 }}
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 3152 }}
| −0.0295
| −0.0295
| 0.0531
| 0.0531
Line 67: Line 68:
| 2.3.5.7.11.13.17.19.23
| 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
| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211
| {{mapping| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)
| −0.0468
| −0.0468
| 0.0699
| 0.0699
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 8388608/7381125
| 8388608/7381125
| [[Fortune]]
| [[Fortune]]
|-
| 1
| 243\742
| 392.992
| 2744/2187
| [[Emmthird]] (7-limit)
|-
|-
| 1
| 1
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|-
|-
| 14
| 14
| 434\742<br />(10\742)
| 434\742<br>(10\742)
| 701.886<br />(16.173)
| 701.886<br>(16.173)
| 3/2<br />(105/104)
| 3/2<br>(105/104)
| [[Silicon]]
| [[Silicon]]
|-
|-
| 53
| 53
| 239\742<br />(1\742)
| 239\742<br>(1\742)
| 386.523<br />(1.617)
| 386.523<br>(1.617)
| 5/4<br />(32805/32768)
| 5/4<br>(32805/32768)
| [[Mercator]]
| [[Mercator]]
|-
|-
| 53
| 53
| 565\742<br />(5\742)
| 565\742<br>(5\742)
| 913.746<br />(8.086)
| 913.746<br>(8.086)
| 441/260<br />(196/195)
| 441/260<br>(196/195)
| [[Iodine]]
| [[Iodine]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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
 
== 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

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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