742edo: Difference between revisions

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== 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 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.


=== Prime harmonics ===
=== Prime harmonics ===
Line 24: Line 24:
| 2.3.5
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| -84 53 }}
| {{monzo| 23 6 -14 }}, {{monzo| -84 53 }}
| [{{val| 742 1176 1723 }}]
| {{mapping| 742 1176 1723 }}
| -0.0157
| -0.0157
| 0.0555
| 0.0555
Line 31: Line 31:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}
| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}
| [{{val| 742 1176 1723 2083 }}]
| {{mapping| 742 1176 1723 2083 }}
| -0.0035
| -0.0035
| 0.0525
| 0.0525
Line 38: Line 38:
| 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
| [{{val| 742 1176 1723 2083 2567 }}]
| {{mapping| 742 1176 1723 2083 2567 }}
| -0.0123
| -0.0123
| 0.0501
| 0.0501
Line 45: Line 45:
| 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
| [{{val| 742 1176 1723 2083 2567 2746 }}]
| {{mapping| 742 1176 1723 2083 2567 2746 }}
| -0.0302
| -0.0302
| 0.0608
| 0.0608
Line 52: Line 52:
| 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
| [{{val| 742 1176 1723 2083 2567 2746 3033 }}]
| {{mapping| 742 1176 1723 2083 2567 2746 3033 }}
| -0.0317
| -0.0317
| 0.0564
| 0.0564
Line 59: Line 59:
| 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
| [{{val| 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 66: Line 66:
| 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
| [{{val| 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
Line 77: Line 77:
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
Line 118: Line 118:
| [[Iodine]]
| [[Iodine]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 12:11, 20 October 2023

← 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 edo, 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.

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)

Subsets and supersets

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.

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

Periods
per 8ve 		Generator* Cents* 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

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

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