742edo: Difference between revisions
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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 has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any edo until [[1178edo|1178]]. It tempers out [[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 [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak tuning]], and is uniquely [[consistent]] in the [[21-odd-limit]]. It has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any edo until [[1178edo|1178]]. It tempers out [[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]]. | 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. | ||
In 5-limit, 742edo supports [[vishnu]] and [[fortune]]. | In 5-limit, 742edo supports [[vishnu]] and [[fortune]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|742}} | {{Harmonics in equal|742}} | ||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] | ||
== Rank-two temperaments by generator == | |||
{| class="wikitable center-all left-5" | |||
!Periods | |||
per octave | |||
!Generator | |||
(reduced) | |||
!Cents | |||
(reduced) | |||
!Associated | |||
ratio | |||
!Temperaments | |||
|- | |||
|1 | |||
|137\742 | |||
|221.563 | |||
|8388608/7381125 | |||
|Fortune | |||
|- | |||
|2 | |||
|44\742 | |||
|71.159 | |||
|25/24 | |||
|Vishnu | |||
|- | |||
|14 | |||
|434\742 | |||
(10\742) | |||
|701.886 | |||
(16.173) | |||
|3/2 | |||
(105/104) | |||
|Silicon | |||
|- | |||
|53 | |||
|565\742 | |||
(5\742) | |||
|913.746 | |||
(8.086) | |||
|441/260 | |||
|Iodine | |||
|}<!-- 3-digit number --> | |||
[[Category:Zeta]] | [[Category:Zeta]] | ||
Revision as of 13:01, 6 September 2022
| ← 741edo | 742edo | 743edo → |
Theory
742edo is a very strong 19-limit system and a zeta peak tuning, and is uniquely consistent in the 21-odd-limit. It has a lower 19-limit relative error than any edo until 1178. It tempers out 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.
In 5-limit, 742edo supports vishnu and fortune.
Prime harmonics
| 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) | |
Rank-two temperaments by generator
| Periods
per octave |
Generator
(reduced) |
Cents
(reduced) |
Associated
ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 137\742 | 221.563 | 8388608/7381125 | Fortune |
| 2 | 44\742 | 71.159 | 25/24 | Vishnu |
| 14 | 434\742
(10\742) |
701.886
(16.173) |
3/2
(105/104) |
Silicon |
| 53 | 565\742
(5\742) |
913.746
(8.086) |
441/260 | Iodine |