742edo: Difference between revisions
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| Prime factorization = 2 × 7 × 53 | | Prime factorization = 2 × 7 × 53 | ||
| Step size = 1.61725¢ | | Step size = 1.61725¢ | ||
| Fifth = 434\742 (701. | | Fifth = 434\742 (701.89¢) (→ [[53edo|31\53]]) | ||
| Semitones = 70:56 (113. | | Semitones = 70:56 (113.21¢ : 90.57¢) | ||
| Consistency = 21 | | Consistency = 21 | ||
}} | }} | ||
| Line 18: | Line 18: | ||
{{Harmonics in equal|742}} | {{Harmonics in equal|742}} | ||
== Regular temperament properties == | |||
=== Rank-2 temperaments === | |||
== Rank- | |||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
!Periods | ! Periods<br>per Octave | ||
per | ! Generator<br>(Reduced) | ||
!Generator | ! Cents<br>(Reduced) | ||
( | ! Associated<br>Ratio | ||
!Cents | ! Temperaments | ||
( | |||
!Associated | |||
!Temperaments | |||
|- | |- | ||
|1 | | 1 | ||
|137\742 | | 137\742 | ||
|221.563 | | 221.563 | ||
|8388608/7381125 | | 8388608/7381125 | ||
|Fortune | | [[Fortune]] | ||
|- | |- | ||
|2 | | 2 | ||
|44\742 | | 44\742 | ||
|71.159 | | 71.159 | ||
|25/24 | | 25/24 | ||
|Vishnu | | [[Vishnu]] | ||
|- | |- | ||
|14 | | 14 | ||
|434\ | | 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 | ||
|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]] | ||
|}<!-- 3-digit number --> | |} | ||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | |||
[[Category:Zeta]] | [[Category:Zeta]] | ||
Revision as of 23:30, 11 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) | |
Regular temperament properties
Rank-2 temperaments
| 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 (196/195) |
Iodine |