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]] | ||