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742edo: Difference between revisions

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The ''742 equal division'' divides the octave into 742 equal parts of  1.617 cents each. It 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|consistent]] in the 21-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.
{{Infobox ET}}
[[Category:zeta]]
{{ED intro}}
 
== Theory ==
742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency|distinctly consistent]] in the [[21-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[vishnuzma]] and the fortune comma in the 5-limit, [[support]]ing [[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 ===
{{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 ===
Since 742 factors into 2 × 7 × 53, 742edo has subset edos {{EDOs| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] ({{nowrap|224 & 518}}) with 14 periods per octave in the 13-limit, and [[iodine]] ({{nowrap|159& 583f}}) with 53 periods per octave in the 17-limit.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| {{Monzo| 23 6 -14 }}, {{monzo| -84 53 }}
| {{Mapping| 742 1176 1723 }}
| −0.0157
| 0.0555
| 3.43
|-
| 2.3.5.7
| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}
| {{Mapping| 742 1176 1723 2083 }}
| −0.0035
| 0.0525
| 3.24
|-
| 2.3.5.7.11
| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)
| −0.0468
| 0.0699
| 4.32
|}
* 742et has a lower 19-limit [[Tenney–Euclidean temperament measures #TE simple badness|relative error]] than any previous equal temperaments, being the first to beat [[581edo|581]], before getting bettered by [[935edo|935]] in absolute error and by [[1178edo|1178]] in 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 [[764edo|764]]; in the 23-limit it beats [[718edo|718]] and is bettered by [[814edo|814]].
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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<br>(10\742)
| 701.886<br>(16.173)
| 3/2<br>(105/104)
| [[Silicon]]
|-
| 53
| 239\742<br>(1\742)
| 386.523<br>(1.617)
| 5/4<br>(32805/32768)
| [[Mercator]]
|-
| 53
| 565\742<br>(5\742)
| 913.746<br>(8.086)
| 441/260<br>(196/195)
| [[Iodine]]
|}
<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
Retrieved from "https://en.xen.wiki/w/742edo"