742edo: Difference between revisions

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== 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 [[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|start=12|collapsed=1|title=Approximation of prime harmonics in 742edo (continued)}}

=== Subsets and supersets ===

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.

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~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 23:

Line 25:

|-

| 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

| 3.43

Line 31:

Line 33:

| 2.3.5.7

| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}

| [{{~~val~~| 742 1176 1723 2083 }}]

| {{Mapping| 742 1176 1723 2083 }}

| -0.0035

| −0.0035

| 0.0525

| 3.24

Line 38:

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| 2.3.5.7.11

| 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

| 3.10

Line 45:

Line 47:

| 2.3.5.7.11.13

| 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

| 3.76

Line 52:

Line 54:

| 2.3.5.7.11.13.17

| 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

| 3.49

Line 59:

Line 61:

| 2.3.5.7.11.13.17.19

| 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

| 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. It is only bettered by [[935edo|935]] in terms of absolute error, and by [[1178edo|1178]] in terms of relative error.

* 742et is also notable in the 17-limit, where it has a lower absolute ~~error~~ than any previous equal temperaments~~, past~~ [[581edo|581]] and ~~followed~~ by [[764edo|764]].

* 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 [[581edo|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 per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

Line 80:

Line 91:

| 8388608/7381125

| [[Fortune]]

|-

| 1

| 243\742

| 392.992

| 2744/2187

| [[Emmthird]] (7-limit)

|-

| 1

Line 111:

Line 128:

| [[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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|742}}
+{{ED intro}}
 == Theory ==
-edo 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.
+edo 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}}
+{{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 ===
-= 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.
+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|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 23: / Line 25: @@
 |-
 | 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
 | 3.43
@@ Line 31: / Line 33: @@
 | 2.3.5.7
 | 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}
-| [{{val| 742 1176 1723 2083 }}]
+| {{Mapping| 742 1176 1723 2083 }}
-| -0.0035
+| −0.0035
 | 0.0525
 | 3.24
@@ Line 38: / Line 40: @@
 | 2.3.5.7.11
 | 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
 | 3.10
@@ Line 45: / Line 47: @@
 | 2.3.5.7.11.13
 | 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
 | 3.76
@@ Line 52: / Line 54: @@
 | 2.3.5.7.11.13.17
 | 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
 | 3.49
@@ Line 59: / Line 61: @@
 | 2.3.5.7.11.13.17.19
 | 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
 | 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. It is only bettered by [[935edo|935]] in terms of absolute error, and by [[1178edo|1178]] in terms of relative error.
-* 742et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past [[581edo|581]] and followed by [[764edo|764]].
+* 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 [[581edo|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<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 80: / Line 91: @@
 | 8388608/7381125
 | [[Fortune]]
+|-
+| 1
+| 243\742
+| 392.992
+| 2744/2187
+| [[Emmthird]] (7-limit)
 |-
 | 1
@@ Line 111: / Line 128: @@
 | [[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