935edo: Difference between revisions

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== Theory ==

935edo is a very strong [[23-limit]] system, and is [[consistent]] through to the [[27-odd-limit]]. It does reasonably well in the higher limits, though the sharply tuned [[11/1|11]] and [[23/1|23]] and the flatly tuned [[29/1|29]] and [[31/1|31]] create inconsistencies together, those being [[29/22]], [[29/23]], [[31/22]], [[31/23]] and their [[octave complement]]s; it is otherwise consistent to the [[39-odd-limit]]. It is a [[zeta peak edo]].

935edo is a very strong [[23-limit]] system, and is [[consistent]] through to the [[27-odd-limit]]. It does reasonably well in the higher limits, though the sharply tuned [[11/1|11]] and [[23/1|23]] and the flatly tuned [[29/1|29]] and [[31/1|31]] create inconsistencies together, those being [[29/22]], [[29/23]], [[31/22]], [[31/23]] and their [[octave complement]]s; it is otherwise consistent to the [[39-odd-limit]]. Its tuning of [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] comes from [[187edo]]. It is a [[zeta peak edo]].

As an equal temperament, it [[tempering out|tempers out]] the {{monzo| 39 -29 3 }} ([[alphatricot comma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), and {{monzo| 91 -12 -31 }} (astro comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[117649/117612]], [[151263/151250]], [[161280/161051]] in the [[11-limit]]; [[2080/2079]], [[4096/4095]], [[4225/4224]] in the [[13-limit]]; [[2058/2057]], [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[2432/2431]], [[3136/3135]], [[3250/3249]], [[4200/4199]] in the 19-limit; and [[2025/2024]], [[2300/2299]], [[2646/2645]] among others in the 23-limit.

As an equal temperament, it [[tempering out|tempers out]] the {{monzo| 39 -29 3 }} ([[alphatricot comma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), and {{monzo| 91 -12 -31 }} ([[astro comma]]) in the [[5-limit]]; [[4375/4374]] and 52734375/52706752 in the [[7-limit]]; [[117649/117612]], [[151263/151250]], [[161280/161051]] in the [[11-limit]]; [[2080/2079]], [[4096/4095]], [[4225/4224]] in the [[13-limit]]; [[2058/2057]], [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[2432/2431]], [[3136/3135]], [[3250/3249]], [[4200/4199]] in the [[19-limit]]; and [[2025/2024]], [[2300/2299]], [[2646/2645]] among others in the 23-limit.

=== Prime harmonics ===

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* 935et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19- and 23-limit. It is the first to beat [[764edo|764]] in the 13-limit, [[814edo|814]] in the 17- and 23-limit, and [[742edo|742]] in the 19-limit, only to be bettered by [[954edo|954h]] in all of those subgroups.

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 103\935

| 132.19

| {{Monzo| -38 5 13 }}

| [[Astro]]

|-

| 1

| 339\935

| 435.08

| 9/7

| [[Supermajor (temperament)|Supermajor]]

|-

| 1

| 442\935

| 567.27

| 104/75

| [[Alphatrillium]]

|-

| 17

| 194\935 (26\935)

| 248.98 (33.37)

| {{Monzo| -23 5 9 -2 }} (100352/98415)

| [[Chlorine]]

|}

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is a very strong [[23-limit]] system, and is [[consistent]] through to the [[27-odd-limit]]. It does reasonably well in the higher limits, though the sharply tuned [[11/1|11]] and [[23/1|23]] and the flatly tuned [[29/1|29]] and [[31/1|31]] create inconsistencies together, those being [[29/22]], [[29/23]], [[31/22]], [[31/23]] and their [[octave complement]]s; it is otherwise consistent to the [[39-odd-limit]]. It is a [[zeta peak edo]].
+edo is a very strong [[23-limit]] system, and is [[consistent]] through to the [[27-odd-limit]]. It does reasonably well in the higher limits, though the sharply tuned [[11/1|11]] and [[23/1|23]] and the flatly tuned [[29/1|29]] and [[31/1|31]] create inconsistencies together, those being [[29/22]], [[29/23]], [[31/22]], [[31/23]] and their [[octave complement]]s; it is otherwise consistent to the [[39-odd-limit]]. Its tuning of [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] comes from [[187edo]]. It is a [[zeta peak edo]].
-As an equal temperament, it [[tempering out|tempers out]] the {{monzo| 39 -29 3 }} ([[alphatricot comma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), and {{monzo| 91 -12 -31 }} (astro comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[117649/117612]], [[151263/151250]], [[161280/161051]] in the [[11-limit]]; [[2080/2079]], [[4096/4095]], [[4225/4224]] in the [[13-limit]]; [[2058/2057]], [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[2432/2431]], [[3136/3135]], [[3250/3249]], [[4200/4199]] in the 19-limit; and [[2025/2024]], [[2300/2299]], [[2646/2645]] among others in the 23-limit.
+As an equal temperament, it [[tempering out|tempers out]] the {{monzo| 39 -29 3 }} ([[alphatricot comma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), and {{monzo| 91 -12 -31 }} ([[astro comma]]) in the [[5-limit]]; [[4375/4374]] and 52734375/52706752 in the [[7-limit]]; [[117649/117612]], [[151263/151250]], [[161280/161051]] in the [[11-limit]]; [[2080/2079]], [[4096/4095]], [[4225/4224]] in the [[13-limit]]; [[2058/2057]], [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[2432/2431]], [[3136/3135]], [[3250/3249]], [[4200/4199]] in the [[19-limit]]; and [[2025/2024]], [[2300/2299]], [[2646/2645]] among others in the 23-limit.
 === Prime harmonics ===
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 |}
 * 935et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19- and 23-limit. It is the first to beat [[764edo|764]] in the 13-limit, [[814edo|814]] in the 17- and 23-limit, and [[742edo|742]] in the 19-limit, only to be bettered by [[954edo|954h]] in all of those subgroups.
+=== 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
+| 103\935
+| 132.19
+| {{Monzo| -38 5 13 }}
+| [[Astro]]
+|-
+| 1
+| 339\935
+| 435.08
+| 9/7
+| [[Supermajor (temperament)|Supermajor]]
+|-
+| 1
+| 442\935
+| 567.27
+| 104/75
+| [[Alphatrillium]]
+|-
+| 17
+| 194\935<br>(26\935)
+| 248.98<br>(33.37)
+| {{Monzo| -23 5 9 -2 }}<br>(100352/98415)
+| [[Chlorine]]
+|}
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct