680edo: Difference between revisions

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==~~= Odd harmonics =~~==

== Theory ==

680edo ~~retains~~ a reasonable 3rd and ~~5th~~ [[harmonic]], though nowhere near the accuracy of the prior ~~multiple~~ [[~~612edo~~]]~~; as a multiple of~~ [[~~34edo~~]]~~, 680edo~~ borrows ~~that edo~~'s accurate representation of the interval [[25/24]], implying that the error on ~~[[prime]] 3~~ is approximately twice that on ~~prime 5~~. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log2([[7/4]]).

680edo contains a reasonable [[3/1|3rd]] and [[5/1|5th harmonic]], though nowhere near the accuracy of the prior superset of [[34edo]] – [[612edo]]; it borrows 34edo's accurate representation of the interval [[25/24]], implying that the error on the 3rd harmonic is approximately twice that on the 5th harmonic. However, 680edo is most notable for its approximation of the [[7/1|7th harmonic]], 680 being the denominator of a semiconvergent to log2([[7/4]]).

Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high [[prime limit]]s, with great representation of prime 23 (inherited from [[170edo]]) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as [[37/29]], of which 680edo is a weak [[circle]].

Its primes [[11/1|11]], [[13/1|13]], [[17/1|17]], and [[19/1|19]] are all approximated rather badly, but 680edo actually shines in very high [[prime limit]]s, with great representation of prime [[23/1|23]] (inherited from [[170edo]]) and accurate representation of prime [[31/1|31]] as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as [[37/29]], of which 680edo is a weak [[circle]].

As an equal temperament, it [[tempering out|tempers out]] the [[quintosec comma]] in the [[5-limit]] and the [[breedsma]] in the [[7-limit]], [[support]]ing [[decoid]]. In the [[11-limit]], the 680e val, better in overall accuracy, tempers out [[4000/3993]], [[12005/11979]] and [[19712/19683]]; the [[patent val]] tempers out [[5632/5625]] and [[9801/9800]]. In the [[13-limit]], the 680ef val tempers out [[1575/1573]], [[2080/2079]], and [[2200/2197]]; the patent val tempers out [[676/675]], [[1001/1000]], [[1716/1715]], 2080/2079, and [[4096/4095]] among others.

=== Prime harmonics ===

{{Harmonics in equal|680|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 680edo (continued)}}

=== Subsets and supersets ===

Since 680 factors into {{~~factorization~~|~~680~~}}, 680edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.

Since 680 factors into primes as {{nowrap| 23 × 5 × 17 }}, 680edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3.5

| {{Monzo| 47 -15 -10 }}, {{monzo| -5 -32 24 }}

| {{Mapping| 680 1078 1579 }}

| -0.1062

| 0.1061

| 6.01

|-

| 2.3.5.7

| 2401/2400, 67108864/66976875, {{monzo| 8 -20 9 1 }}

| {{Mapping| 680 1078 1579 1909 }}

| -0.0795

| 0.1029

| 5.83

|}

{~~{todo~~:~~expand}~~}

=== 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

|-

| 5

| 63\612

| 111.18

| 16/15

| [[Quintosec]] (680)

|-

| 10

| 5\612

| 8.82

| 225/224

| [[Decoid]] (680)

|}

<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|680}}
+{{ED intro}}
-=== Odd harmonics ===
+== Theory ==
-edo retains a reasonable 3rd and 5th [[harmonic]], though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on [[prime]] 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]).
+edo contains a reasonable [[3/1|3rd]] and [[5/1|5th harmonic]], though nowhere near the accuracy of the prior superset of [[34edo]] – [[612edo]]; it borrows 34edo's accurate representation of the interval [[25/24]], implying that the error on the 3rd harmonic is approximately twice that on the 5th harmonic. However, 680edo is most notable for its approximation of the [[7/1|7th harmonic]], 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]).
-Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high [[prime limit]]s, with great representation of prime 23 (inherited from [[170edo]]) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as [[37/29]], of which 680edo is a weak [[circle]].
+Its primes [[11/1|11]], [[13/1|13]], [[17/1|17]], and [[19/1|19]] are all approximated rather badly, but 680edo actually shines in very high [[prime limit]]s, with great representation of prime [[23/1|23]] (inherited from [[170edo]]) and accurate representation of prime [[31/1|31]] as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as [[37/29]], of which 680edo is a weak [[circle]].
+As an equal temperament, it [[tempering out|tempers out]] the [[quintosec comma]] in the [[5-limit]] and the [[breedsma]] in the [[7-limit]], [[support]]ing [[decoid]]. In the [[11-limit]], the 680e val, better in overall accuracy, tempers out [[4000/3993]], [[12005/11979]] and [[19712/19683]]; the [[patent val]] tempers out [[5632/5625]] and [[9801/9800]]. In the [[13-limit]], the 680ef val tempers out [[1575/1573]], [[2080/2079]], and [[2200/2197]]; the patent val tempers out [[676/675]], [[1001/1000]], [[1716/1715]], 2080/2079, and [[4096/4095]] among others.
+=== Prime harmonics ===
 {{Harmonics in equal|680|columns=11}}
 {{Harmonics in equal|680|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 680edo (continued)}}
 === Subsets and supersets ===
-Since 680 factors into {{factorization|680}}, 680edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.
+Since 680 factors into primes as {{nowrap| 2<sup>3</sup> × 5 × 17 }}, 680edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.
+== 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| 47 -15 -10 }}, {{monzo| -5 -32 24 }}
+| {{Mapping| 680 1078 1579 }}
+| -0.1062
+| 0.1061
+| 6.01
+|-
+| 2.3.5.7
+| 2401/2400, 67108864/66976875, {{monzo| 8 -20 9 1 }}
+| {{Mapping| 680 1078 1579 1909 }}
+| -0.0795
+| 0.1029
+| 5.83
+|}
-{{todo:expand}}
+=== 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
+|-
+| 5
+| 63\612
+| 111.18
+| 16/15
+| [[Quintosec]] (680)
+|-
+| 10
+| 5\612
+| 8.82
+| 225/224
+| [[Decoid]] (680)
+|}
+<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]