1178edo: Difference between revisions

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== Theory ==

1178edo is a very strong 19-limit system, ~~and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also~~ [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[~~comma basis~~|~~basis~~]] ~~for its~~ 19-limit [[~~comma~~]]s ~~consists of~~ [[~~2500/2499~~]], [[3025/3024]], ~~3250~~/~~3249~~, ~~4200~~/~~4199~~, [[4225/4224]], [[~~4375~~/~~4374~~]], ~~and~~ [[4914/4913]]~~. It~~ [[~~support~~]]s and ~~provides a great tuning for~~ [[~~semihemienneadecal~~]].

1178edo is a very strong [[19-limit]] system, [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It is a [[zeta peak edo|zeta peak]], [[zeta integral edo|integral]] and [[zeta gap edo|gap edo]].

As an equal temperament, it tempers out {{monzo| -14 -19-19 }} ([[enneadeca]]) in the [[5-limit]], 4375/4374 ([[ragisma]]) and 703125/702464 ([[meter]]) in the [[7-limit]], so that it [[support]]s [[enneadecal]]. In the [[11-limit]] it tempers out [[3025/3024]], [[9801/9800]], and [[234375/234256]], supporting [[hemienneadecal]], and in the [[13-limit]] [[4225/4224]] and [[10648/10647]], supporting and providing a great tuning for [[semihemienneadecal]]. It further tempers out [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[3250/3249]], [[4200/4199]] in the 19-limit; and [[2025/2024]] among others in the [[23-limit]].

=== Prime harmonics ===

{{Harmonics in equal|1178|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 1178edo (continued)}}

=== Subsets and supersets ===

Since 1178 factors into {{~~factorization~~|~~1178~~}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

Since 1178 factors into primes as {{nowrap| 2 × 19 × 31 }}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==

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| 0.0318

| 3.12

|-

| 2.3.5.7.11.13.17.19.23

| 2025/2024, 2500/2499, 3025/3024, 3060/3059, 3250/3249, 3520/3519, 4200/4199, 4375/4374

| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 5004 5329 }}

| +0.0292

| 0.0436

| 4.28

|}

* 1178et holds the record of lowest relative error in the 19-limit, being the first to beat [[742edo|742]] in relative error and [[954edo|954h]] in absolute error, before getting superseded by [[1578edo|1578]] in both.

* It holds the record of lowest absolute errors in the 13-, 17-, and 23-limit, after 954, 954, [[1106edo|1106]] and before [[1236edo|1236]], 1236, and [[1308edo|1308]], respectively.

=== Rank-2 temperaments ===

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|-

| 19

| ~~489\1178 (~~7\1178)

| 7\1178

| ~~498.13 (~~7.13)

| 7.13

| ~~4/3 (~~225/224)

| 225/224

| [[Enneadecal]]

|-

~~| 31~~

~~| 581\1178 (11\1178)~~

~~| 591.851 (11.205)~~

~~| 936/665 (?)~~

~~| [[31st-octave temperaments#217 & 1178|217 & 1178]]~~

|-

| 38

| ~~260\1178 (~~12\1178)

| 12\1178

| ~~264.86 (~~12.22)

| 12.22

| ~~500/429 (~~144/143)

| 144/143

| [[Semihemienneadecal]]

|-

| 38

| ~~489\1178 (~~7\1178)

| 7\1178

| ~~498.13 (~~7.13)

| 7.13

| ~~4/3 (~~225/224)

| 225/224

| [[Hemienneadecal]]

|}

<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~~

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

== Music ==

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]].
+edo is a very strong [[19-limit]] system, [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It is a [[zeta peak edo|zeta peak]], [[zeta integral edo|integral]] and [[zeta gap edo|gap edo]].
+As an equal temperament, it tempers out {{monzo| -14 -19-19 }} ([[enneadeca]]) in the [[5-limit]], 4375/4374 ([[ragisma]]) and 703125/702464  ([[meter]]) in the [[7-limit]], so that it [[support]]s [[enneadecal]]. In the [[11-limit]] it tempers out [[3025/3024]], [[9801/9800]], and [[234375/234256]], supporting [[hemienneadecal]], and in the [[13-limit]] [[4225/4224]] and [[10648/10647]], supporting and providing a great tuning for [[semihemienneadecal]]. It further tempers out [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[3250/3249]], [[4200/4199]] in the 19-limit; and [[2025/2024]] among others in the [[23-limit]].
 === Prime harmonics ===
 {{Harmonics in equal|1178|columns=11}}
+{{Harmonics in equal|1178|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 1178edo (continued)}}
 === Subsets and supersets ===
-Since 1178 factors into {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.
+Since 1178 factors into primes as {{nowrap| 2 × 19 × 31 }}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.
 == Regular temperament properties ==
@@ Line 71: / Line 74: @@
 | 0.0318
 | 3.12
+|-
+| 2.3.5.7.11.13.17.19.23
+| 2025/2024, 2500/2499, 3025/3024, 3060/3059, 3250/3249, 3520/3519, 4200/4199, 4375/4374
+| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 5004 5329 }}
+| +0.0292
+| 0.0436
+| 4.28
 |}
+* 1178et holds the record of lowest relative error in the 19-limit, being the first to beat [[742edo|742]] in relative error and [[954edo|954h]] in absolute error, before getting superseded by [[1578edo|1578]] in both.
+* It holds the record of lowest absolute errors in the 13-, 17-, and 23-limit, after 954, 954, [[1106edo|1106]] and before [[1236edo|1236]], 1236, and [[1308edo|1308]], respectively.
 === Rank-2 temperaments ===
@@ Line 90: / Line 102: @@
 |-
 | 19
-| 489\1178<br>(7\1178)
+| 7\1178
-| 498.13<br>(7.13)
+| 7.13
-| 4/3<br>(225/224)
+| 225/224
 | [[Enneadecal]]
-|-
-| 31
-| 581\1178<br>(11\1178)
-| 591.851<br>(11.205)
-| 936/665<br>(?)
-| [[31st-octave temperaments#217 & 1178|217 & 1178]]
 |-
 | 38
-| 260\1178<br>(12\1178)
+| 12\1178
-| 264.86<br>(12.22)
+| 12.22
-| 500/429<br>(144/143)
+| 144/143
 | [[Semihemienneadecal]]
 |-
 | 38
-| 489\1178<br>(7\1178)
+| 7\1178
-| 498.13<br>(7.13)
+| 7.13
-| 4/3<br>(225/224)
+| 225/224
 | [[Hemienneadecal]]
 |}
-<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
+<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
 == Music ==