Xenharmonic Wiki - User contributions [en]

13edt

2026-05-22T16:47:26Z

Zhenlige: “3/1-equave-7-limit” doesn't worth a page

{{Infobox ET}}
[[File:13edt.png|thumb|alt=13edt.png|A plot of the [[the Riemann zeta function and tuning#Removing primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]

'''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.

It provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as [[Sensamagic_clan#Bohpier|bohpier temperament]]. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 ([[26edt]], [[39edt]], and [[52edt]]) come to the fore.

13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].

In the [[no-2]] [[3/1]]-[[equave]]-[[7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].

== Theory ==
{{Harmonics in equal|13|3|1|prec=2|intervals=odd}}
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|start=12}}

* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]

== Intervals ==
{{Main|Intervals of BP}}

{| class="wikitable center-all right-2 right-3"
|-
! Steps
! [[Cent]]s
! [[Hekt]]s
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
! Corresponding 3.5.7 subgroup intervals
! [[Lambda ups and downs notation|Lambda]] (sLsLsLsLs, {{nowrap|E {{=}} 1/1}})
|-
| 0
| 0
| 0
| P1
| 1/1
| E
|-
| 1
| 146.3
| 100
| A1/m2
| [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}})
| F
|-
| 2
| 292.6
| 200
| M2/d3
| [[25/21]] (−9.2{{c}})
| F#, Gb
|-
| 3
| 438.9
| 300
| A2/P3/d4
| [[9/7]] (+3.8{{c}})
| G
|-
| 4
| 585.2
| 400
| A3/m4/d5
| [[7/5]] (+2.7{{c}})
| H
|-
| 5
| 731.5
| 500
| M4/m5
| [[75/49]] (−5.4{{c}})
| H#, Jb
|-
| 6
| 877.8
| 600
| A4/M5
| [[5/3]] (−6.5{{c}})
| J
|-
| 7
| 1024.1
| 700
| A5/m6/d7
| [[9/5]] (+6.5{{c}})
| A
|-
| 8
| 1170.4
| 800
| M6/m7
| [[49/25]] (+5.4{{c}})
| A#, Bb
|-
| 9
| 1316.7
| 900
| A6/M7/d8
| [[15/7]] (−2.7{{c}})
| B
|-
| 10
| 1463.0
| 1000
| P8/d9
| [[7/3]] (−3.8{{c}})
| C
|-
| 11
| 1609.3
| 1100
| A8/m9
| [[63/25]] (+9.2{{c}})
| C#, Db
|-
| 12
| 1755.7
| 1200
| M9/d10
| [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}})
| D
|-
| 13
| 1902.0
| 1300
| A9/P10
| [[3/1]]
| E
|}

== JI approximation ==
[[File:13ed3-17-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal Equave stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 3.5.7
| 245/243, 3125/3087
| [{{val| 13 19 23 }}] (b13)
| +1.393
| 1.150
| 0.79
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all right-3 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per tritave
! Generator (reduced)
! Cents (reduced)
! Associated ratio
! Temperament
|-
| 1
| 1\13
| 146.30
| 49/45
| [[Procyon]]
|-
| 1
| 2\13
| 292.61
| 25/21
| [[Sirius]]
|-
| 1
| 3\13
| 438.91
| 9/7
| [[BPS]]
|-
| 1
| 4\13
| 585.22
| 7/5
| [[Canopus]]
|-
| 1
| 5\13
| 731.63
| 75/49
|
|-
| 1
| 6\13
| 877.83
| 5/3
| [[Arcturus]]
|}

== See also ==
* [[Bohlen-p_et]]
* [[Catalog of 3.5.7 subgroup rank two temperaments]]
* [[No-twos subgroup temperaments#3.5.7 subgroup temperaments]]
* [[19ed5|19ED5]]: relative ED5
* [[23ed7|23ED7]]: relative ED7

[[Category:Tritave]]
[[Category:Macrotonal]]
[[Category:Nonoctave]]
[[Category:Bohlen–Pierce]]

User:Zhenlige/Chromatic notation/zh-CN

2026-05-08T16:50:00Z

Zhenlige: /* 应用 */

{{Foreign language|Simplified Chinese}}

本页介绍一种基于[[5L 7s]]半音音阶的记谱法。通常不需要增加更多的还原音，因为在纯律（JI）中12音差（[[12-comma]]）远小于[[256/243|3限变化半音]]。对于中庸全音律（meantone）半音音阶[[7L 5s]]，存在两种方式使用本记谱法，类似于使用标准记谱法记录反自然音阶（[[2L 5s|antidiatonic]]）的两种方式。

== 音程 ==
级数使用半音数表示，也可以使用[[12edo]]自然音阶级数，其中增四度和减五度合并为三全音。为避免歧义，[[5L 7s]]的大小增减分别重新命名为宽（'''W'''ide）、窄（'''n'''arrow）、超（'''S'''uper）、次（'''s'''ub）。可以注意到，自然音阶的大音程在此体系下是宽音程，小音程是窄音程。这一点对于四五度同样适用，假如将其看作和其他音程相同的大小宽窄关系。中立音程应缩写为“m”或“~”，避免与表示窄音程的“n”混淆。
{| class="wikitable"
|+ 半音音阶音级
! colspan="3" | 音程
! rowspan="2" | 英语缩写
! colspan="2" | 包含步数
! rowspan="2" | 29edo调音（步）
! rowspan="2" | 对应自然音阶音级
! colspan="2" | 近似比值
|-
! 半音数 !! 基于[[12edo]]自然音阶的 级数名称 !! 性质
! L !! s
! [[Schismatic]] !! [[Garibaldi]]的额外映射
|-
| 0
| 一度
| 纯 || P0s / P1 || 0 || 0 || 0 || '''[[1/1|纯一度]]''' || [[1/1]] ||
|-
| rowspan="2" | 1
| rowspan="2" | 小二度
| 窄 || n1s / nm2 || 0 || 1 || 2 || '''小二度''' || [[135/128]], [[256/243]] || [[21/20]]
|-
| 宽 || W1s / Wm2 || 1 || 0 || 3 || 增一度 || [[16/15]] || [[15/14]]
|-
| rowspan="2" | 2
| rowspan="2" | 大二度
| 窄 || n2s / nM2 || 0 || 2 || 4 || 减三度 || [[10/9]] ||
|-
| 宽 || W2s / WM2 || 1 || 1 || 5 || '''大二度''' || [[9/8]] ||
|-
| rowspan="2" | 3
| rowspan="2" | 小三度
| 窄 || n3s / nm3 || 1 || 2 || 7 || '''小三度''' || [[32/27]] ||
|-
| 宽 || W3s / Wm3 || 2 || 1 || 8 || 增二度 || [[6/5]] ||
|-
| rowspan="2" | 4
| rowspan="2" | 大三度
| 窄 || n4s / nM3 || 1 || 3 || 9 || 减四度 || [[5/4]] || [[56/45]]
|-
| 宽 || W4s / WM3 || 2 || 2 || 10 || '''大三度''' || [[81/64]] || [[80/63]], [[63/50]]
|-
| rowspan="2" | 5
| rowspan="2" | 四度
| 纯 || P5s / P4 || 2 || 3 || 12 || '''纯四度''' || [[4/3]] ||
|-
| 超 || S5s / S4 || 3 || 2 || 13 || 增三度 || [[27/20]] ||
|-
| rowspan="2" | 6
| rowspan="2" | 三全音（[[Tritone]]）
| 窄 || n6s / nT || 2 || 4 || 14 || '''减五度''' || [[45/32]] || [[7/5]]
|-
| 宽 || W6s / WT || 3 || 3 || 15 || '''增四度''' || [[64/45]] || [[10/7]]
|-
| rowspan="2" | 7
| rowspan="2" | 五度
| 次 || s7s / s5 || 2 || 5 || 16 || 减六度 || [[40/27]] ||
|-
| 纯 || P7s / P5 || 3 || 4 || 17 || '''[[Perfect fifth (diatonic interval category)|纯五度]]''' || [[3/2]] ||
|-
| rowspan="2" | 8
| rowspan="2" | 小六度
| 窄 || n8s / nm6 || 3 || 5 || 19 || '''小六度''' || [[128/81]] || [[63/40]], [[100/63]]
|-
| 宽 || W8s / Wm6 || 4 || 4 || 20 || 增五度 || [[8/5]] || [[45/28]]
|-
| rowspan="2" | 9
| rowspan="2" | 大六度
| 窄 || n9s / nM6 || 3 || 6 || 21 || 减七度 || [[5/3]] ||
|-
| 宽 || W9s / WM6 || 4 || 5 || 22 || '''大六度''' || [[27/16]] ||
|-
| rowspan="2" | 10
| rowspan="2" | 小七度
| 窄 || n10s / nm7 || 4 || 6 || 24 || '''小七度''' || [[16/9]] ||
|-
| 宽 || W10s / Wm7 || 5 || 5 || 25 || 增六度 || [[9/5]] ||
|-
| rowspan="2" | 11
| rowspan="2" | 大七度
| 窄 || n11s / nM7 || 4 || 7 || 26 || 减八度 || [[15/8]] || [[28/15]]
|-
| 宽 || W11s / WM7 || 5 || 6 || 27 || '''大七度''' || [[243/128]], [[256/135]] || [[40/21]]
|-
| 12
| 八度
| 纯 || P12s / P8 || 5 || 7 || 29 || '''[[2/1|纯八度]]''' || [[2/1]] ||
|}

使用基于12edo的音程名称，五度链可表示为：P1-P5-WM2-WM6-WM3-WM7-WT-Wm2-Wm6-Wm3-Wm7-S4-S1-S5-SM2-……。音程性质依次为：纯—宽大—宽小—超—超大—超小—倍超等等。

== 音名 ==
使用拉丁字母ABCDEFG和希腊字母αβγδε。拉丁字母表示的音高与标准记谱法相同。希腊字母表示相应拉丁字母下方[[2187/2048|增一度]]的音高。使用上下音差符号表示变化[[12-comma|12点差]]。
{| class="wikitable"
|-
! 半音音阶 !! 自然音阶
|-
| '''D''' || '''D'''
|-
| '''ε''' || ♭E
|-
| ^ε || ♯D
|-
| '''E''' || '''E'''
|-
| '''F''' || '''F'''
|-
| '''γ''' || ♭G
|-
| ^γ || ♯F
|-
| '''G''' || '''G'''
|-
| '''α''' || ♭A
|-
| ^α || ♯G
|-
| '''A''' || '''A'''
|-
| '''β''' || ♭B
|-
| ^β || ♯A
|-
| '''B''' || '''B'''
|-
| '''C''' || '''C'''
|-
| '''δ''' || ♭D
|-
| ^δ || ♯C
|-
| '''D''' || '''D'''
|}

== 谱表 ==
使用九线谱。为提升可读性，谱线绘制为粗细交替。第一线和第九（最高）线为粗线。只使用中音（C）谱号。C永远位于粗线上。标准谱号下，谱线上的音高为：'''E'''γ'''α'''β'''C'''D'''E'''γ'''α'''。

== 应用 ==
半音音阶记谱法适合以下性质的调律：需要通过叠加很多纯五度以得到某些质数音程，使得自然音阶记谱法不直观，或部分音程位于自然音级之间，如半八度。

{| class="wikitable"
|+ 部分调律的音程映射（12edo自然音阶命名）
! 音程 !! [[Ponta]] !! [[Cassandra]] !! [[Leapday]] !! [[Diaschismic]]
|-
| [[5/4]] || 窄大三度（nM3） || 窄大三度（nM3） || 超小三度（Sm3） || 中大三度（~M3）
|-
| [[7/4]] || 三重超大六度（3SM6） || 次小七度（sm7） || 超大六度（SM6） || 半次小七度（1/2-sm7）
|-
| [[11/8]] || 七重次五度（7s5） || 倍超四度（2S4） || 超四度（S4） || 半次三全音（1/2-sT）
|-
| [[13/8]] || || 超小六度（Sm6） || 宽小六度（Wm6） || 半次大六度（1/2-sM6）
|-
| [[17/16]] || || || 倍超一度（2S1） || 中小二度（~m2）
|}

{| class="wikitable"
|+ 部分调律的音程映射（半音数命名）
! 音程 !! [[Ponta]] !! [[Cassandra]] !! [[Leapday]] !! [[Diaschismic]]
|-
| [[5/4]] || 窄4半音（n4s） || 窄4半音（n4s） || 超3半音（S3s） || 中4半音（~4s）
|-
| [[7/4]] || 三重超9半音（3S9s） || 次10半音（s10s） || 超9半音（S9s） || 半次10半音（1/2-s10s）
|-
| [[11/8]] || 七重次7半音（7s7s） || 倍超5半音（2S5s） || 超5半音（S5s） || 半次6半音（1/2-s6s）
|-
| [[13/8]] || || 超8半音（S8s） || 宽8半音（W8s） || 半次9半音（1/2-s9s）
|-
| [[17/16]] || || || 倍超0半音（2S0s） || 中小1半音（~1s）
|}

User:Zhenlige/Chromatic notation/zh-CN

2026-04-26T10:59:41Z

Zhenlige: /* 音名 */

{{Foreign language|Simplified Chinese}}

本页介绍一种基于[[5L 7s]]半音音阶的记谱法。通常不需要增加更多的还原音，因为在纯律（JI）中12音差（[[12-comma]]）远小于[[256/243|3限变化半音]]。对于中庸全音律（meantone）半音音阶[[7L 5s]]，存在两种方式使用本记谱法，类似于使用标准记谱法记录反自然音阶（[[2L 5s|antidiatonic]]）的两种方式。

== 音程 ==
级数使用半音数表示，也可以使用[[12edo]]自然音阶级数，其中增四度和减五度合并为三全音。为避免歧义，[[5L 7s]]的大小增减分别重新命名为宽（'''W'''ide）、窄（'''n'''arrow）、超（'''S'''uper）、次（'''s'''ub）。可以注意到，自然音阶的大音程在此体系下是宽音程，小音程是窄音程。这一点对于四五度同样适用，假如将其看作和其他音程相同的大小宽窄关系。中立音程应缩写为“m”或“~”，避免与表示窄音程的“n”混淆。
{| class="wikitable"
|+ 半音音阶音级
! colspan="3" | 音程
! rowspan="2" | 英语缩写
! colspan="2" | 包含步数
! rowspan="2" | 29edo调音（步）
! rowspan="2" | 对应自然音阶音级
! colspan="2" | 近似比值
|-
! 半音数 !! 基于[[12edo]]自然音阶的 级数名称 !! 性质
! L !! s
! [[Schismatic]] !! [[Garibaldi]]的额外映射
|-
| 0
| 一度
| 纯 || P0s / P1 || 0 || 0 || 0 || '''[[1/1|纯一度]]''' || [[1/1]] ||
|-
| rowspan="2" | 1
| rowspan="2" | 小二度
| 窄 || n1s / nm2 || 0 || 1 || 2 || '''小二度''' || [[135/128]], [[256/243]] || [[21/20]]
|-
| 宽 || W1s / Wm2 || 1 || 0 || 3 || 增一度 || [[16/15]] || [[15/14]]
|-
| rowspan="2" | 2
| rowspan="2" | 大二度
| 窄 || n2s / nM2 || 0 || 2 || 4 || 减三度 || [[10/9]] ||
|-
| 宽 || W2s / WM2 || 1 || 1 || 5 || '''大二度''' || [[9/8]] ||
|-
| rowspan="2" | 3
| rowspan="2" | 小三度
| 窄 || n3s / nm3 || 1 || 2 || 7 || '''小三度''' || [[32/27]] ||
|-
| 宽 || W3s / Wm3 || 2 || 1 || 8 || 增二度 || [[6/5]] ||
|-
| rowspan="2" | 4
| rowspan="2" | 大三度
| 窄 || n4s / nM3 || 1 || 3 || 9 || 减四度 || [[5/4]] || [[56/45]]
|-
| 宽 || W4s / WM3 || 2 || 2 || 10 || '''大三度''' || [[81/64]] || [[80/63]], [[63/50]]
|-
| rowspan="2" | 5
| rowspan="2" | 四度
| 纯 || P5s / P4 || 2 || 3 || 12 || '''纯四度''' || [[4/3]] ||
|-
| 超 || S5s / S4 || 3 || 2 || 13 || 增三度 || [[27/20]] ||
|-
| rowspan="2" | 6
| rowspan="2" | 三全音（[[Tritone]]）
| 窄 || n6s / nT || 2 || 4 || 14 || '''减五度''' || [[45/32]] || [[7/5]]
|-
| 宽 || W6s / WT || 3 || 3 || 15 || '''增四度''' || [[64/45]] || [[10/7]]
|-
| rowspan="2" | 7
| rowspan="2" | 五度
| 次 || s7s / s5 || 2 || 5 || 16 || 减六度 || [[40/27]] ||
|-
| 纯 || P7s / P5 || 3 || 4 || 17 || '''[[Perfect fifth (diatonic interval category)|纯五度]]''' || [[3/2]] ||
|-
| rowspan="2" | 8
| rowspan="2" | 小六度
| 窄 || n8s / nm6 || 3 || 5 || 19 || '''小六度''' || [[128/81]] || [[63/40]], [[100/63]]
|-
| 宽 || W8s / Wm6 || 4 || 4 || 20 || 增五度 || [[8/5]] || [[45/28]]
|-
| rowspan="2" | 9
| rowspan="2" | 大六度
| 窄 || n9s / nM6 || 3 || 6 || 21 || 减七度 || [[5/3]] ||
|-
| 宽 || W9s / WM6 || 4 || 5 || 22 || '''大六度''' || [[27/16]] ||
|-
| rowspan="2" | 10
| rowspan="2" | 小七度
| 窄 || n10s / nm7 || 4 || 6 || 24 || '''小七度''' || [[16/9]] ||
|-
| 宽 || W10s / Wm7 || 5 || 5 || 25 || 增六度 || [[9/5]] ||
|-
| rowspan="2" | 11
| rowspan="2" | 大七度
| 窄 || n11s / nM7 || 4 || 7 || 26 || 减八度 || [[15/8]] || [[28/15]]
|-
| 宽 || W11s / WM7 || 5 || 6 || 27 || '''大七度''' || [[243/128]], [[256/135]] || [[40/21]]
|-
| 12
| 八度
| 纯 || P12s / P8 || 5 || 7 || 29 || '''[[2/1|纯八度]]''' || [[2/1]] ||
|}

使用基于12edo的音程名称，五度链可表示为：P1-P5-WM2-WM6-WM3-WM7-WT-Wm2-Wm6-Wm3-Wm7-S4-S1-S5-SM2-……。音程性质依次为：纯—宽大—宽小—超—超大—超小—倍超等等。

== 音名 ==
使用拉丁字母ABCDEFG和希腊字母αβγδε。拉丁字母表示的音高与标准记谱法相同。希腊字母表示相应拉丁字母下方[[2187/2048|增一度]]的音高。使用上下音差符号表示变化[[12-comma|12点差]]。
{| class="wikitable"
|-
! 半音音阶 !! 自然音阶
|-
| '''D''' || '''D'''
|-
| '''ε''' || ♭E
|-
| ^ε || ♯D
|-
| '''E''' || '''E'''
|-
| '''F''' || '''F'''
|-
| '''γ''' || ♭G
|-
| ^γ || ♯F
|-
| '''G''' || '''G'''
|-
| '''α''' || ♭A
|-
| ^α || ♯G
|-
| '''A''' || '''A'''
|-
| '''β''' || ♭B
|-
| ^β || ♯A
|-
| '''B''' || '''B'''
|-
| '''C''' || '''C'''
|-
| '''δ''' || ♭D
|-
| ^δ || ♯C
|-
| '''D''' || '''D'''
|}

== 谱表 ==
使用九线谱。为提升可读性，谱线绘制为粗细交替。第一线和第九（最高）线为粗线。只使用中音（C）谱号。C永远位于粗线上。标准谱号下，谱线上的音高为：'''E'''γ'''α'''β'''C'''D'''E'''γ'''α'''。

== 应用 ==
半音音阶记谱法适合以下性质的调律：需要通过叠加很多纯五度以得到某些质数音程，使得自然音阶记谱法不直观，或部分音程位于自然音级之间，如半八度。

{| class="wikitable"
|+ 部分调律的音程映射（12edo自然音阶命名）
! 音程 !! [[Ponta]] !! [[Cassandra]] !! [[Leapday]] !! [[Diaschismic]]
|-
| [[5/4]] || 窄大三度（nM3） || 窄大三度（nM3） || 超小三度（Sm3） || 中大三度（~M3）
|-
| [[7/4]] || 三重超大六度（3SM6） || 次小七度（sm7） || 超大三度（SM6） || 半次小七度（1/2-sm7）
|-
| [[11/8]] || 七重次五度（7s5） || 倍超四度（2S4） || 超四度（S4） || 半次三全音（1/2-sT）
|-
| [[13/8]] || || 超小六度（Sm6） || 宽小六度（Wm6） || 半次大六度（1/2-sM6）
|-
| [[17/16]] || || || 倍超一度（2S1） || 中小二度（~m2）
|}

{| class="wikitable"
|+ 部分调律的音程映射（半音数命名）
! 音程 !! [[Ponta]] !! [[Cassandra]] !! [[Leapday]] !! [[Diaschismic]]
|-
| [[5/4]] || 窄4半音（n4s） || 窄4半音（n4s） || 超3半音（S3s） || 中4半音（~4s）
|-
| [[7/4]] || 三重超9半音（3S9s） || 次10半音（s10s） || 超9半音（S9s） || 半次10半音（1/2-s10s）
|-
| [[11/8]] || 七重次7半音（7s7s） || 倍超5半音（2S5s） || 超5半音（S5s） || 半次6半音（1/2-s6s）
|-
| [[13/8]] || || 超8半音（S8s） || 宽8半音（W8s） || 半次9半音（1/2-s9s）
|-
| [[17/16]] || || || 倍超0半音（2S0s） || 中小1半音（~1s）
|}

Superpyth

2026-04-24T18:07:27Z

Zhenlige: /* Tuning considerations and optima */simplify explanation

{{interwiki
| en = Superpyth
| de = Superpyth
| es =
| ja =
}}
{{Infobox regtemp
| Title = Archy; superpyth
| Subgroups = 2.3.7, 2.3.5.7, 2.3.5.7.11
| Comma basis = [[64/63]] (2.3.7); [[64/63]], [[245/243]] (7-limit); [[64/63]], [[100/99]], [[245/243]] (11-limit)
| Mapping = 1; 1 9 -2 16
| Edo join 1 = 22 | Edo join 2 = 27e
| Generators = 3/2
| Generators tuning = 710.1
| Optimization method = CWE
| Pergen = (P8, P5)
| Color name = Ruti
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[5L 17s]]
| Odd limit 1 = 2.3.7 7 | Mistuning 1 = 9.09 | Complexity 1 = 5
| Odd limit 2 = 9 | Mistuning 2 = 15.27 | Complexity 2 = 12
}}
'''Superpyth''', sometimes called '''archy''' in the [[2.3.7 subgroup]], is a [[regular temperament|temperament]] where the [[generator]] is a [[3/2|perfect fifth]], tuned sharp such that a stack of two perfect fifths [[octave reduction|octave-reduced]] gives a whole tone that represents both [[9/8]] and [[8/7]], [[tempering out]] the septimal comma, [[64/63]]. Likewise, two perfect fourths give a minor seventh that represents both [[7/4]] and [[16/9]], so that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, three fourths reach a minor third that approximates [[7/6]], while four fifths reach a major third that approximates [[9/7]].

Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[22edo|13\22]] (~1/4 septimal comma) and [[27edo|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.

If intervals of [[5/1|5]] are desired, the 5th harmonic is canonically mapped to +9 generators through tempering out [[245/243]], so [[5/4]] is an augmented second (e.g. C–D♯). This mapping equates the pythagorean limma, [[256/243]], to the syntonic [[81/80]], tempering out [[20480/19683]], so that 5/4 is mapped to a major third minus a limma. Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of [[7/1|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.

If intervals of 11 are desired, the canonical way is to map [[11/8]] to +16 generators, or a doubly augmented second (C–D𝄪), tempering out [[100/99]]. A simpler but less accurate way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out [[99/98]]. The latter is called '''suprapyth''', a name coined by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group | ''A few full 11-limit 896/891 temperaments'']</ref>. The [[2.3.7.11 subgroup]] restriction of suprapyth, known as [[supra]], is also notable. The two mappings unite on [[22edo]].

If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out [[91/90]]. In practice, however, this mapping only works in [[27edo]], as flatter tunings swap the sizes of [[13/12]] and [[14/13]]. An alternative mapping is –14 generators (known as [[uberpyth]]), or a doubly diminished octave (C–C𝄫), by tempering out [[144/143]]. This has a more flexible range, but the 13 tends to be tuned very sharp except in 27edo, as [[13/8]] is equated with [[18/11]]. A more practical option is to split the sharp ~3/2 into two ~[[16/13]]'s, which results in [[beatles]], and has an alternative mapping of primes 5 and 11. Alternatively, one can keep the superpyth mappings of 5 and 11 to get [[archytas clan #Thomas|thomas]].

[[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.

For more technical data, see [[Archytas clan #Superpyth]].

== Interval chains ==
In these tables, odd harmonics and subharmonics 1–11 are in '''bold'''.

<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Archy (2.3.7)
|-
! # !! Cents* !! Approximate ratios
|-
| 0 || 0.0 || '''1/1'''
|-
| 1 || 709.4 || '''3/2'''
|-
| 2 || 218.8 || '''8/7''', '''9/8'''
|-
| 3 || 928.2 || 12/7
|-
| 4 || 437.6 || 9/7
|-
| 5 || 1147.0 || 27/14
|-
| 6 || 656.3 || 72/49, 81/56
|-
| 7 || 165.7 || 54/49
|}
<nowiki/>* In 2.3.7-subgroup [[CWE]] tuning, octave reduced
</div></div>
<div><div style="display: inline-grid;">
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Full 7-limit superpyth
|-
! rowspan="3" | # !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
|-
! rowspan="2" | 7-limit !! colspan="2" | 11-limit extensions
|-
! Superpyth !! Suprapyth
|-
| 0 || 0.0 || '''1/1''' || ||
|-
| 1 || 710.1 || '''3/2''' || ||
|-
| 2 || 220.2 || '''8/7''', '''9/8''' || ||
|-
| 3 || 930.4 || 12/7 || ||
|-
| 4 || 440.5 || 9/7 || || 14/11
|-
| 5 || 1150.6 || 27/14, 35/18 || 88/45 || 21/11, 64/33
|-
| 6 || 660.7 || 35/24, 40/27 || 22/15 || '''16/11'''
|-
| 7 || 170.8 || 10/9 || 11/10 || 12/11
|-
| 8 || 881.0 || 5/3 || 33/20 || 18/11
|-
| 9 || 391.1 || '''5/4''' || || 27/22
|-
| 10 || 1101.2 || 15/8, 40/21 || ||
|-
| 11 || 611.3 || 10/7 || ||
|-
| 12 || 121.4 || 15/14 || ||
|-
| 13 || 831.6 || 45/28 || 44/27 ||
|-
| 14 || 341.7 || 60/49 || 11/9 || 40/33
|-
| 15 || 1051.8 || 50/27 || 11/6 || 20/11
|-
| 16 || 561.9 || 25/18 || '''11/8''' || 15/11
|-
| 17 || 72.0 || 25/24 || 22/21, 33/32 || 45/44
|-
| 18 || 782.1 || 25/16 || 11/7 ||
|-
| 19 || 292.3 || 25/21 || 33/28 ||
|-
| 20 || 1002.4 || 25/14 || ||
|-
| 21 || 512.5 || 75/56 || ||
|-
| 22 || 22.6 || 50/49, 225/224 || 99/98 ||
|}
<nowiki/>* In 7-limit CWE tuning, octave reduced
</div></div>

== Notation ==
Because superpyth is generated by an octave and a fifth, [[chain-of-fifths notation]] can be used. However, if one wishes to use the 5-limit triads as bases for harmony, then much of the logic which is used in [[meantone]] cannot be used in superpyth, as superpyth does not temper out [[81/80]]. For example, the [[4:5:6|classical major triad]] on C is written as C–D♯–G rather than C–E–G as in meantone, which is awkward to notate and conceptualize. To solve this, one may want to adopt a pair of accidentals (such as ^ and v) to represent modifications by 81/80, thus notating the major triad as C–vE–G and the minor triad as C–^E♭–G. 81/80 is equated to [[28/27]], [[36/35]], and [[256/243]] in superpyth, leading to the equivalences {{nowrap| ^C {{=}} D♭ }}, {{nowrap| E {{=}} vF }}, etc. The limma (C–D♭) thus becomes the most important interval for note alterations, being around a quartertone in size and representing so many important ratios, rather than the apotome (C–C♯) as in meantone, which is a submajor second in size in superpyth.

== Chords and harmony ==
{{See also| Chords of superpyth }}

Superpyth contains a version of the [[5L 2s|diatonic]] scale where the major third represents [[9/7]], and the minor third represents [[7/6]]. {{W|Tertian harmony}} can thus be used, with the major and minor triads representing [[14:18:21|1–9/7–3/2]] and [[6:7:9|1–7/6–3/2]] respectively, rather than the 1–5/4–3/2 and [[10:12:15|1–6/5–3/2]] triads in meantone. However, the contrast between these triads is not as effective as the contrast between the meantone triads, as the interval between 7/6 and 9/7 is too wide, being ~140–180{{c}} in size rather than the ideal ~60–80{{c}} semitone in meantone.

Perhaps a more interesting approach is for the tonic chords of superpyth to be considered the tetrad 1–7/6–4/3–3/2 ([[6:7:8:9]]) and its utonal inverse 1–9/8–9/7–3/2 (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of {{w|Steely Dan}} fame). Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add 1–9/8–4/3–3/2 (a sus2-4 chord). These three chords comprise the three ways to divide the superpyth perfect fifth into two whole tones and one septimal minor third. In the diatonic major scale, the 1–7/6–4/3–3/2 chord occurs on II, III, and VI, while its inverse occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense, though in a different way from in [[mavila]]. [[Chromatic]] or [[enharmonic]] alterations of them also exist, for example, the 1–9/8–9/7–3/2 chord may be altered to 1–9/8–11/8–3/2 (8:9:11:12), which is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 1–9/8–9/7–3/2.

Another approach takes account of the fact that, in the 5-limit, the major triad can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, with the minor triad being its utonal inversion. A similar construction of septimal chords gives us 1–7/6–4/3 ([[6:7:8]]) and its inversion 1–8/7–4/3 ([[21:24:28]]). These intervals contrast by [[49/48]], similarly to how 5-limit thirds contrast by [[25/24]]. There are some issues, however. For example, the 6:7:8 chord has the root on the top rather than the bottom, and the notes may clash from being too close to each other. However, the wide voicing of these chords, those being 1–7/4–3 (4:7:12) and 1–12/7–3 (7:12:21), solve both of these issues. These triads span a twelfth. In terms of the [[chain of fifths]], these chords are simpler in superpyth than the 5-limit triads in meantone.

Therefore, it may be helpful to also consider the [[9-odd-limit]] [[anomalous saturated suspension|saturated suspensions]], 1–7/6–3/2–7/4 ([[12:14:18:21]]) and 1–9/7–3/2–12/7 ([[14:18:21:24]]), which extend the chords above and are good for creating tensions and resolutions: 1–9/7–3/2–12/7 on the fifth degree creates a leading tone that wants to go to the tonic; 1–7/6–3/2–7/4 on the fourth degree creates a flat sixth that wants to go to the fifth.

In meantone, the dominant seventh chord, a tempering of [[20:25:30:36|1–5/4–3/2–9/5]], is often used on the dominant to resolve to the tonic. A similar chord in superpyth is [[28:36:42:49|1–9/7–3/2–7/4]], with a leading tone at 9/7 above the perfect fifth, or [[27/14]]. This chord contains a [[49/36]] [[tritone]] between the 9/7 and 7/4, which creates tension in the chord. (However, 11-limit superpyth maps it to [[15/11]], making it a [[15-odd-limit]] [[swetismic chords|swetismic essentially tempered chord]].)

== Scales ==
; 5-note mos ([[2L 3s]], proper)

In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around [[5edo]]. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~[[28/27]] becoming a chroma. See [[Pentatonic Functional Just System]] for a further explanation of what such a system would look like.

; 7-note mos ([[5L 2s]], improper)

In contrast to the meantone diatonic scale, the superpyth diatonic is improper. Since the fifth is sharp rather than flat in meantone, the large steps (major seconds) are wider, being around 220{{c}} in size. The small steps (minor seconds) are thus narrower, being around 50{{c}} (a quartertone) wide. This has the effect of large and small steps being more distinct compared to meantone diatonic, as well as stronger leading tones due to narrower small steps, though one may want to bend the leading tone down by a small step to avoid it being too close to the tonic.

; 12-note mos ([[5L 7s]], improper)

The superpyth chromatic scale is also improper (the boundary of propriety is [[17edo]]). This scale is the first that contains much beyond the 2.3.7-subgroup, with three [[4:5:6]] chords, three [[10:12:15]] chords, one [[4:5:6:7:9]] chord, and one [[140:180:210:252:315|1/(9:7:6:5:4)]] chord.

=== Scala files ===
* [[Archy5]] – in 49edo tuning
* [[Archy7]] – in 49edo tuning
* [[Archy12]] – in 49edo tuning
* [[12-22a]] – in 22edo tuning

== Tunings ==
=== Tuning considerations and optima ===
The fifth of superpyth is supposed to be tuned sharp of just for the accuracy of the overall temperament. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of superpyth's optimal range.

Without tempered octaves, superpyth is of considerably higher damage than meantone, despite it being seen as the "counterpart" of meantone for sharp fifths and septimal thirds. The vanishing comma, 64/63, is not only larger than 81/80, but it must be split over only three intervals (one minor seventh and two perfect fifths), rather than five as in meantone (one major third and four fifths). This can be shown by the fact that 1/5-comma meantone, the meantone tuning with the minimum damage to harmonics 3 and 5, has a tuning error on 3 and 5 of 4.3{{c}}, while 1/3-comma superpyth, the superpyth tuning with the minimum damage to harmonics 3 and 7 (the minimax tuning for the no-5 [[7-odd-limit]] [[tonality diamond]]) has a tuning error on 3 and 7 of 9.1{{c}}, over twice as much. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing others. An additional consideration is the use of tertian triads in conventional diatonic harmony, whereby the 9/7 supermajor third may be more important than it looks from the bare math.

If we focus purely on the [[2.3.7 subgroup]] for now, and as a starting point adopt an approach based on the example of [[quarter-comma meantone]], treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma and 1/4-comma turn out to be logical solutions. In 1/3-comma superpyth, the whole tone leans towards 8/7 so that 3 and 7 are equally sharp and the minor third is tuned to exactly 7/6; 27edo is extremely close to a closed system of 1/3-comma. In 1/4-comma tuning, which is the minimax tuning for the no-5 [[9-odd-limit]], the whole tone is midway between 8/7 and 9/8 so that the 7 is twice as sharp as 3 and that the major third is exactly 9/7; 22edo is very close to a closed circle of 1/4-comma.

In general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; [[71edo]] (709.859{{c}}) and [[93edo]] with its sharp fifth of 709.677{{c}} come very close to forming closed systems of 2/7-comma.

27edo is also the point where superpyth tunes 5/4 to the familiar 400{{c}} major third of [[12edo]], and in sharper tunings different mappings of 5/4 arise with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to [[19edo]] (which represents [[1/3-comma meantone]] and is on the edge between septimal meantone and [[flattone]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]), which map [[7/5]] wider than [[10/7]] in the superpyth mapping. Furthermore, the [[11-limit]] canonical extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11 as well as 7/5 and 10/7, and 27e-edo conflating 11/8 with 7/5. Suprapyth, on the other hand, only works in 22edo, as sharper tunings map [[11/10]] wider than [[12/11]].

Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical "modified meantones" ([[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], …), as they prioritize the tuning of 3/2 more than the accuracy of septimal harmony. The [[supra]] mapping of the [[2.3.7.11 subgroup]], and the quasisuper mapping of 5, work best for tunings in the range of 17c-edo to 22-edo.

A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth.

Finally, it may be noted that the {{w|plastic ratio}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. In fact, it is the tuning that makes ~6:7:8 become +1+1 [[delta-rational]].

=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~3/2 = 712.8606{{c}} (2/5-comma)
| CSEE: ~3/2 = 711.9997{{c}} (7/19-comma)
| POEE: ~3/2 = 709.6343{{c}}
|-
! Tenney
| CTE: ~3/2 = 709.5948{{c}}
| CWE: ~3/2 = 709.3901{{c}}
| POTE: ~3/2 = 709.3213{{c}}
|-
! Benedetti, Wilson
| CBE: ~3/2 = 707.7286{{c}} (18/85-comma)
| CSBE: ~3/2 = 707.9869{{c}} (25/113-comma)
| POBE: ~3/2 = 708.6428{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~3/2 = 709.7805{{c}}
| CSEE: ~3/2 = 710.2428{{c}}
| POEE: ~3/2 = 710.4936{{c}}
|-
! Tenney
| CTE: ~3/2 = 709.5907{{c}}
| CWE: ~3/2 = 710.1193{{c}}
| POTE: ~3/2 = 710.2910{{c}}
|-
! Benedetti, Wilson
| CBE: ~3/2 = 709.4859{{c}}
| CSBE: ~3/2 = 710.0321{{c}}
| POBE: ~3/2 = 710.2421{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 709.5143{{c}}
| CWE: ~3/2 = 710.0129{{c}}
| POTE: ~3/2 = 710.1747{{c}}
|}

=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5 superpyth): ~2 = 1200.000{{c}}, ~3/2 = 709.758{{c}}
* DKW (2.3.7 archy): ~2 = 1200.000{{c}}, ~3/2 = 712.585{{c}}

=== Tuning spectra ===
==== Archy ====
{| class="wikitable center-all left-4 left-5"
|-
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Extension
! Comments
|-
| '''[[7edo|4\7]]'''
|
| '''685.714'''
|
| '''Lower bound of 2.3.7-subgroup 9-odd-limit diamond monotone'''
|-
| [[12edo|7\12]]
|
| 700.000
| ↓ Dominant
|
|-
|
| 3/2
| 701.955
|
| Pythagorean tuning
|-
| [[17edo|10\17]]
|
| 705.882
| ↑ Dominant ↓ Quasisuper
|
|-
|
| 81/56
| 706.499
|
| 1/6 comma
|-
|
| 27/14
| 707.408
|
| 1/5 comma
|-
| [[39edo|23\39]]
|
| 707.692
|
| 39d val
|-
|
| 9/7
| 708.771
|
| 1/4 comma, 2.3.7-subgroup 9-odd-limit minimax
|-
| [[22edo|13\22]]
|
| 709.091
| ↑ Quasisuper ↓ Superpyth
|
|-
|
| 49/27
| 709.745
|
| 2/7 comma
|-
| [[49edo|29\49]]
|
| 710.204
|
|
|-
|
| 7/6
| 711.043
|
| 1/3 comma, 2.3.7-subgroup 7-odd-limit minimax
|-
| [[27edo|16\27]]
|
| 711.111
| ↑ Superpyth ↓ Quasiultra
|
|-
| [[59edo|35\59]]
|
| 711.864
|
|
|-
| [[32edo|19\32]]
|
| 712.500
| ↑ Quasiultra ↓ Ultrapyth
|
|-
|
| 49/48
| 712.861
|
| 2/5 comma, 2.3.7-subgroup CEE tuning
|-
| [[37edo|22\37]]
|
| 713.514
|
|
|-
| [[42edo|25\42]]
|
| 714.286
|
|
|-
|
| 7/4
| 715.587
|
| 1/2 comma
|-
| '''[[5edo|3\5]]'''
|
| '''720.000'''
| ↑ Ultrapyth
| '''Upper bound of 2.3.7-subgroup 7- and 9-odd-limit diamond monotone'''
|-
|
| 21/16
| 729.219
|
| Full comma
|}
<nowiki/>* Besides the octave

==== Superpyth ====
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 3/2
| 701.955
| Pythagorean tuning
|-
| [[17edo|10\17]]
|
| 705.882
| 17e val
|-
|
| 9/7
| 708.771
|
|-
|
| 15/8
| 708.807
|
|-
|
| 11/10
| 709.286
|
|-
| '''[[22edo|13\22]]'''
|
| '''709.091'''
| '''Lower bound of 7-, 9-, and 11-odd-limit diamond monotone'''
|-
|
| 11/8
| 709.457
|
|-
|
| 5/4
| 709.590
| 9-odd-limit minimax
|-
| [[71edo|42\71]]
|
| 709.859
| 71d val
|-
|
| 15/14
| 709.954
|
|-
|
| 11/6
| 709.958
|
|-
|
| 25/24
| 710.040
|
|-
|
| 11/7
| 710.138
|
|-
| [[49edo|29\49]]
|
| 710.204
|
|-
|
| 15/11
| 710.508
|
|-
| [[76edo|45\76]]
|
| 710.526
| 76bcdee val
|-
|
| 11/9
| 710.529
|
|-
|
| 5/3
| 710.545
|
|-
|
| 21/11
| 710.620
|
|-
|
| 7/5
| 710.681
| 7-odd-limit minimax
|-
|
| 7/6
| 711.043
|
|-
| '''[[27edo|16\27]]'''
|
| '''711.111'''
| 27e val, '''upper bound of 11-odd-limit diamond monotone'''
|-
|
| 21/20
| 711.553
|
|-
|
| 9/5
| 711.772
|
|-
| [[32edo|19\32]]
|
| 712.500
| 32cee val
|-
|
| 7/4
| 715.587
|
|-
| '''[[5edo|3\5]]'''
|
| '''720.000'''
| 5e val, '''upper bound of 7- and 9-odd-limit diamond monotone'''
|-
|
| 21/16
| 729.219
|
|}
<nowiki/>* Besides the octave

== Music ==
; [[Flora Canou]]
* [https://soundcloud.com/floracanou/prelude-the-triad-challenge?in=floracanou/sets/totmc-suite "Prelude: the Triad Challenge"] from [https://soundcloud.com/floracanou/sets/totmc-suite ''TOTMC Suite''] (2023–2025) – in superpyth, [[70ed6]] tuning

; [[Lillian Hearne]]
* [https://soundcloud.com/lillianhearne/superpyth12-chromatic-riff ''Superpyth{{lbrack}}12{{rbrack}} chromatic riff''] (2015)
* [https://soundcloud.com/lillianhearne/trio-in-superpyth-temperament-for-irish-whistle-cello-and-piano ''Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello''] (2015)
: Both in 22edo tuning

; [[Joel Grant Taylor]]
* [https://web.archive.org/web/20201127013613/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 ''12of22studyPentUp4thsMstr'']
* [https://web.archive.org/web/20201127013450/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 ''12of22gamelan1b'']
* [https://web.archive.org/web/20201127015919/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 ''12of22study3 (children's story)'']
* [https://web.archive.org/web/20201127012539/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 ''12of22study7'']
: All in Superpyth[12], 22edo tuning.

== References ==

[[Category:Superpyth| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Archytas clan]]
[[Category:Sensamagic clan]]
[[Category:Orwellismic temperaments]]

User:Zhenlige/Earth171

2026-04-09T05:07:42Z

Zhenlige: /* History */

{{worldbuilding}}

'''NOTICE: Unless otherwise stated, all texts below this bolded paragraph describes a fictional world that shares the history of the real world until 2025. It may not reflect the development of the real world since 2026.'''
<hr>

The standard tuning is [[171edo]].

== History ==
[[12edo]] was the standard tuning back in the early 21st century. In 2027, a group of microtonalists made a series of videos and articles introducing microtonal music theory. A few months later, they became popular online. More and more people, including musicians, were becoming interested in alternate tunings besides 12edo.

In 2030s, after hundreds of years composing mostly in 12edo, with AI speeding up composition, it became difficult to write new 12edo music. Therefore, many composers thought of microtonal music almost at the same time. Microtonal melody and harmony, especially [[7-limit]] harmony, became common in popular music.

However, there was not a widely-accepted system that describes the new intervals well enough back then. [[Cent]] values appear to be more physical than musical, since they say little about whether an interval is consonant, and [[JI]] ratios are sometimes unflexible for describing the continuous set of possible pitches. The musicians decided to choose a large [[EDO]] that approximates simple JI intervals well as the new standard tuning. Different EDOs were proposed, namely [[53edo|53]], [[72edo|72]], [[99edo|99]], [[130edo|130]], [[140edo|140]], [[159edo|159]], [[171edo|171]], [[224edo|224]], [[270edo|270]] and [[311edo|311]]. Finally 171edo was chosen because it has very accurate 7-limit and its step size is close to melodic [[JND]].

In 2050, a new standard of digital music based on 171edo was released. Since traditional note names were too inconvenient to represent the full 171edo, a new system of naming absolute pitches was released. Notes are named as “note ''x'' octave ''y''”, where ''x'' ranges from 0 to 170. The frequency of the note is <math>16\cdot2^{y+{x\over171}}</math> Hz. The traditional note names ABCDEFG are considered movable.

User:Zhenlige/Earth171

2026-04-09T04:29:25Z

Zhenlige: /* History */

{{worldbuilding}}

'''NOTICE: Unless otherwise stated, all texts below this bolded paragraph describes a fictional world that shares the history of the real world until 2025. It may not reflect the development of the real world since 2026.'''
<hr>

The standard tuning is [[171edo]].

== History ==
[[12edo]] was the standard tuning back in the early 21st century. In 2027, a group of microtonalists made a series of videos and articles introducing microtonal music theory. A few months later, they became popular online. More and more people, including musicians, were becoming interested in alternate tunings besides 12edo.

In 2030s, after hundreds of years composing mostly in 12edo, with AI speeding up composition, it became difficult to write new 12edo music. Therefore, many composers thought of microtonal music almost at the same time. Microtonal melody and harmony, especially [[7-limit]] harmony, became common in popular music.

However, there was not a widely-accepted system that describes the new intervals well enough back then. [[Cent]] values appear to be more physical than musical, since they say little about whether an interval is consonant, and [[JI]] ratios are sometimes unflexible for describing the continuous set of possible pitches. The musicians decided to choose a large [[EDO]] that approximates simple JI intervals well as the new standard tuning. Different EDOs were proposed, namely [[53edo|53]], [[72edo|72]], [[99edo|99]], [[130edo|130]], [[140edo|140]], [[159edo|159]], [[171edo|171]], [[224edo|224]], [[270edo|270]] and [[311edo|311]]. Finally 171edo was chosen because it has very accurate 7-limit and its step size is close to melodic [[JND]].

In 2050, a new standard of digital music based on 171edo was released. Since traditional note names were too inconvenient to represent the full 171edo, a new system of naming absolute pitches was released. Notes are named as “note ''x'' octave ''y''”, where ''x'' ranges from 0 to 170. The frequency of the note is <math>16\cdot2^{x+{y\over171}}</math> Hz. The traditional note names ABCDEFG are considered movable.

User:Zhenlige/Earth171

2026-04-09T04:15:12Z

Zhenlige: /* History */

User:Zhenlige/Earth171

2026-04-09T03:26:30Z

Zhenlige: Created page with "{{worldbuilding}} '''NOTICE: Unless otherwise stated, all texts below this bolded paragraph describes a fictional world that shares the history of the real world until 2025. It may not reflect the development of the real world since 2026.''' <hr> The standard tuning is 171edo. == History == 12edo was the standard..."

User:Zhenlige/171edo scales

2026-03-30T11:59:32Z

Zhenlige:

[[171edo]] has both very accurate [[7-limit]] and a step size just small enough to make intervals off by 1 step still quite acceptable. Therefore, it is very suitable for making scales which can be used as [[well temperament]]s.

== 12wt ==
<pre>
13\171
28\171
42\171
55\171
71\171
84\171
100\171
113\171
127\171
142\171
155\171
2/1
</pre>

== 31wt ==
<pre>
6\171
11\171
17\171
22\171
28\171
33\171
38\171
44\171
50\171
55\171
62\171
66\171
71\171
77\171
83\171
89\171
94\171
100\171
105\171
110\171
116\171
121\171
127\171
133\171
138\171
143\171
149\171
155\171
161\171
165\171
2/1
</pre>

== 34wt ==
<pre>
5\171
10\171
15\171
20\171
25\171
29\171
35\171
40\171
45\171
50\171
55\171
60\171
65\171
71\171
75\171
80\171
85\171
90\171
95\171
100\171
106\171
110\171
116\171
120\171
126\171
130\171
135\171
141\171
145\171
151\171
155\171
161\171
165\171
2/1
</pre>

== Alpha WT ==
<pre>
11\171
22\171
33\171
45\171
55\171
67\171
78\171
89\171
100\171
</pre>

User:Zhenlige/171edo scales

2026-03-30T11:52:48Z

Zhenlige:

2026-03-12T04:40:08Z

Zhenlige:

This is a list of 12-tone [[NEJI]]s designed by me. The goal is to make a scale that satisfy the restrictions below:
*The [[octave]] is just, and all [[3/2|fifths]] are acceptable (≤8 [[cent]]s off from just).
*The least common denominator is small enough (≤256).
*Most notes are in a manageable [[prime limit]].
*It contains some simple just intervals.

== 1 ==
<pre>
(* 12neji-1 *)
(* 192:204:216:228:243:256:272:288:305:324:342:363:384 *)

17/16 "G♯"
9/8 "A"
19/16 "A♯"
81/64 "B"
4/3 "C"
17/12 "C♯"
3/2 "D"
305/192 "D♯"
27/16 "E"
57/32 "F"
121/64 "F♯"
2/1 "G"
</pre>

=== Fifths ===
Assuming G centered:
*One fifth flatten by [[243/242]] (~7.139 cents) at B-F♯
*Two fifths flatten by approximately half of [[153/152]] at G♯-D♯-A♯
**One fifth flatten by 305/304 (~5.686 cents) at D♯-A♯
**One fifth flatten by 306/305 (~5.667 cents) at G♯-D♯
*One fifth flatten by [[513/512]] (~3.378 cents) at F-C
*One fifth flatten by [[1089/1088]] (~1.590 cents) at F♯-C♯
*Seven just fifths at C-G-D-A-E-B, C♯-G♯ and A♯-F

== 2 ==
<pre>
(* 12neji-2 *)
(* 192:204:216:228:242:256:272:288:305:323:342:363:384 *)

17/16 "G♯"
9/8 "A"
19/16 "A♯"
121/96 "B"
4/3 "C"
17/12 "C♯"
3/2 "D"
305/192 "D♯"
323/192 "E"
57/32 "F"
121/64 "F♯"
2/1 "G"
</pre>

=== Fifths ===
Assuming G centered:
*Two fifths flatten by approximately half of [[153/152]] at G♯-D♯-A♯
**One fifth flatten by 305/304 (~5.686 cents) at D♯-A♯
**One fifth flatten by 306/305 (~5.667 cents) at G♯-D♯
*One fifth flatten by [[324/323]] (~5.352 cents) at A-E
*One fifth flatten by [[513/512]] (~3.378 cents) at F-C
*One fifth flatten by [[969/968]] (~1.788 cents) at E-B
*One fifth flatten by [[1089/1088]] (~1.590 cents) at F♯-C♯
*Six just fifths at C-G-D-A, B-F♯, C♯-G♯ and A♯-F

[[Category:Scales]]
[[Category:12-tone scales]]
[[Category:Just intonation scales]]
[[Category:Tempered scales]]

2026-02-19T19:39:14Z

Zhenlige: /* Applications */

This page introduces a proposed notation system centering the [[5L 7s]] chromatic scale. This system is suitable for temperaments with consonant intervals far away in the [[chain of fifths]], such as [[schismatic]]. Adding more naturals is usually unnecessary, since the [[12-comma]] is much smaller than the [[256/243|limma]] in JI. For the meantone chromatic scale [[7L 5s]], there are two approches to use this system, in anology to using the standard notation system for the [[2L 5s|antidiatonic]] scale.

== Intervals ==
The degrees are named as 0-indexed numbers of “sem”, which stands for “'''sem'''itone”, referring to the chromatic step. [[12edo]] diatonic interval names can also be used for representing degrees, with A4 and d5 merged into “tritone”. The [[5L 7s]] major, minor, augmented and diminished are renamed to wide (W), narrow (n), super (S) and sub (s) respectively, to avoid ambiguity. Note that diatonic major is always wide and diatonic minor is always narrow, which also applies to fourths and fifths as if they were named the same as other intervals. Mid or neutral degrees should be abbreviated as “m” or “~” to avoid confusion with “n” which stands for narrow.
{| class="wikitable"
|+ Chromatic interval degrees
! colspan="3" | Interval
! rowspan="2" | Abbrev.
! colspan="2" | Steps
! rowspan="2" | 29edo tuning (steps)
! rowspan="2" | Equivalent diatonic degree
! colspan="2" | Approximate ratios
|-
! Degree !! Degree name from [[12edo]] diatonic !! Quality
! L !! s
! [[schismatic]] !! additional in [[garibaldi]]
|-
| Unison 0-tone 0-sem
| Unison
|Perfect|| P0s / P1 || 0 || 0 || 0 || '''[[1/1|P1]]''' || [[1/1]] ||
|-
| rowspan="2" | Semitone Sem / 1-sem
| rowspan="2" | Minor second
|Narrow|| n1s / nm2 || 0 || 1 || 2 || '''m2''' || [[135/128]], [[256/243]] || [[21/20]]
|-
|Wide|| W1s / Wm2 || 1 || 0 || 3 || A1 || [[16/15]] || [[15/14]]
|-
| rowspan="2" | Tone / 1-tone 2-sem
| rowspan="2" | Major second
|Narrow|| n2s / nM2 || 0 || 2 || 4 || d3 || [[10/9]] ||
|-
|Wide|| W2s / WM2 || 1 || 1 || 5 || '''M2''' || [[9/8]] ||
|-
| rowspan="2" | 3-sem
| rowspan="2" | Minor third
|Narrow|| n3s / nm3 || 1 || 2 || 7 || '''m3''' || [[32/27]] ||
|-
|Wide|| W3s / Wm3 || 2 || 1 || 8 || A2 || [[6/5]] ||
|-
| rowspan="2" | Ditone / 2-tone 4-sem
| rowspan="2" | Major third
|Narrow|| n4s / nM3 || 1 || 3 || 9 || d4 || [[5/4]] || [[56/45]]
|-
|Wide|| W4s / WM3 || 2 || 2 || 10 || '''M3''' || [[81/64]] || [[80/63]], [[63/50]]
|-
| rowspan="2" | 5-sem
| rowspan="2" | Fourth
|Perfect|| P5s / P4 || 2 || 3 || 12 || '''P4''' || [[4/3]] ||
|-
|Super|| S5s / S4 || 3 || 2 || 13 || A3 || [[27/20]] ||
|-
| rowspan="2" | [[Tritone]] / 3-tone 6-sem
| rowspan="2" | Tritone
|Narrow|| n6s / nT || 2 || 4 || 14 || '''d5''' || [[45/32]] || [[7/5]]
|-
|Wide|| W6s / WT || 3 || 3 || 15 || '''A4''' || [[64/45]] || [[10/7]]
|-
| rowspan="2" | 7-sem
| rowspan="2" | Fifth
|Sub|| s7s / s5 || 2 || 5 || 16 || d6 || [[40/27]] ||
|-
|Perfect|| P7s / P5 || 3 || 4 || 17 || '''[[Perfect fifth (diatonic interval category)|P5]]''' || [[3/2]] ||
|-
| rowspan="2" | Tetratone / 4-tone 8-sem
| rowspan="2" | Minor sixth
|Narrow|| n8s / nm6 || 3 || 5 || 19 || '''m6''' || [[128/81]] || [[63/40]], [[100/63]]
|-
|Wide|| W8s / Wm6 || 4 || 4 || 20 || A5 || [[8/5]] || [[45/28]]
|-
| rowspan="2" | 9-sem
| rowspan="2" | Major sixth
|Narrow|| n9s / nM6 || 3 || 6 || 21 || d7 || [[5/3]] ||
|-
|Wide|| W9s / WM6 || 4 || 5 || 22 || '''M6''' || [[27/16]] ||
|-
| rowspan="2" | Pentatone / 5-tone 10-sem
| rowspan="2" | Minor seventh
|Narrow|| n10s / nm7 || 4 || 6 || 24 || '''m7''' || [[16/9]] ||
|-
|Wide|| W10s / Wm7 || 5 || 5 || 25 || A6 || [[9/5]] ||
|-
| rowspan="2" | 11-sem
| rowspan="2" | Major seventh
|Narrow|| n11s / nM7 || 4 || 7 || 26 || d8 || [[15/8]] || [[28/15]]
|-
|Wide|| W11s / WM7 || 5 || 6 || 27 || '''M7''' || [[243/128]], [[256/135]] || [[40/21]]
|-
| Hexatone / 6-tone 12-sem
| Octave
|Perfect|| P12s / P8 || 5 || 7 || 29 || '''[[2/1|P8]]''' || [[2/1]] ||
|}

The 12edo diatonic based interval names along the chain of fifths is P1-P5-WM2-WM6-WM3-WM7-WT-Wm2-Wm6-Wm3-Wm7-S4-S1-S5-SM2-… with the quality goes as perfect - wide major - wide minor - super perfect - super major - super minor - double super perfect….

== Note names ==
Latin letters ABCDEFG and Greek letters αβγδε are used. Latin letters represent the same notes as diatonic. Each Greek letter represents the note a [[2187/2048|diatonic chroma]] below the note marked with the corresponding Latin letter. Ups and downs are used for altering [[12-comma]]s.
{| class="wikitable"
|-
! Chromatic !! Diatonic
|-
| '''D''' || '''D'''
|-
| '''ε''' || E♭
|-
| ^ε || D♯
|-
| '''E''' || '''E'''
|-
| '''F''' || '''F'''
|-
| '''γ''' || G♭
|-
| ^γ || F♯
|-
| '''G''' || '''G'''
|-
| '''α''' || A♭
|-
| ^α || G♯
|-
| '''A''' || '''A'''
|-
| '''β''' || B♭
|-
| ^β || A♯
|-
| '''B''' || '''B'''
|-
| '''C''' || '''C'''
|-
| '''δ''' || D♭
|-
| ^δ || C♯
|-
| '''D''' || '''D'''
|}

== Staves ==
A 9-lined staff is used. To improve readability, the lines alter between thin and thick. The first and last lines are thick. Only C-clef is used. C is always on a thick line. For the standard clef, the notes on the lines are '''E'''γ'''α'''β'''C'''D'''E'''γ'''α'''.

== Applications ==
The chromatic notation is useful for notating temperaments that require many fifths to get some primes, making diatonic notation very unintuitive, or that have interdiatonic degrees such as the half octave.

{| class="wikitable"
|+ Interval mappings in some temperaments (12edo diatonic based names)
! Interval !! [[Ponta]] !! [[Cassandra]] !! [[Leapday]] !! [[Diaschismic]]
|-
| [[5/4]] || nM3 || nM3 || Sm3 || ~M3
|-
| [[7/4]] || 3SM6 || sm7 || SM6 || 1/2-sm7
|-
| [[11/8]] || 7s5 || 2S4 || S4 || 1/2-sT
|-
| [[13/8]] || || Sm6 || Wm6 || 1/2-sM6
|-
| [[17/16]] || || || 2S1 || ~m2
|}

{| class="wikitable"
|+ Interval mappings in some temperaments (n-sem names)
! Interval !! [[Ponta]] !! [[Cassandra]] !! [[Leapday]] !! [[Diaschismic]]
|-
| [[5/4]] || n4s || n4s || S3s || ~4s
|-
| [[7/4]] || 3S9s || s10s || S9s || 1/2-s10s
|-
| [[11/8]] || 7s7s || 2S5s || S5s || 1/2-s6s
|-
| [[13/8]] || || S8s || W8s || 1/2-s9s
|-
| [[17/16]] || || || 2S0s || ~1s
|}