Xenharmonic Wiki - User contributions [en]

Talk:Pythrabian comma

2026-04-30T17:08:46Z

Aura:

== Type of the name ==
Should we treat ''Pythrabian'' as a special comma name like ''Pythagorean''/''Alpharabian comma'' and give it a distinct common name to derive temp names, or also as a common name, which means it can use the second declension (pythrabia – pythrabic – pythrabian – pythrabian comma)? —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:04, 30 April 2026 (UTC)

: Okay, I definitely think that comma deserves to be treated like both the Pythagorean and Alpharabian commas in that it should have a distinct common name from which we can derive temperament names. However, I'm not sure which common name we should use for this. I think we'll have to brainstorm at some point. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:08, 30 April 2026 (UTC)

Frameshift comma

2026-04-09T17:22:28Z

Aura:

{{Infobox Interval
| Ratio = 22876792454961/22866405883904
| Name = frameshift comma
| Color name = Quadla-trilu comma
| Comma = yes
}}

The '''frameshift comma''' is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (2.3.11) comma. It is the difference between:

* 4 [[Apotome|apotomes]] and [[1331/1024]]
* The [[243/242|rastma]] and the [[pythrabian comma]]
* The [[gamelisma]] and 3 [[Symbiotic comma|symbiotic commas]]

== Temperaments ==
Tempering out this comma results in the [[frameshift]] temperament in the 2.3.11. This allows reaching [[11/8]] by a stepwise motion consisting of four apotomes and one [[128/121]] with astoundingly small error. [[Aura]] dubs this the '''frameshift progression''' (see below).

It also splits the fifth into three 22528/19683's – an interval sharp of [[8/7]] by the [[symbiotic comma]]. Naturally, tempering it out results in a [[slendric]] extension.
If both rastma and pythrabian comma are tempered out, this results in [[41edo]] – a trivial tuning of frameshift.

== Etymology ==
Discovered by Aura in 2021, it comes from the fact that both the [[rastmic]] and [[pythrabian]] temperaments are ways of conceptualizing how the 11-prime relates to the 3-prime, these two frameworks for viewing the 11-prime relative to the 3-prime become linked when this comma is tempered out – hence the name "frameshift" comma.[[File:Frameshift Cadence.mp3|thumb|Frameshift progression sample in [[159edo]].]]

[[File:Frameshift Cadence Score.png|thumb|Score for the above sample of the Frameshift progression in 159edo.]]

== See also ==
* [[Unnoticeable comma]]
* [[Frameshift microtemperaments]]

[[Category:Alpharabian]]
[[Category:Pythrabian]]
[[Category:Frameshift]]
[[Category:Chord progressions]]
[[Category:Commas named for their regular temperament properties]]

Schismatic family

2026-04-05T09:00:23Z

Aura: /* Hemiterm */Corrected wrong information

{{interwiki
| en = Schismatic family
| de = Schismatische Temperaturen
| es =
| ja =
}}
{{Technical data page}}
The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]].

== Schismic, schismatic, a.k.a. helmholtz ==
{{Main| Schismic }}

The 5-limit version of the temperament is a [[microtemperament]], called ''schismic'', ''schismatic'', or ''helmholtz''. The generator is a fifth, flattened by a fraction of a schisma, and 5/4 is represented by a diminished fourth. This defies the tradition of {{w|tertian harmony}}, as the [[just major triad]] on C is C–F♭–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.

As a 5-limit system, schismic is far more accurate than [[meantone]] but still with manageable [[complexity]]. [[53edo]] is a possible tuning for schismic, but you need [[118edo]] if you want to get the full effect. In exact analogy with [[1/4-comma meantone]] there is also 1/8 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better fifth, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut. Simply leaving the fifths just would also make for a viable tuning, thus collapsing schismic to a simple relabeling of the 3-limit.

[[Subgroup]]: 2.3.5

[[Comma list]]: 32805/32768

{{Mapping|legend=1| 1 0 15 | 0 1 -8 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}
: [[error map]]: {{val| +0.075 -0.100 -0.027 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}}
: error map: {{val| 0.000 -0.224 -0.160 }}

[[Tuning ranges]]:
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955] (1/8-comma to untempered)

{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }}

[[Badness]] (Sintel): 0.0999

=== Overview to extensions ===
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], [[#Grackle|grackle]] adds {{monzo| -44 26 0 1 }}, [[#Pontiac|pontiac]] adds {{monzo| -59 39 0 -1 }}, and [[#Schism|schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.

[[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.

Temperaments involving larger splits include [[#Tsaharuk|tsaharuk]], [[#Quanharuk|quanharuk]], [[#Quintilipyth|quintilipyth]], [[#Quintaschis|quintaschis]], [[#Altinex|altinex]], [[Stearnsmic clan #Pogo|pogo]], [[#Sextilifourths|sextilifourths]], [[#Octant|octant]], [[#Nonant|nonant]], [[#Septant|septant]], [[#Septiquarschis|septiquarschis]], and [[#Tridecafifths|tridecafifths]]. Those split the schismic structure into five to thirteen parts.

Temperaments discussed elsewhere include:
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]

Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, octant, nonant, septant, septiquarschis, and tridecafifths.

The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].

== Garibaldi ==
{{Main| Garibaldi }}

Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}
: [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}}
: error map: {{val| 0.000 +0.122 -2.933 +2.090 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
: {{monzo list| 1 0 0 0 | 5/3 1/15 0 -1/15 | 5/3 -8/15 0 8/15 | 5/3 -14/15 0 14/15 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: {{monzo list| 1 0 0 0 | 25/16 1/8 0 -1/16 | 5/2 -1 0 1/2 | 25/8 -7/4 0 7/8 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915]

{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94 }}

[[Badness]] (Sintel): 0.548

=== Cassandra ===
Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 2200/2187

Mapping: {{mapping| 1 0 15 25 -33 | 0 1 -8 -14 23 }}

Optimal tunings:
* WE: ~2 = 1200.3089{{c}}, ~3/2 = 702.3377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1562{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]

{{Optimal ET sequence|legend=0| 12e, 41, 53, 94, 229c }}

Badness (Sintel): 0.906

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 325/324, 385/384

Mapping: {{mapping| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}

Optimal tunings:
* WE: ~2 = 1200.1703{{c}}, ~3/2 = 702.2122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1135{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
: unchanged-interval (eigenmonzo) basis: 2.13/7

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]

{{Optimal ET sequence|legend=0| 41, 53, 94, 429ccdeef, 523ccdeef }}

Badness (Sintel): 0.854

===== Cassie =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 120/119, 154/153, 225/224, 273/272, 325/324

Mapping: {{mapping| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}

Optimal tunings:
* WE: ~2 = 1199.8140{{c}}, ~3/2 = 701.9833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0909{{c}}

{{Optimal ET sequence|legend=0| 12e, 41, 53, 94g }}

Badness (Sintel): 1.19

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272

Mapping: {{mapping| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}

Optimal tunings:
* WE: ~2 = 1199.9556{{c}}, ~3/2 = 702.0530{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0787{{c}}

{{Optimal ET sequence|legend=0| 12e, 41, 53 }}

Badness (Sintel): 1.11

===== Cassandric =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 275/273, 325/324, 375/374, 385/384

Mapping: {{mapping| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}

Optimal tunings:
* WE: ~2 = 1200.0046{{c}}, ~3/2 = 702.2167{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0962{{c}}

{{Optimal ET sequence|legend=0| 41g, 53, 94 }}

Badness (Sintel): 1.18

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374

Mapping: {{mapping| 1 0 15 25 -33 -28 77 9 | 0 1 -8 -14 23 20 -46 -3 }}

Optimal tunings:
* WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0967{{c}}

{{Optimal ET sequence|legend=1| 41g, 53, 94 }}

Badness (Sintel): 1.07

====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 209/208, 225/224, 253/252, 275/273, 325/324, 375/374

Mapping: {{mapping| 1 0 15 25 -33 -28 77 9 60 | 0 1 -8 -14 23 20 -46 -3 -35 }}

Optimal tunings:
* WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}}

{{Optimal ET sequence|legend=0| 41g, 53, 94 }}

Badness (Sintel): 1.08

===== Cassander =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 225/224, 275/273, 325/324, 385/384

Mapping: {{mapping| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}

Optimal tunings:
* WE: ~2 = 1200.1986{{c}}, ~3/2 = 702.2598{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}

{{Optimal ET sequence|legend=0| 41, 53g, 94 }}

Badness (Sintel): 1.14

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324

Mapping: {{mapping| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}

Optimal tunings:
* WE: ~2 = 1200.3057{{c}}, ~3/2 = 702.3138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1373{{c}}

{{Optimal ET sequence|legend=0| 41, 53g, 94 }}

Badness (Sintel): 1.07

=== Andromeda ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/242

Mapping: {{mapping| 1 0 15 25 32 | 0 1 -8 -14 -18 }}

Optimal tunings:
* WE: ~2 = 1200.1917{{c}}, ~3/2 = 702.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3599{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]

{{Optimal ET sequence|legend=0| 12, 29, 41 }}

Badness (Sintel): 0.779

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 196/195, 245/242

Mapping: {{mapping| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}

Optimal tunings:
* WE: ~2 = 1200.3031{{c}}, ~3/2 = 702.7368{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5420{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [702.439, 703.448] (24\41 to 17\29)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]

{{Optimal ET sequence|legend=0| 12f, 29, 41 }}

Badness (Sintel): 0.857

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 189/187, 196/195

Mapping: {{mapping| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}

Optimal tunings:
* WE: ~2 = 1199.1984{{c}}, ~3/2 = 701.8424{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3384{{c}}

{{Optimal ET sequence|legend=0| 12f, 29, 41 }}

Badness (Sintel): 1.19

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195

Mapping: {{mapping| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}

Optimal tunings:
* WE: ~2 = 1199.5242{{c}}, ~3/2 = 702.0783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3711{{c}}

{{Optimal ET sequence|legend=0| 12f, 29, 41 }}

Badness (Sintel): 1.17

===== Schisicosiennic =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 154/153, 170/169, 196/195

Mapping: {{mapping| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}

Optimal tunings:
* WE: ~2 = 1200.6122{{c}}, ~3/2 = 703.0830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6968{{c}}

{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}

Badness (Sintel): 1.11

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189

Mapping: {{mapping| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}

Optimal tunings:
* WE: ~2 = 1200.7981{{c}}, ~3/2 = 703.2199{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7221{{c}}

{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}

Badness (Sintel): 1.09

===== Schisicosiennoid =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 85/84, 100/99, 105/104, 119/117, 221/220

Mapping: {{mapping| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}

Optimal tunings:
* WE: ~2 = 1201.3146{{c}}, ~3/2 = 703.4864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6491{{c}}

{{Optimal ET sequence|legend=0| 12f, 29g, 41g }}

Badness (Sintel): 1.06

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152

Mapping: {{mapping| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1201.3140{{c}}, ~3/2 = 703.4860{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6578{{c}}

{{Optimal ET sequence|legend=1| 12f, 29g, 41g }}

Badness (Sintel): 1.02

=== Helenus ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 3125/3087

Mapping: {{mapping| 1 0 15 25 51 | 0 1 -8 -14 -30 }}

Optimal tunings:
* WE: ~2 = 1199.7097{{c}}, ~3/2 = 701.5554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7370{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9

{{Optimal ET sequence|legend=0| 12, 41e, 53, 118d }}

Badness (Sintel): 1.18

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 275/273, 847/845

Mapping: {{mapping| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}

Optimal tunings:
* WE: ~2 = 1199.7370{{c}}, ~3/2 = 701.5937{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7570{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9

{{Optimal ET sequence|legend=0| 12f, …, 41ef, 53, 118d }}

Badness (Sintel): 1.09

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 99/98, 120/119, 176/175, 275/273, 442/441

Mapping: {{mapping| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}

Optimal tunings:
* WE: ~2 = 1199.2895{{c}}, ~3/2 = 701.2643{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6967{{c}}

{{Optimal ET sequence|legend=0| 12f, 53, 65d, 118dg }}

Badness (Sintel): 1.21

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245

Mapping: {{mapping| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}

Optimal tunings:
* WE: ~2 = 1199.5280{{c}}, ~3/2 = 701.4290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7149{{c}}

{{Optimal ET sequence|legend=0| 12f, 53, 65d }}

Badness (Sintel): 1.18

=== Karadeniz ===
{{See also| Turkish maqam music temperaments #Karadeniz temperament }}

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 3125/3087

Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}
: mapping generators: ~2, ~11/9

Optimal tunings:
* WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}}

{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106, 147 }}

Badness (Sintel): 1.37

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 325/324, 640/637

Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}

Optimal tunings:
* WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}}

{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106f }}

Badness (Sintel): 1.34

=== Hemigari ===
Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 3125/3087

Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
: mapping generators: ~2, ~110/63

Optimal tunings:
* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}

{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}

Badness (Sintel): 1.68

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 225/224, 275/273

Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}

Optimal tunings:
* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}

{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}

Badness (Sintel): 1.13

=== Sanjaab ===
Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323, 3125/3087

Mapping: {{mapping| 1 2 -1 -3 0 | 0 -3 24 42 25 }}
: mapping generators: ~2, ~11/10

Optimal tunings:
* WE: ~2 = 1200.1997{{c}}, ~11/10 = 166.0018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9786{{c}}

{{Optimal ET sequence|legend=0| 29, 65d, 94 }}

Badness (Sintel): 1.92

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 847/845, 1331/1323

Mapping: {{mapping| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}

Optimal tunings:
* WE: ~2 = 1200.1224{{c}}, ~11/10 = 165.9800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9659{{c}}

{{Optimal ET sequence|legend=0| 29, 65d, 94 }}

Badness (Sintel): 1.40

== Pontiac ==
{{Main| Pontiac }}

Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 32805/32768

{{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
: [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}}
: error map: {{val| 0.000 -0.197 -0.377 -0.268 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955]

{{Optimal ET sequence|legend=1| 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd }}

[[Badness]] (Sintel): 0.358

=== Helenoid ===
Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7.

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 4375/4374

Mapping: {{mapping| 1 0 15 -59 51 | 0 1 -8 39 -30 }}

Optimal tunings:
* WE: ~2 = 1200.3277{{c}}, ~3/2 = 701.9135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7223{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }}
: unchanged-interval (eigenmonzo) basis: 2.11/7

{{Optimal ET sequence|legend=0| 53, 118, 289e, 407de }}

Badness (Sintel): 1.28

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 625/624, 729/728

Mapping: {{mapping| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}

Optimal tunings:
* WE: ~2 = 1200.1780{{c}}, ~3/2 = 701.8491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7446{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }}
: unchanged-interval (eigenmonzo) basis: 2.13/7

{{Optimal ET sequence|legend=0| 53, 118, 171e }}

Badness (Sintel): 1.39

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 385/384, 561/560, 625/624, 729/728

Mapping: {{mapping| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}

Optimal tunings:
* WE: ~2 = 1200.1645{{c}}, ~3/2 = 701.8385{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7425{{c}}

Minimax tuning:
* 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }}
: unchanged-interval (eigenmonzo) basis: 2.17/13

{{Optimal ET sequence|legend=0| 53, 118, 171e }}

Badness (Sintel): 1.47

==== Helena ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 385/384, 3146/3125

Mapping: {{mapping| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}

Optimal tunings:
* WE: ~2 = 1200.5227{{c}}, ~3/2 = 702.0456{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7418{{c}}

{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}

Badness (Sintel): 1.50

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125

Mapping: {{mapping| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}

Optimal tunings:
* WE: ~2 = 1200.4988{{c}}, ~3/2 = 702.0218{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7332{{c}}

{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}

Badness (Sintel): 1.56

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625

Mapping: {{mapping| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}

Optimal tunings:
* WE: ~2 = 1200.5185{{c}}, ~3/2 = 702.0323{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7318{{c}}

{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}

Badness (Sintel): 1.33

=== Ponta ===
Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 32805/32768

Mapping: {{mapping| 1 0 15 -59 135 | 0 1 -8 39 -83 }}

Optimal tunings:
* WE: ~2 = 1199.9814{{c}}, ~3/2 = 701.7725{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7834{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
: unchanged-interval (eigenmonzo) basis: 2.11/7

{{Optimal ET sequence|legend=0| 53, 171, 224 }}

Badness (Sintel): 1.61

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2200/2197

Mapping: {{mapping| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}

Optimal tunings:
* WE: ~2 = 1199.9601{{c}}, ~3/2 = 701.7610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7845{{c}}

Minimax tuning:
* 13 and 15-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
: unchanged-interval (eigenmonzo) basis: 2.11/7

{{Optimal ET sequence|legend=0| 53, 171, 224 }}

Badness (Sintel): 0.976

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197

Mapping: {{mapping| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}

Optimal tunings:
* WE: ~2 = 1199.8850{{c}}, ~3/2 = 701.7101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7775{{c}}

Minimax tuning:
* 17-odd-limit: ~3/2 = {{monzo| 83/143 0 0 0 -1/143 0 1/143 }}
: unchanged-interval (eigenmonzo) basis: 2.17/11

{{Optimal ET sequence|legend=0| 53, 171, 224, 395e, 619eg }}

Badness (Sintel): 1.16

=== Pontic ===
Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 32805/32768

Mapping: {{mapping| 1 0 15 -59 -136 | 0 1 -8 39 88 }}

Optimal tunings:
* WE: ~2 = 1200.1259{{c}}, ~3/2 = 701.7980{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7256{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }}
: unchanged-interval (eigenmonzo) basis: 2.11

{{Optimal ET sequence|legend=0| 53e, 118, 289, 407d }}

Badness (Sintel): 1.64

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 625/624, 729/728, 3584/3575

Mapping: {{mapping| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}

Optimal tunings:
* WE: ~2 = 1199.9254{{c}}, ~3/2 = 701.6945{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7378{{c}}

Minimax tuning:
* 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11

{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }}

Badness (Sintel): 1.87

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873

Mapping: {{mapping| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}

Optimal tunings:
* WE: ~2 = 1199.9454{{c}}, ~3/2 = 701.7085{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7401{{c}}

Minimax tuning:
* 17-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11

{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }}

Badness (Sintel): 1.51

==== Pontoid ====
Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 4375/4374, 32805/32768

Mapping: {{mapping| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}

Optimal tunings:
* WE: ~2 = 1200.0897{{c}}, ~3/2 = 701.7874{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7356{{c}}

{{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289 }}

Badness (Sintel): 2.07

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768

Mapping: {{mapping| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}

Optimal tunings:
* WE: ~2 = 1200.1045{{c}}, ~3/2 = 701.7962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}}

{{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289, 460e, 749defg }}

Badness (Sintel): 1.50

=== Bipont ===
Bipont tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. It may be described as {{nowrap| 118 & 224 }}. It has a period of half octave and a ploidacot signature of diploid monocot. [[342edo]] may be recommended as a tuning.

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 32805/32768

Mapping: {{mapping| 2 0 30 -118 -85 | 0 1 -8 39 29 }}
: mapping generators: ~99/70, ~3

Optimal tunings:
* WE: ~99/70 = 600.0500{{c}}, ~3/2 = 701.8153{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7584{{c}}

{{Optimal ET sequence|legend=0| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }}

Badness (Sintel): 0.484

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 4096/4095

Mapping: {{mapping| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}

Optimal tunings:
* WE: ~99/70 = 599.9939{{c}}, ~3/2 = 701.7657{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7728{{c}}

{{Optimal ET sequence|legend=0| 106, 118, 224, 566f, 790f }}

Badness (Sintel): 1.25

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873

Mapping: {{mapping| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}

Optimal tunings:
* WE: ~99/70 = 599.9839{{c}}, ~3/2 = 701.7463{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7649{{c}}

{{Optimal ET sequence|legend=0| 106g, 118, 224, 342, 566f }}

Badness (Sintel): 1.38

==== Counterbipont ====
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768

Mapping: {{mapping| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}

Optimal tunings:
* WE: ~99/70 = 600.0405{{c}}, ~3/2 = 701.8160{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7697{{c}}

{{Optimal ET sequence|legend=0| 106f, 118f, 224, 342f, 566, 1356cf }}

Badness (Sintel): 1.06

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768

Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}

Optimal tunings:
* WE: ~99/70 = 600.0336{{c}}, ~3/2 = 701.8031{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7647{{c}}

{{Optimal ET sequence|legend=0| 106fg, 118f, 224, 342f, 566 }}

Badness (Sintel): 1.29

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864

Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}

Optimal tunings:
* WE: ~99/70 = 600.0243{{c}}, ~3/2 = 701.7891{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7613{{c}}

{{Optimal ET sequence|legend=0| 106fgh, 118f, 224, 342f, 566h, 908fgh }}

Badness (Sintel): 1.35

==== Quadrapont ====
Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768

Mapping: {{mapping| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}
: mapping generators: ~208/175, ~3

Optimal tunings:
* WE: ~208/175 = 300.0229{{c}}, ~3/2 = 701.8097{{c}}
* CWE: ~208/175 = 300.0000{{c}}, ~3/2 = 701.7578{{c}}

{{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde }}

Badness (Sintel): 0.869

== Grackle ==
Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 32805/32768

{{Mapping|legend=1| 1 0 15 44 | 0 1 -8 -26 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
: [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}}
: error map: {{val| 0.000 -0.709 +3.715 -1.234 }}

[[Minimax tuning]]:
* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7

{{Optimal ET sequence|legend=1| 12, …, 65, 77, 166c }}

[[Badness]] (Sintel): 1.78

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 32805/32768

Mapping: {{mapping| 1 0 15 44 70 | 0 1 -8 -26 -42 }}

Optimal tunings:
* WE: ~2 = 1199.7077{{c}}, ~3/2 = 701.0017{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1804{{c}}

{{Optimal ET sequence|legend=0| 12, 65e, 77, 89, 166c }}

Badness (Sintel): 1.62

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 5445/5408

Mapping: {{mapping| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}

Optimal tunings:
* WE: ~2 = 1199.7782{{c}}, ~3/2 = 701.0966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2319{{c}}

{{Optimal ET sequence|legend=0| 12f, 65ef, 77, 166cf }}

Badness (Sintel): 1.56

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873

Mapping: {{mapping| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}

Optimal tunings:
* WE: ~2 = 1199.5839{{c}}, ~3/2 = 700.9632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2137{{c}}

{{Optimal ET sequence|legend=0| 12f, 77, 89f, 166cf }}

Badness (Sintel): 1.52

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323

Mapping: {{mapping| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}

Optimal tunings:
* WE: ~2 = 1199.7146{{c}}, ~3/2 = 701.0500{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2212{{c}}

{{Optimal ET sequence|legend=0| 12f, 77, 166cf }}

Badness (Sintel): 1.40

==== Grackloid ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 729/728, 1287/1280

Mapping: {{mapping| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}

Optimal tunings:
* WE: ~2 = 1200.0060{{c}}, ~3/2 = 701.2202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2167{{c}}

{{Optimal ET sequence|legend=0| 12, 77, 166c }}

Badness (Sintel): 2.00

=== Grack ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 896/891

Mapping: {{mapping| 1 0 15 44 51 | 0 1 -8 -26 -30 }}

Optimal tunings:
* WE: ~2 = 1199.8388{{c}}, ~3/2 = 701.3071{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4068{{c}}

{{Optimal ET sequence|legend=0| 12, 53d, 65, 77e }}

Badness (Sintel): 1.85

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 245/242, 832/825

Mapping: {{mapping| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}

Optimal tunings:
* WE: ~2 = 1199.7329{{c}}, ~3/2 = 701.1918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.3555{{c}}

{{Optimal ET sequence|legend=0| 12f, 53dff, 65f, 77e }}

Badness (Sintel): 1.84

==== Catahelenic ====
Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 126/125, 245/242, 352/351

Mapping: {{mapping| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}

Optimal tunings:
* WE: ~2 = 1199.8928{{c}}, ~3/2 = 701.4664{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5327{{c}}

{{Optimal ET sequence|legend=0| 12f, …, 53d, 65 }}

Badness (Sintel): 2.01

== Quasipyth ==
Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 390625/387072

{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}

{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}

[[Badness]] (Sintel): 5.04

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 19712/19683, 78125/77616

Mapping: {{mapping| 1 0 15 109 -117 | 0 1 -8 -67 76 }}

Optimal tunings:
* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}

{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}

Badness (Sintel): 3.83

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 2200/2197, 19712/19683

Mapping: {{mapping| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}

Optimal tunings:
* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}

{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}

Badness (Sintel): 2.13

== Schism ==
See [[Archytas clan #Schism]].

Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.

== Bischismic ==
Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 32805/32768

{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }}
: mapping generators: ~567/400, ~3

[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 600.0072{{c}}, ~3/2 = 701.6005{{c}}
: [[error map]]: {{val| +0.014 -0.340 +0.982 -0.629 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5915{{c}}
: error map: {{val| 0.000 -0.364 +0.954 -0.656 }}

[[Minimax tuning]]:
* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7

{{Optimal ET sequence|legend=1| 12, …, 106d, 118, 130, 248, 378 }}

[[Badness]] (Sintel): 1.39

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 8019/8000

Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}

Optimal tunings:
* WE: ~99/70 = 600.0165{{c}}, ~3/2 = 701.6316{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6110{{c}}

{{Optimal ET sequence|legend=0| 12, …, 106de, 118, 130, 248 }}

Badness (Sintel): 0.931

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 3136/3125

Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}

Optimal tunings:
* WE: ~99/70 = 599.9610{{c}}, ~3/2 = 701.5445{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5908{{c}}

{{Optimal ET sequence|legend=0| 12, 118, 130, 248, 378 }}

Badness (Sintel): 1.19

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125

Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}

Optimal tunings:
* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}

{{Optimal ET sequence|legend=0| 12, 118, 130, 248g }}

Badness (Sintel): 1.49

==== Bischis ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 3136/3125

Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}

Optimal tunings:
* WE: ~55/39 = 599.9766{{c}}, ~3/2 = 701.5380{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5670{{c}}

{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }}

Badness (Sintel): 1.21

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125

Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}

Optimal tunings:
* WE: ~55/39 = 600.0997{{c}}, ~3/2 = 701.7114{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}

{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}

Badness (Sintel): 1.37

== Kleischismic ==
Kleischismic tempers out 1500625/1492992, the [[uniwiz comma]], and may be described as the {{nowrap| 94 & 118 }} temperament. The generator is a infrafifth, two of which plus a semi-octave period make the [[3/1|3rd]] [[harmonic]]; its [[ploidacot]] is thus diploid alpha-dicot. In schismic, 10 fifths make the interval class of [[9/5]]. Kleischismic then finds [[7/4]] by that minus a [[36/35]] quartertone, which is the aforementioned generator minus a semi-octave period. The generator stands in for [[16/11]] and the quartertone stands in for [[33/32]] in the [[11-limit]]. [[212edo]] and [[330edo]] in the 330e val may be recommended as tunings.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 1500625/1492992

{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
: mapping generators: ~1225/864, ~35/24

[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.1246{{c}}, ~35/24 = 651.0550{{c}} (~36/35 = 50.9304{{c}})
: [[error map]]: {{val| +0.249 +0.280 -0.453 -0.650 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~35/24 = 650.9204{{c}} (~36/35 = 50.9204{{c}})
: error map: {{val| 0.000 -0.114 -1.041 -1.338 }}

{{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}

[[Badness]] (Sintel): 2.80

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 9801/9800, 14641/14580

Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}

Optimal tunings:
* WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}})

{{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}

Badness (Sintel): 1.21

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 729/728, 1575/1573

Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}

Optimal tunings:
* WE: ~99/70 = 600.0696{{c}}, ~35/24 = 651.0136{{c}} (~36/35 = 50.9440{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}})

{{Optimal ET sequence|legend=0| 24, 94, 118, 212f }}

Badness (Sintel): 1.56

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 289/288, 352/351, 385/384, 561/560

Mapping: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}

Optimal tunings:
* WE: ~99/70 = 600.1134{{c}}, ~35/24 = 651.0646{{c}} (~36/35 = 50.9512{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}})

{{Optimal ET sequence|legend=0| 24, 94, 118 }}

Badness (Sintel): 1.30

==== Kleischis ====
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1573/1568, 14641/14580

Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}

Optimal tunings:
* WE: ~99/70 = 600.1909{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}})

{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212 }}

Badness (Sintel): 1.55

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580

Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}

Optimal tunings:
* WE: ~99/70 = 600.2190{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9518{{c}} (~36/35 = 50.9518{{c}})

{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212g }}

Badness (Sintel): 1.26

== Salsa ==
Salsa tempers out 245/243, the [[sensamagic comma]], and may be described as the {{nowrap| 41 & 65 }} temperament. It has a neutral third as a generator; its [[ploidacot]] is dicot. In fact it is related to [[hemififths]], from which this less accurate temperament only differs by the mapping of [[5/1|5]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 245/243, 32805/32768

{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
: mapping generators: ~2, ~128/105

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7707{{c}}, ~128/105 = 351.2748{{c}}
: [[error map]]: {{val| +0.771 +1.365 -1.315 -3.024 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 351.0471{{c}}
: error map: {{val| 0.000 +0.139 -3.068 -5.213 }}

{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}

[[Badness]] (Sintel): 2.03

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 243/242, 245/242, 385/384

Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}

Optimal tunings:
* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}

{{Optimal ET sequence|legend=0| 17, 24, 41, 106d }}

Badness (Sintel): 1.30

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 243/242, 245/242

Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}

Optimal tunings:
* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}

{{Optimal ET sequence|legend=0| 17, 24, 41 }}

Badness (Sintel): 1.27

== Hemischis ==
Hemischis tempers out 6144/6125, the [[porwell comma]], as well as 19683/19600, the [[cataharry comma]], and may be described as the {{nowrap| 53 & 130 }} temperament. Its [[ploidacot]] is alpha-dicot.

The [[S-expression]]-based comma list for 13-limit hemischis is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]], ([[4225/4224|S65]])}. Tempering out [[169/168]] ({{S|13}}), [[225/224]] ({{S|15}}) or [[625/624]] ({{S|25}}) leads to [[53edo]] while tempering out [[24192/24167]] ([[S-expression|S12/S13]]), [[10985/10976]] ([[S-expression|S13/S14]]), [[43904/43875]] ([[S-expression|S14/S15]]) or [[2401/2400]] ([[S-expression|S49]]) leads to [[130edo]] and implies S12, S13, S14, and S15 are tempered together.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 6144/6125, 19683/19600

{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}
: mapping generators: ~2, ~140/81

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8579{{c}}, ~140/81 = 951.6847{{c}}
: [[error map]]: {{val| -0.142 -0.586 +0.600 +0.708 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.7966{{c}}
: error map: {{val| 0.000 -0.362 +0.941 +1.088 }}

{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}

[[Badness]] (Sintel): 1.16

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625, 8019/8000

Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}

Optimal tunings:
* WE: ~2 = 1199.8482{{c}}, ~140/81 = 950.6809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}}

{{Optimal ET sequence|legend=0| 53, 130, 183, 313, 809cd }}

Badness (Sintel): 1.20

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 4096/4095

Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}

Optimal tunings:
* WE: ~2 = 1199.9140{{c}}, ~140/81 = 950.7324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8010{{c}}

{{Optimal ET sequence|legend=0| 53, 130, 183, 313 }}

Badness (Sintel): 0.860

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095

Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}

Optimal tunings:
* WE: ~2 = 1199.9740{{c}}, ~26/15 = 950.7894{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8100{{c}}

{{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}

Badness (Sintel): 1.07

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095

Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}

Optimal tunings:
* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}}

{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}

Badness (Sintel): 1.11

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095

Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}

Optimal tunings:
* WE: ~2 = 1200.0215{{c}}, ~26/15 = 950.8239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}}

{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}

Badness (Sintel): 1.06

; Music
* ''HemischisMatic EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify] | [https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play

== Term ==
Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 250047/250000

{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
: mapping generators: ~63/50, ~3

[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}

[[Minimax tuning]]:
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7

{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}

[[Badness]] (Sintel): 0.505

=== Terminal ===
Terminal tempers out 441/440 and 4375/4356, and may be described as {{nowrap| 159 & 171 }}. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 32805/32768

Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}

Optimal tunings:
* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}

{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}

Badness (Sintel): 1.97

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 13720/13689

Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}

Optimal tunings:
* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}

{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}

Badness (Sintel): 1.53

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619

Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}

Optimal tunings:
* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}

{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}

Badness (Sintel): 1.38

=== Terminator ===
Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768, 137781/137500

Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}

Optimal tunings:
* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}

{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}

Badness (Sintel): 2.21

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095, 31250/31213

Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}

Optimal tunings:
* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}

{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}

Badness (Sintel): 1.47

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095

Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}

Optimal tunings:
* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}

{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}

Badness (Sintel): 1.04

=== Semiterm ===
The semiterm temperament tempers out [[9801/9800]] (kalisma) as well as [[151263/151250]] (odiheim comma), and may be described as {{nowrap| 12 & 342 }}. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 151263/151250

Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
: mapping generators: ~55/49, ~3

Optimal tunings:
* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}

{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}

Badness (Sintel): 0.973

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375

Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}

Optimal tunings:
* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}

{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *

<nowiki>*</nowiki> optimal patent val: [[354edo|354]]

Badness (Sintel): 1.85

=== Hemiterm ===
The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 32805/32768, 102487/102400

Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
: mapping generators: ~63/50, ~693/400

Optimal tunings:
* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})

{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}

Badness (Sintel): 0.684

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712

Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}

Optimal tunings:
* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})

{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}

Badness (Sintel): 1.30

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264

Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}

Optimal tunings:
* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})

{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}

Badness (Sintel): 1.14

== Altinex ==
Named by [[Aura]] in 2021, altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 367653125/362797056

{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
: mapping generators: ~1536/1225, ~34300/19683

[[Optimal tuning]]s:
* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}}
: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }}
* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}}
: error map: {{val| 0.000 -0.027 -1.735 -2.441 }}

{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}

[[Badness]] (Sintel): 10.7

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 14700/14641, 19712/19683

Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}

Optimal tunings:
* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}

{{Optimal ET sequence|legend=0| 24, 135, 159 }}

Badness (Sintel): 3.35

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384, 676/675, 19712/19683

Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}

Optimal tunings:
* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}

{{Optimal ET sequence|legend=0| 24, 135f, 159 }}

Badness (Sintel): 2.27

== Squirrel ==
Squirrel tempers out 686/675, the [[sengic comma]], and may be described as {{nowrap| 29 & 36 }}. It has a [[~]][[11/10]] generator, three of which give the fourth ([[4/3]]), and thirteen of which give [[7/4]] with octave reduction. Its [[ploidacot]] is omega-tricot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 686/675, 32805/32768

{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}
: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 166.1597{{c}}
: error map: {{val| 0.000 -0.434 +1.518 -8.750 }}

{{Optimal ET sequence|legend=1| 29, 36, 65 }}

[[Badness]] (Sintel): 4.42

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 245/242, 686/675, 896/891

Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}

Optimal tunings:
* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}

{{Optimal ET sequence|legend=0| 29, 36, 65 }}

Badness (Sintel): 2.26

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 245/242, 896/891

Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}

Optimal tunings:
* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}

{{Optimal ET sequence|legend=0| 29, 65f, 94df }}

Badness (Sintel): 1.81

== Tertiaschis ==
Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 1071875/1062882

{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}}
: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}}
: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}

{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}

[[Badness]] (Sintel): 5.36

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 19712/19683

Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}

Optimal tunings:
* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}

{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}

Badness (Sintel): 2.07

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 10985/10976

Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}

Optimal tunings:
* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}

{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}

Badness (Sintel): 1.52

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}

Optimal tunings:
* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}

{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}

Badness (Sintel): 1.35

== Countertertiaschis ==
Named by [[Flora Canou]] in 2021, Countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 244140625/243045684

{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}

{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}

[[Badness]] (Sintel): 4.76

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 32805/32768

Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}

Optimal tunings:
* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}

{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}

Badness (Sintel): 1.62

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976

Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}

Optimal tunings:
* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}

{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}

Badness (Sintel): 1.01

== Quadrant ==
Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 390625/388962

{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
: mapping generators: ~25/21, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}}
: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }}
* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}}
: error map: {{val| 0.000 -0.137 -0.858 +0.268 }}

{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}

[[Badness]] (Sintel): 2.79

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 32805/32768

Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}

Optimal tunings:
* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}

{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}

Badness (Sintel): 1.51

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647

Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}

Optimal tunings:
* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}

{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}

Badness (Sintel): 1.13

== Sesquiquartififths ==
Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 32805/32768

{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
: mapping generators: ~2, ~448/405

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0846{{c}}, ~448/405 = 175.4460{{c}}
: [[error map]]: {{val| +0.085 -0.086 +0.007 -0.093 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 175.4320{{c}}
: error map: {{val| 0.000 -0.227 -0.137 -0.306 }}

[[Minimax tuning]]:
* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7

{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}

[[Badness]] (Sintel): 0.285

=== Sesquart ===
Sesquart is the main [[11-limit|11-]] and [[13-limit]] extension of sesquiquartififths of practical interest, as it identifies the neutral third with [[11/9]], which is realized in [[41edo]], [[89edo]], [[130edo]], and [[171edo]] also makes for a possible tuning.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 16384/16335

Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}

Optimal tunings:
* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}

{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}

Badness (Sintel): 0.969

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 3584/3575

Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}

Optimal tunings:
* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}

{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}

Badness (Sintel): 0.925

===== Heartia =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 256/255, 273/272, 364/363, 441/440

Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}

Optimal tunings:
* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}

{{Optimal ET sequence|legend=0| 41, 89, 130g }}

Badness (Sintel): 1.45

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440

Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}

Optimal tunings:
* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}

{{Optimal ET sequence|legend=0| 41, 89, 130g }}

Badness (Sintel): 1.40

===== Sesquartia =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575

Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}

Optimal tunings:
* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}

{{Optimal ET sequence|legend=0| 41, 130, 171 }}

Badness (Sintel): 1.18

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594

Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}

Optimal tunings:
* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}

{{Optimal ET sequence|legend=0| 41, 130, 171 }}

Badness (Sintel): 1.24

====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594

Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}

Optimal tunings:
* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}

{{Optimal ET sequence|legend=0| 41i, 130, 171 }}

Badness (Sintel): 1.36

===== Hearty =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625

Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}

Optimal tunings:
* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}

{{Optimal ET sequence|legend=0| 41g, 89, 130 }}

Badness (Sintel): 1.56

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455

Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}

Optimal tunings:
* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}

{{Optimal ET sequence|legend=0| 41g, 89, 130 }}

Badness (Sintel): 1.39

====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440

Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}

Optimal tunings:
* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}

{{Optimal ET sequence|legend=0| 41g, 89, 130 }}

Badness (Sintel): 1.37

=== Bisesqui ===
Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 32805/32768

Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
: mapping generators: ~99/70, ~448/405

Optimal tunings:
* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}

{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}

Badness (Sintel): 0.561

== Tsaharuk ==
{{Main| Tsaharuk }}

Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2|perfect fifth]], so its [[ploidacot]] is pentacot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 420175/419904

{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
: mapping generators: ~2, ~243/224

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}
: [[error map]]: {{val| +0.104 -0.041 -0.067 -0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3496{{c}}
: error map: {{val| 0.000 -0.207 -0.296 -0.436 }}

{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}

[[Badness]] (Sintel): 0.777

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 19712/19683

Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}

Optimal tunings:
* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}

{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}

Badness (Sintel): 2.10

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 729/728, 1331/1323

Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}

Optimal tunings:
* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}

{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}

Badness (Sintel): 1.57

== Quanharuk ==
Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd]] [[harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16875/16807, 32805/32768

{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
: mapping generators: ~2, ~56/45

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}}
: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 380.3546{{c}}
: error map: {{val| 0.000 -0.182 -0.498 +0.890 }}

{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}

[[Badness]] (Sintel): 1.82

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 32805/32768

Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}

Optimal tunings:
* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}

{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}

Badness (Sintel): 1.04

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 4096/4095

Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}

Optimal tunings:
* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}

{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}

Badness (Sintel): 0.884

== Quintilipyth ==
Named by [[Xenllium]] in 2021, quintilipyth (formerly ''quintilischis'') slices the [[4/3|perfect fourth]] into five semitones and tempers out the [[compass comma]] (9765625/9680832) in the [[7-limit]]. It may be described as the {{nowrap| 12 & 253 }} temperament, and its [[ploidacot]] is omega-pentacot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 9765625/9680832

{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
: mapping generators: ~2, ~625/588

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1138{{c}}, ~625/588 = 99.6347{{c}}
: [[error map]]: {{val| +0.114 +0.099 -1.041 +0.761 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/588 = 99.6265{{c}}
: error map: {{val| 0.000 -0.087 -1.255 +0.544 }}

{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}

[[Badness]] (Sintel): 6.43

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4356, 32805/32768

Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}

Optimal tunings:
* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}

{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}

Badness (Sintel): 3.74

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647

Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}

Optimal tunings:
* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}

{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}

Badness (Sintel): 2.86

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619

Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}

Optimal tunings:
* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}

{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }}

Badness (Sintel): 2.34

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971

Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}

Optimal tunings:
* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}

{{Optimal ET sequence|legend=0| 12f, 253, 265 }}

Badness (Sintel): 2.32

== Quintaschis ==
Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 49009212/48828125

{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}}
: error map: {{val| 0.000 -0.277 +0.266 -0.363 }}

{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}

[[Badness]] (Sintel): 3.36

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 32805/32768, 1953125/1951488

Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}

Optimal tunings:
* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}

{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}

Badness (Sintel): 3.69

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 32805/32768, 109512/109375

Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}

Optimal tunings:
* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}

{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}

Badness (Sintel): 3.07

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768

Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}

Optimal tunings:
* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}

{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}

Badness (Sintel): 2.58

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859

Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}

Optimal tunings:
* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}

{{Optimal ET sequence|legend=0| 12f, 289 }}

Badness (Sintel): 2.56

=== Quintahelenic ===
Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 8019/8000, 151263/151250

Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}

Optimal tunings:
* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}

{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}

Badness (Sintel): 2.72

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000

Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}

Optimal tunings:
* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}

{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}

Badness (Sintel): 2.30

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750

Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}

Optimal tunings:
* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}

{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}

Badness (Sintel): 2.06

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700

Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}

Optimal tunings:
* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}

{{Optimal ET sequence|legend=0| 12f, 301 }}

Badness (Sintel): 2.24

==== Quintahelenoid ====
Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436

Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}

Optimal tunings:
* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}

{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}

Badness (Sintel): 2.73

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157

Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}

Optimal tunings:
* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}

{{Optimal ET sequence|legend=0| 12, 301 }}

Badness (Sintel): 2.44

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137

Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}

Optimal tunings:
* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}

{{Optimal ET sequence|legend=0| 12, 301 }}

Badness (Sintel): 2.41

== Sextilifourths ==
Named by [[Xenllium]] in 2021, sextilifourths (also known as ''sextilischis'', formerly ''sextilififths'') slices the [[4/3|perfect fourth]] into six small semitones, which serves as both [[21/20]] and [[22/21]]. It may be described as {{nowrap| 130 & 159 }}, and its [[ploidacot]] is omega-hexacot. [[289edo]] gives a highly recommendable tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 235298/234375

{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
: mapping generators: ~2, ~21/20

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}}
: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}}
: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}

{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}

[[Badness]] (Sintel): 2.75

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 235298/234375

Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}

Optimal tunings:
* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}

{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}

Badness (Sintel): 1.50

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 10985/10976

Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}

Optimal tunings:
* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}

{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}

Badness (Sintel): 1.04

== Octant ==
Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 2259436291848/2251875390625

{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
: mapping generators: ~42875/39366, ~3

[[Optimal tuning]]s:
* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}

{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}

[[Badness]] (Sintel): 3.98

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 46656/46585

Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}

Optimal tunings:
* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}

{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}

Badness (Sintel): 1.48

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655

Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}

Optimal tunings:
* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}

{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}

Badness (Sintel): 1.26

== Nonant ==
Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 40353607/40310784

{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
: mapping generators: ~2592/2401, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}

{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}

[[Badness]] (Sintel): 1.77

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768, 42875/42592

Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}

Optimal tunings:
* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}

{{Optimal ET sequence|legend=0| 36, 135, 171 }}

Badness (Sintel): 4.20

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095, 16807/16731

Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}

Optimal tunings:
* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}

{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}

Badness (Sintel): 3.15

== Septant ==
Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 516560652/514714375

{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
: mapping generators: ~8575/7776, ~3

[[Optimal tuning]]s:
* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}

{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}

[[Badness]] (Sintel): 2.81

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 24057/24010, 32805/32768

Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}

Optimal tunings:
* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}

{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}

Badness (Sintel): 1.46

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024

Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}

Optimal tunings:
* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}

{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}

Badness (Sintel): 1.02

== Septiquarschis ==
Named by [[Xenllium]] in 2021, septiquarschis tempers out [[829440/823543]] (mynaslender comma) and [[67108864/66706983]] (septiness comma), and may be described as the {{nowrap| 89 & 94 }} temperament. It splits septimal minor seventh ([[7/4]]) into four generators. Note that in the data below, the generator is the [[octave complement]] so that seven of them minus five octaves make a [[3/2|perfect fifth]]; its [[ploidacot]] is thus epsilon-heptacot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 829440/823543

{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
: mapping generators: ~2, ~256/147

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}

{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}

[[Badness]] (Sintel): 4.73

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 15488/15435, 32805/32768

Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }}

Optimal tunings:
* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}

{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}

Badness (Sintel): 1.72

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1573/1568, 4096/4095

Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}

Optimal tunings:
* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}

{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}

Badness (Sintel): 1.46

== Tridecafifths ==
Named by [[Eliora]] in 2023, tridecafifths may be described as the {{nowrap| 89 & 200 }} temperament. It divides the [[3/2|perfect fifth]] into thirteen quartertones, so its [[ploidacot]] is 13-cot. [[289edo]] gives a highly recommendable tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}

{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
: mapping generators: ~2, ~1323/1280

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}
: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}
: error map: {{val| 0.000 -0.261 -0.221 -0.421 }}

{{Optimal ET sequence|legend=1| 89, 200, 289 }}

[[Badness]] (Sintel): 10.9

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 32805/32768, 55296000/55240493

Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}

Optimal tunings:
* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}

{{Optimal ET sequence|legend=0| 89, 200, 289 }}

Badness (Sintel): 4.23

== Subgroup extensions ==
=== Maqamschismic (2.3.5.11) ===
Proposed by [[Eufalesio]] in 2026, maqamschismic is equivalent to the no-7 [[cassandra]]. The 2.3.5.11.13 subgroup adds [[352/351]] to the comma list and tempers 11/9~39/32 together (and 16/13~27/22), providing a very simple framework for tuning [[maqam]]at (especially the Turkish version), as outlined by [[Ozan Yarman]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[helenus]].

Subgroup: 2.3.5.11

Comma list: 2200/2187, 4125/4096

Subgroup-val mapping: {{mapping| 1 0 15 -33 | 0 1 -8 23 }}

Optimal tunings:
* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}}

{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e, 241ce, 335ce }}

Badness (Sintel): 1.34

==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13

Comma list: 325/324, 352/351, 4125/4096

Subgroup-val mapping: {{mapping| 1 0 15 -33 -28 | 0 1 -8 23 20 }}

Optimal tunings:
* WE: ~2 = 1200.4565{{c}} ~3/2 = 702.3057{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0485{{c}}

{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e }}

Badness (Sintel): 0.862

=== Tridecaschismic (2.3.5.13) ===
Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, benefitting from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is has among the lowest badness of schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].

Subgroup: 2.3.5.13

Comma list: 325/324, 32805/32768

Subgroup-val mapping: {{mapping| 1 0 15 -28 | 0 1 -8 20 }}

Optimal tunings:
* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}

{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}

Badness (Sintel): 0.582

==== 2.3.5.13.19 subgroup ====
Subgroup: 2.3.5.13.19

Comma list: 325/324, 361/360, 513/512

Subgroup-val mapping: {{mapping| 1 0 15 -28 9 | 0 1 -8 20 -3 }}

Optimal tunings:
* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}

{{Optimal ET sequence|legend=0| 12, …, 41, 53 }}

Badness (Sintel): 0.354

=== Photia (2.3.5.17) ===
{{See also| No-elevens subgroup temperaments #Garibaldia }}

[[Subgroup]]: 2.3.5.17

[[Comma list]]: 256/255, 1458/1445

{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}

{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}
: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}
: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}

{{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }}

[[Badness]] (Sintel): 0.479

==== 2.3.5.17.19 subgroup ====
Subgroup: 2.3.5.17.19

Comma list: 171/170, 256/255, 324/323

Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}

Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}

Optimal tunings:
* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}

{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }}

Badness (Sintel): 0.332

=== Nestoria (2.3.5.19) ===
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''

Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]].

[[Subgroup]]: 2.3.5.19

[[Comma list]]: 361/360, 513/512

{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}

{{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}
: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}
: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}

{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}

[[Badness]] (Sintel): 0.126

=== Taylor (2.3.5.13) ===
This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.

[[Subgroup]]: 2.3.5.13

[[Comma list]]: 676/675, 32805/32768

{{Mapping|legend=2| 1 0 15 14 | 0 2 -16 -13 }}

{{Mapping|legend=3| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
: mapping generators: ~2, ~26/15

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}}
: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}}
: error map: {{val| 0.000 -0.256 +0.098 -1.568 }}

{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}

[[Badness]] (Sintel): 0.334

==== Dakota (2.3.5.13.19) ====
Subgroup: 2.3.5.13.19

Comma list: 361/360, 513/512, 676/675

Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }}

Optimal tunings:
* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}

{{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}

Badness (Sintel): 0.262

===== 2.3.5.13.19.37 subgroup =====
Subgroup: 2.3.5.13.19.37

Comma list: 361/360, 481/480, 513/512, 676/675

Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}

Optimal tunings:
* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}

{{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}

Badness (Sintel): 0.223

=== Quintilischis (2.3.5.17) ===
: ''For full 17- and 19-limit extensions, see [[#Quintilipyth]] or [[#Quintaschis]].''

[[Subgroup]]: 2.3.5.17

[[Comma list]]: 32805/32768, 1419857/1417176

{{Mapping|legend=2| 1 2 -1 5 | 0 -5 40 -11 }}

{{Mapping|legend=3| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}
: mapping generators: ~2, ~18/17

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}}
: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}}
: error map: {{val| 0.000 -0.205 -0.317 -1.104 }}

{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}

[[Badness]] (Sintel): 1.34

==== 2.3.5.17.19 subgroup ====
Subgroup: 2.3.5.17.19

Comma list: 4624/4617, 6144/6137, 6885/6859

Subgroup-val mapping: {{mapping| 1 2 -1 5 4 | 0 -5 40 -11 3 }}

Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}

Optimal tunings:
* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}

{{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }}

Badness (Sintel): 1.17

[[Category:Temperament families]]
[[Category:Schismatic family| ]] 
[[Category:Rank 2]]

User:Aura/Aura's Ideas on Functional Harmony (Part 1)

2026-03-02T20:11:37Z

Aura: /* The Realm of the Paradiatonic and the Parachormatic */

One construct from Western Classical music with potential implications for Microtonalists is '''[[Wikipedia:Function (music)|harmonic function]]'''- especially as it pertains to the [[5L 2s|diatonic]] MOS scale and its various relatives. In Mainstream Music Theory there were once two prevailing schools of thought in regards to diatonic functional harmony- '''[[Wikipedia:Function (music) #German functional theory|German Theory]]''' and '''[[Wikipedia:Function (music) #Viennese theory of the degrees|Viennese Theory]]'''- however, in a conversation with [[User:Mousemambo|Mousemambo]] on Discord, it has been revealed to me that in modern practice, the old ideas of functional harmony have largely disintegrated due firstly to the conviction that after around 1900 CE, art music took a turn away from Common Practice Period foundations and those old analyses just don't work anymore as originally formulated, and secondly due to a suspicion that they never really existed beyond the pareidolia of minds trying to see patterns in noise. Mousemambo has also pointed out to me that modern writers have moved away from the convoluted depths of the two Germanic schools, now more often simply identifying scale degrees and the chords for which they are the root as either Tonic; Dominant, which is basically anything leading to Tonic; and Predominant, which is basically anything leading to Dominant. However, upon listening to the ways in which Plagal cadences get used, and how the chords on the perfect 4th above the Tonic get used as a sort of "home away from home" in some tunings, it is obvious to me that the stance taken by modern writers is an oversimplification, and that there are more remnants of the ideas of the two schools in modern music than one would initially think. Furthermore, the genres of music I write call for a reconstruction of at least some of the ideals of the old Germanic schools from the ground up. Thus, ideas from both schools, as well as a number of other ideas, find a home in my microtonal theory and practice. If the reader will bear with me, I shall use narrative set-ups and character metaphors to describe how the various harmonic functions act in composition and the way they relate to one another, and, furthermore, I'll eventually be looking at ways to extend this reconstruction of functional harmony into the microtonal realm. However, before I get into that, I must answer a few questions about how function relates to different parts of the chord.

== Harmonic Information, Tonic, and Chord Structure ==

In modern theory, it is often contended that the third is the most important scale degree for determining harmonic information, followed by the root and the seventh while the fifth is the least important, however, I disagree with this assessment.

Instead, I contend that it's the root of a chord and the relationship between the chord root and the [[Tonic]] that dictates the bulk of the context for the function of the other notes in a given chord, with other bits of information being dictated by the relationship of other notes in the chord to both the Tonic and the actual chord root- do note that which note is considered to be the Tonic can in fact change based on additional context, such as the location of tritones- and, to a lesser extent, wolf fifths and wolf fourths- in a scale, as these, in combination with a tonality's direction of construction, can tonicize certain notes. From there, I think that only perfect fourths or perfect fifths that are either above or below a chord root can actually create stable frameworks for building chords, while dividing such intervals in two pieces without causing crowding creates the notes that impart character and color to chords. However, because perfect fifths are larger than perfect fourths, there's greater ease and a greater selection of options in dividing a perfect fifth without causing crowding than there is in doing the same with a perfect fourth.

I must also admit that I think additional harmonic information can be supplied by the likes of both otonal and utonal [[primodality]], albeit my approach is a bit more unusual. While primes other than 2 can form the basis of tonality, it should be mentioned that the higher the prime involved as common numerator and or common denominator, the weaker the tonicization effect. Furthermore, I'm of the opinion that if you want to add intervals from segments with higher-prime denominators such as /3 or /5 to an otherwise /2^n segment to help flesh out what is essentially a Bass-Up tonality, it will usually work out best if that /3 or /5 interval is also a 2^n/ type of interval- in this case, 4/3 or 8/5. Similarly, I'm of the opinion that if you want to add intervals from segments with higher-prime denominators such as /5 or /7 to an otherwise /3*2^n segment to help flesh out what is essentially a Bass-Up tonality, it will usually work out best if that /5 or /7 interval is also a 3*2^n/ type of interval- in this case, 6/5 or 12/7.

== Facets Derived from German Theory ==

Among the chief ideas that come from German Theory is that there are three basic, or primary functions, and that there are multiple operations that can be applied to these three basic functions in order to derive new functions. However, compared to those ideal functions, one of the three is in part original to my work.

[[File:Diatonic Function Map (Version 4).png|thumb|Diagram of diatonic and paradiatonic function locations as of an older edition of this article- note the use of the now obsolete terms "Interregnant" and "Intersubiant" for what are now the Imponent and Subient functions respectively. Apart from these, and the new spelling standardization of the term "Servient", there have only been minor changes, and a new diagram should be forthcoming.]]

=== Basic Diatonic Functions ===

The three basic functions have their roots in [[LCJI]]. The functions themselves are labeled as follows:

'''[[Tonic]]''' - This functionality has its roots in the fundamental at the root of both the harmonic and subharmonic series, which for all intents and purposes, can be thought of as [[1/1]], and, in [[octave equivalence|octave equivalent]] systems, [[2/1]]. To use a character metaphor for how the Tonic acts in functional harmony, the Tonic is the king of the Kingdom of Tonality- a very good king who not only exercises the highest authority in matters of governing the kingdom and does not tolerate challenges to his leadership, but also knows how be a top-notch confidante to his subjects both wherever and whenever possible. For more specifics on the functions of the Tonic, see the linked article.

'''Dominant''' - As per the name, and as noted on the [[Wikipedia:Dominant (music)|Wikipedia article]], the Dominant is the second most important after the Tonic, However, in contrast to what is stated about the Dominant in the article, there are several caveats which must be addressed in the realm of microtonality. Firstly, there's the matter of its origins- specifically, it is generated by the tonic as the first [[3-limit]] interval and indeed the first nontonic note in the "majoresque" direction. Secondly, there's the matter of just what it does, as it typically serves as one, or often more, of the following- a primary creator of instability in the "majoresque" direction that requires the Tonic for resolution, the second most important melodic and or harmonic anchor after the Tonic- a function that requires it to be tuned with a great deal of accuracy in order to blend well with the Tonic- as well as a generator of many of the "majoresque" notes in a [[5L 2s|diatonic MOS]], and or a discourager against the usage of other microtonally nearby pitches. Thirdly, one must take stock of the fact that, aside from the Unison and Octave, each octave-reduced harmonic and corresponding subharmonic interval come together to generate their own axis which has a preferred direction of travel<ref>[https://www.youtube.com/watch?v=HBdWxSxxe1M Quartertone Harmony - Beyond the Circle of Fifths SD 480p]</ref> which is determined by a Tonality's direction of construction. Finally, one must take stock of the fact that when you take the notes that occur before the Tonic on each of these axes when moving in the preferred direction of travel and place them in a sequence, one finds that a clear hierarchy of functional strength based on the closeness of harmonic and subharmonic connection to the Tonic becomes apparent, with the [[3/2]] Perfect 5th away from the Tonic in a tonality's direction of construction naturally emerging as the note with the strongest connection to the Tonic, though it should be noted that the relationships in this hierarchy are quite sensitive to detuning, and can even be scrambled by such detuning. Thus, the term "Dominant"- in its most basic form as referred to in this article, and specifically at the root level- is restricted to where it only refers to such notes that occur roughly at a 3/2 interval away from the Tonic in the scale's direction of construction, with acceptable detuning levels being at around 3.5 cents from JI on either side. On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that the Dominant does not see the Tonic occurring in the proximal tertian structure of its chord- that is, as a third or fifth. To use a character metaphor for how the Dominant acts in functional harmony, the Dominant is both the Head Steward of the Tonic's castle, and the one that executes the Tonic's directives as a Manager of Civil Service in the Kingdom of Tonality.

'''Servient''' - Compared to the term "''Subdominant''" from music theory of old, the term "Servient" (or, in older articles and manuscripts "Serviant"), specifically at the root level, is restricted to those notes that occur roughly at a [[4/3]] interval away from the Tonic in the scale's direction of construction since the Servient function is essentially the inverse of the Dominant function, and acts as a sort of counterweight to the Dominant relative to the Tonic. It typically serves as one, or often more, of the following— a primary creator of instability in the "minoresque" direction that can either be intensified with the dominant or resolved with the tonic, the third most important melodic and or harmonic anchor after the tonic and the dominant, a generator of many of the "minoresque" notes in a diatonic MOS, and or a discourager against the usage of other microtonally nearby pitches. To use a character metaphor for how the Servient acts in functional harmony, the Servient is a Servant who goes above and beyond the call of duty and acts as a confidante that observes things and reports back to the Dominant and Tonic about the way things are working both inside and outside the Tonic's castle due to its relationships to various Nontonic functions. Although one might think that the term "Subdominant" would be eligible for getting a similar treatment to the term "Dominant" here, the problems with such an option are threefold. Firstly, not all possible "Subdominant" harmonies have the same harmonic properties relative to the Tonic, as there is an extremely close connection between the Tonic and the 4/3 Perfect 4th. Secondly, in music built from the Treble downwards, the notes with these sorts of functions are actually located ''above'' the Dominant. Thirdly, in common parlance, "Subdominant" is often equated with "Predominant", however, while the Serviant does tend to resolve towards the Dominant, or else some other note that acts as a surrogate for the Dominant, the fact remains that it can also create plagal cadences and even semiplagal cadences, which break the modern Tonic-Dominant-Predominant paradigm and are better explained in part by the ideals of the two Germanic schools. On the chord level, not only is the root level definition of the Servient function at play, but it should also be noted that a Servient chord often sees the Tonic occurring in the proximal structure of its chord- that is, as either a third or fifth- which explains why Servient chords are weaker than their Dominant counterparts in both Bass-Up and Treble-Down Tonalities.

=== Basic Diatonic Function-Deriving Operations ===

The way I see it, there are seven known operations which can be used to derive additional diatonic functions from the three basic functions listed above.

'''Stacking''' - The notes that are arrived at through stacking multiple instances of either 3/2 or 4/3 (or their tempered counterparts) are dubbed according to the number of instances stacked, and the nature of the notes separated by the interval being stacked. Thus, stacking two instances of the Dominant or the Servient results in the creation of the "Bidominant" or "Biservient" respectively. This concept comes from the German language's way of referring to the chord built on the second scale degree of the Diatonic scale as the "Doppeldominante", which literally means "Double Dominant".

'''Parallelism''' - Notes located in the same primary [[tetrachord]] as either the Tonic, the Dominant, or the Servient but that don't crowd them tend to take on similar functions to said notes, with the caveat that functions derived from the Tonic in this fashion are still technically Nontonic functions. This process is one of two that create what in traditional music theory are referred to as [[Wikipedia:Parallel and counter parallel|"''parallels''" and "''counter parallels''"]]. It should be noted that the ability of an interval to relate to the Tonic through Parallelism, as well as the surrounding of more dissonant intervals by consonant intervals in the same region displaying such relationships to the Tonic, results in a tendency towards harmonic stagnation.

'''Adjacency''' - Notes within a suitable voice leading distance from either the Dominant or Servient tend to have the opposite function relative to the Tonic- this process even extends to the relationship between the Dominant and Servient themselves. On the other hand, notes within this same kind of distance from the Tonic often tend to have their functions colored more by their relationships to both the Dominant and Servient. This process is one of two that create what in traditional music theory are referred to as "''parallels''" and "''counter parallels''", however, unlike Parallelism proper, this process can establish these kinds of relationships outside the primary tetrachord.

'''Antipodism''' - Notes that are either opposite in pitch hue or nearly so due to being approximately half an octave away from the starting point are harmonically opposed to the starting point. Non-tonic notes related through this process tend to have the opposite function relative to the Tonic. For the notes related to the Tonic by this process, see Antitonic below.

'''Preparation''' - Notes that "prepare the way" for either a Dominant or a Servient through any of the above operations, or through some other mechanism, relate to said notes by this process. Functions which have this kind of role relative to some other function are denoted with a "pre-" prefix here.

'''Detempering''' - When the comma or subchroma that separates a note from one of the three primary functions is not tempered out, it results in the appearance of notes with either similar functions to the Dominant or Servient, or, in the case of this function being applied to the Tonic, a Nontonic function similar to one of the Chromatic functions. Often, though not always, these notes fall within the uncanny valleys of the three primary functions.

'''Neutralization''' - When one hybridizes Major and Minor Diatonic scale degrees of the same class or even tempers them together, one is using this process.

== Facets Derived from Viennese Theory ==

Among the chief ideas that come from Viennese Theory is the idea that each degree has its own function relative to the Tonic. However, while in Viennese Theory proper, the degrees are strictly defined only relative to the cycle of fifths, I, for the realm of Microtonality, not only take stacks of 3/2 to form a key navigational axis called the "'''Diatonic Axis'''", but also additionally take things like Bass-Up tonality (that is, tonal music built from the low pitches upwards) and Treble-Down tonality (that is, tonal music built from the high pitches downwards) into consideration. On top of that, I also contend that virtually all of the functions described by Viennese Theory find their roots in specific combinations of the different operations described above on the basic functions from German Theory.

=== Derivative Diatonic Functions ===

I should point out that all of the scale degree functions described in Viennese Theory, as well as a few additional functions listed on this page, can be classified as '''first derivative''' functions because only one instance of any given derivational process is needed to reach them. Do note that of all the original Viennese functions, only the Mediant remains virtually unchanged from the original theories to this reconstruction, as another has been renamed, while others have been grouped together under new functions.

'''Supervicinant''' - This is any note that maps to 1\7 above the Tonic, and in fact, intervals in this vicinity have both Tonic Adjacent and Serviant Parallel functions, beyond that, however, the precise behavior of notes in this range is determined by both mappings and concrete tuning ranges, see the Supervicinant and Subvicinant Subtypes section for more discussion of this function.

'''Mediant''' - This is the note that maps to 2\7 from the Tonic in the scale's direction of construction and is named due to being roughly halfway between the Tonic and the Dominant. This is the first of the two diatonic scale degrees that are located relatively far from the Tonic along the Circle of Third Harmonics, and, as a consequence, not only has the most possibilities for realization, but also a tendency towards stagnation, leading to the historical designation as a "weak harmony". Aside from these, the properties that are central to the Mediant function are all most easily derived through the Tonic Parallel function and the Servient Adjacent function, and in addition, Mediants have both Preservient and Predominant functions. Furthermore, since there's such a large range of Mediants, there are a multitude of subtypes, see the Mediant Subtypes section for more discussion of this function.

'''Antitonic''' - This is a special case, see the Antitonic Subtypes section for more discussion of this function.

'''Contramediant''' - Compared to the term "''Submediant''" from traditional music theory, the term "Contramediant" may have a slightly different frame of reference, as while a "Submediant" is halfway between the Tonic and a "Subdominant", the "Contramediant" is halfway between the Tonic and the Serviant. The Contramediant is the note that maps to 5\7 from the Tonic in the scale's direction of construction, and is the second of two that are located relatively far from the Tonic along the Circle of Third Harmonics, and, as a consequence, is not only tied with the Mediant for having the most possibilities for realization, but also has a tendency towards stagnation, leading to the historical designation as a "weak harmony". From a functional standpoint, the properties that are central to the Contramediant function are most easily derived through the Tonic Parallel function and the Dominant Adjacent function, and in addition, Contramediants have both Preservient and Predominant functions. Furthermore, since there's such a large range of Contramediants, there are a multitude of subtypes, see the Contramediant Subtypes section for more discussion of this function.

'''Subvicinant''' - This is any note that maps to 1\7 below the Tonic, and in fact, intervals in this vicinity have both Tonic Adjacent and Dominant Parallel functions, beyond that, however, the precise behavior of notes in this range is determined by both mappings and concrete tuning ranges, see the Supervicinant and Subvicinant Subtypes section for more discussion of this function.

== Supervicinant and Subvicinant Subtypes ==

The Supervicinant and Subvicinant functions each have two subtypes that are common to traditional music theory and one subtype that's not diatonic at all, although leading tones have been specified according to their position and have been given distinct functions.

=== Specific Types of Supervicinant ===

The four subtypes of Supervicinant are as follows...

'''Supercollocant''' - This subtype of Supervicinant, although not found in Viennese Theory proper, is an interval that usually maps to both 1\7 and 2\24 in Bass-Up tonality. Additionally, it has the Antidominant function as typified by the root of the [[Wikipedia:Neapolitan chord|Neapolitan chord]] in Bass-Up tonality and the Antiservient function in Treble-Down tonality. In actuality, two subtypes of Supercollocant exist- the "'''Proximosupercollocant'''", which always maps to both 1\7 and 2\24 in Bass-Up tonality, and the "'''Distosupercollocant'''", which maps to 1\7 and either 2\24 or 3\24 depending on a variety of factors, and is distinct from both Proximosupercollocant and Superabrogant only in finer tuning systems where it is usually more consonant, and thus, weaker. An example of a Proximosupercollocant is [[256/243]] while an example of a Distosupercollocant is [[16/15]].

'''Superabrogant''' - This subtype of Supervicinant is not a traditional diatonic function at all since it is an interval that maps to both 1\7 and 3\24 in Bass-Up tonality. As per this function's name, the intervals in this range are like Supercollocants in that they may cause listeners to forget the ending pitch's relationship to the starting pitch in voice-leading. However, they are noticeably too wide for them to convincingly pass off as being totally semitone-like since they feel as if they're rather disconnected from the Tonic, and furthermore, the occurrence of two successive instances of these sorts of intervals in the same melodic line in the same direction is liable to come across as jarring.

'''Superaequient''' - This subtype of Supervicinant is an interval that maps to 1\7 and either 3\24 or 4\24 depending on a variety of factors. It is so named for the [[equable heptatonic]] region above the Tonic, which comprises the bulk of its range, and for its capacity to straddle the line between the Supertonic and Superabrogant functions in voice-leading. Conspicuously, it is distinct from Supertonic and Superabrogant functions only in finer tuning systems, with the approximation ranges for [[11/10]] and [[10/9]] being located at its borders.

'''Supertonic''' - This subtype of Supervicinant is an interval that maps to both 1\7 and 4\24 in Bass-Up tonality and occurs above the Tonic as the second scale degree. However, it should be noted that the lower boundary of this function is situated at roughly 119/108 above the Tonic, since notes located at smaller distances from the Tonic are liable to cause listeners to forget the ending pitch's relationship to the starting pitch- something which Supertonics don't do under any circumstances. Conspicuously, one type of Supertonic is the "'''Bidominant'''", which is the function specifically of [[9/8]].

=== Specific Types of Subvicinant ===

The four subtypes of Subvicinant are as follows...

'''Subtonic''' - This subtype of Subvicinant is an interval that maps to both 6\7 and 20\24 in Bass-Up tonality and occurs above the Tonic as the seventh scale degree. However, it should be noted that the upper boundary of this function is situated at roughly 216/119 above the Tonic, since notes located at larger distances from the Tonic are liable to cause listeners to forget the ending pitch's relationship to the starting pitch- something which Subtonics don't do under any circumstances. Conspicuously, one type of Supertonic is the "'''Biservient'''", which is the function specifically of [[16/9]].

'''Subaequient''' - This subtype of Subvicinant is an interval that maps to 6\7 and either 20\24 or 21\24 depending on a variety of factors. It is so named for the equable heptatonic region below the Tonic, which comprises the bulk of its range, and for its capacity to straddle the line between the Subtonic and Subabrogant functions in voice-leading. Conspicuously, it is distinct from Subtonic and Subabrogant functions only in finer tuning systems, with the approximation ranges for [[9/5]] and [[20/11]] being located at its borders.

'''Subabrogant''' - This subtype of Subvicinant is not a traditional diatonic function at all since it is an interval that maps to both 6\7 and 21\24 in Bass-Up tonality. As per this function's name, the intervals in this range are like Subcollocants in that they may cause listeners to forget the ending pitch's relationship to the starting pitch in voice-leading. However, they, like Superabrogants, feel as if they're rather disconnected from the Tonic, and furthermore, the occurrence of two successive instances of these sorts of intervals in the same melodic line in the same direction is liable to come across as jarring.

'''Subcollocant''' - This subtype of Subvicinant is the note typically referred to when people say "the leading tone", and an interval that usually maps to both 6\7 and 22\24 in Bass-Up tonality. Additionally, it has the Antiserviant function in Bass-Up tonality and the Antidominant function in Treble-Down tonality. Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, my own analysis, while acknowledging the functional similarities between the Subcollocant and the Dominant in Bass-Up tonality, sees this interval as functionally distinct from the Dominant due to the Subcollocant also being potentially related to the Mediant in the same way that the Dominant is related to the Tonic- a key functionality that is often exploited in [[Wikipedia:Vi–ii–V–I|circle progression]]s. In actuality, two subtypes of Subcollocant exist- the "'''Proximosubcollocant'''", which always maps to both 6\7 and 22\24 in Bass-Up tonality, and the "'''Distosubcollocant'''", which maps to 6\7 and either 21\24 or 22\24 depending on a variety of factors, and is distinct from both Proximosubcollocant and Subabrogant only in finer tuning systems where it is usually more consonant, and thus, weaker. An example of a Proximosubcollocant is [[243/128]] while an example of a Distosubcollocant is [[15/8]].

== Mediant and Contramediant Subtypes ==

There are five basic types of Mediant, although only three of them are distinct in [[24edo]]- you'd need to go to [[41edo]] or [[53edo]] to see all five. Similarly, there are five basic types of Contramediant, with only three of them being distinct in [[24edo]], so you'd need to go to [[41edo]] or [[53edo]] to see all five.

=== Specific Types of Mediant ===

The five subtypes of Mediant are as follows...

'''Proximomediant''' - This type of Mediant is mapped to 2\7 and 6\24 and is so named due to being located on the end of the Mediant range closest to the Tonic. Conspicuously, it is one of the possible the types of Mediant seen in Minor keys and is only distinct from the Mesoproximomediant in finer tuning systems where it is usually more dissonant. One subtype of Proximomediant is the "'''Triservient'''", which is the function specifically of [[32/27]].

'''Mesoproximomediant''' - This type of Mediant is mapped to both 2\7 and either 6\24 or 7\24 depending on a variety of factors. It is so named due to being located between the proximal end of the Mediant range and the midline between the Tonic and the Dominant. Conspicuously, it is one of the possible the types of Mediant seen in Minor keys and is only distinct from the Proximomediant and Mesomediant in finer tuning systems where it's notable for containing the approximation range for [[6/5]] at its lower border and for being the [[fourth complement]] of the Superaequient, hence why it's not merely a simple compositing of the adjacent functions.

'''Mesomediant''' - This type of Mediant is mapped to both 2\7 and 7\24 and is so named due to being located along the midline between the Tonic and the Dominant. This type of Mediant is arguably the most dissonant, and is not a traditional diatonic function at all, and in fact, it doesn't serve well at phrase endings- rather, its Tonic Parallel function is only appropriate during the middle of musical phrases. Intervals with this type of function include [[11/9]] and [[27/22]].

'''Mesodistomediant''' - This type of Mediant is mapped to both 2\7 and either 7\24 or 8\24 depending on a variety of factors. It is so named due to being located between the distal end of the Mediant range and the midline between the Tonic and the Dominant. Conspicuously, it is one of the possible the types of Mediant seen in Major keys and is only distinct from the Mesomediant and Distomediant in finer tuning systems where it's notable for containing the approximation range for [[5/4]] at its upper border and for being the fourth complement of the Distosupercollocant, hence why it's not merely a simple compositing of the adjacent functions.

'''Distomediant''' - This type of Mediant is mapped to both 2\7 and 8\24 and is so named due to being located on the end of the Mediant range furthest from the Tonic. Conspicuously, it is one of the possible the types of Mediant seen in Major keys and is only distinct from the Mesodistomediant in finer tuning systems where it is usually more dissonant. One subtype of Distomediant is the "'''Quadridominant'''", which is the function specifically of [[81/64]].

=== Specific Types of Contramediant ===

The five subtypes of Contramediant are as follows...

'''Proximocontramediant''' - This type of Contramediant is mapped to 5\7 and 18\24 and is so named due to being located on the end of the Contramediant range closest to the Tonic. Conspicuously, it is one of the possible the types of Contraediant seen in Major keys and is only distinct from the Mesoproximocontramediant in finer tuning systems where it is usually more dissonant. One subtype of Proximocontramediant is the "'''Tridominant'''", which is the function specifically of [[27/16]].

'''Mesoproximocontramediant''' - This type of Contramediant is mapped to 5\7 and either 17\24 or 18\24 depending on a variety of factors. It is so named due to being located between the proximal end of the Contramediant range and the midline between the Tonic and the Servient. Conspicuously, it is one of the possible the types of Contramediant seen in Major keys and is only distinct from the Proximocontramediant and Mesocontramediant in finer tuning systems where it's notable for containing the approximation range for [[5/3]] at its upper border, hence why it's not merely a simple compositing of the adjacent functions.

'''Mesocontramediant''' - This type of Contramediant is mapped to both 5\7 and 17\24 and is so named due to being located along the midline between the Tonic and the Servient. This type of Contramediant is arguably the most dissonant, and is not a traditional diatonic function at all, and in fact, it doesn't serve well at phrase endings- rather, its Tonic Parallel function is only appropriate during the middle of musical phrases. Intervals with this type of function include [[18/11]] and [[44/27]].

'''Mesodistocontramediant''' - This type of Contramediant is mapped to both 5\7 and either 16\24 or 17\24 depending on a variety of factors. It is so named due to being located between the distal end of the Contramediant range and the midline between the Tonic and the Servient. Conspicuously, it is one of the possible the types of Contramediant seen in Minor keys and is only distinct from the Mesocontramediant and Distocontramediant in finer tuning systems where it's notable for containing the approximation range for [[8/5]] at its lower border, hence why it's not merely a simple compositing of the adjacent functions.

'''Distocontramediant''' - This type of Contramediant is mapped to both 5\7 and 16\24 and is so named due to being located on the end of the Contramediant range furthest from the Tonic. Conspicuously, it is one of the possible the types of Contramediant seen in Minor keys and is only distinct from the Mesodistocontramediant in finer tuning systems where it is usually more dissonant. One subtype of Distomediant is the "'''Quadriservient'''", which is the function specifically of [[128/81]].

== Antitonic Subtypes ==

Notes that occur around half an octave away from the Tonic, on account of harmonies built on notes in this area tending to oppose that of the Tonic, are referred to by the term "Antitonic" by myself and others. It should be noted that the Antitonic is basically a first derivative function as it is derived from the Tonic through either perfect or imperfect Antipodism. In addition, the term "Antitonic" acts as a generic term for any of a group of diatonic functions found in this region. While some microtonal theorists insist that the Antitonic functionality is more fundamental than perhaps even the Dominant or Servient, others, such as myself, disagree.

=== Specific Types of Antitonic ===

The exact outcome and specific function of any given Antitonic depends on whether or not the interval in question is an augmented fourth or a diminished fifth.

'''Sycophant''' - This type of Antitonic is mapped to both 3\7 and 12\24, and is named as such on account of it having a tendency to "kiss up to" and tonicize the Dominant- that is, to cause the Dominant to become a new Tonic- unless followed up by a different note such as some type of Mediant. A prototypical example of this type of Antitonic is [[45/32]].

'''Tyrant''' - This type of Antitonic is mapped to both 4\7 and 12\24, and it tends to contrast with the Tonic in a manner somewhat akin to that of a Dominant, but by sheer brute force and contrary harmonic nature, and indeed these brute force Dominant-esque tendencies are the source of the name "Tyrant". For example, if the Tonic harmony is Minor in nature, the Antitonic harmony will be Major- or more rarely, Supermajor- in nature. Furthermore, in scales such as the Locrian scale, any type of Serviant harmony tends to resolve towards some other type of substitute for a Dominant, often bypassing this type of Antitonic, though on rare occasions, a Tyrant will act as a leading tone to the Servient. A prototypical example of this type of Antitonic is [[64/45]].

== Chromatic Functions ==

These functions are either derived through Adjacency relative to some First Derivative Diatonic function, or else, are derived from the Primary Diatonic functions through a Chromatic function-deriving operation.

=== Primary Chromatic functions ===

The four basic Chromatic functions are as follows...

'''Superdislocant''' - This is a note that is to the Supercollocant what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone upwards and thus being displaced by a Nontonic function which leads away from the Tonic proper.

'''Subdislocant''' - This is a note that is to the Subcollocant what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone downwards and thus being displaced by a Nontonic function which leads away from the Tonic proper.

'''Protosycophant''' - This is a note which can tonicize the Dominant, but because its harmonies fail to completely oppose those of the Tonic, it fails to count as a true Sycophant. A prototypical example of this type of interval is [[7/5]].

'''Prototyrant''' - This is a note which can tonicize the Serviant, but because its harmonies fail to completely oppose those of the Tonic, it fails to count as a true Tyrant. A prototypical example of this type of interval is [[10/7]].

=== Basic Chromatic Function-Deriving Operations ===

The way I see it, there is at least one known operation which can be used to derive additional Chromatic functions from various Diatonic functions.

'''Displacement''' - This process results in a note that is near the Dominant or Serviant taking on similar functions to said notes rather than the opposite function, or, when applied to the Tonic, results in a Nontonic function that tends to want to lead away from the Tonic rather than towards it. This process is distinct from Detempering in that the notes created by this process are located at further distances from the note to which this process is applied.

== Wolf Fifths, Wolf Fourths and the Uncanny Valleys of Harmony ==

As per Flora Canou's analysis<ref>''[[User:FloraC/Analysis on the 13-limit just intonation space: episode ii #Chapter V. Things Repel the Similar but Not Identical| Analysis on the 13-limit just intonation space: episode ii]]''. Flora Canou. Xenharmonic Wiki.</ref>, there are uncanny valleys on either side of each of the three primary functions. In this analysis, she states the following:

<div style="font-style: italic; border: 1px solid silver; margin: 15px; padding: 15px;">

: From a historical perspective, meantone was discovered for a good reason. Letting it do its duty is of the greatest mercy. Not accounting for the discordance of wolf fifths or melodic bumps is equivalent to holding that meantone need not exist, and that most recent to modern theories are either failure or flukes based on wrong principles.

</div>

However, my contention is that while meantone doing its duty around the Tonic makes sense as [[81/80]] is really quite discordant, it seems that under certain circumstances, the average listener acquainted with [[12edo]] music will not notice wolf fourths or wolf fifths unless they are explicitly pointed out, thus not only are there other methods of accounting for the discordance of wolf fourths and wolf fifths besides shunning them, but there are additional functions hiding in the shadows of the three primary functions, along with rules governing their usage, which shall be covered in this section.

=== Circumtonic Regions ===

The Circumtonic regions are the two main regions on either side of the Tonic, outside the Tonic's "event horizon". These intervals are invariably inside the Tonic's uncanny valleys, and the uncanny valleys around the Tonic, unlike those around the other two functions, are very deep, and as a result, wolf intervals around the Tonic are tolerable in ornamentation but not melody or harmony. Thus, these intervals cannot be directly approached from the Tonic, even melodically, and so they're usually avoided outside of modulation.

'''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 35 cents above the Tonic. These intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance, or for a slightly less resolved version of a Unison or Octave.

'''Subcommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 35 cents below the Tonic. As with Supercommatic intervals, these intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance.

=== Circumdominant Regions ===

The circumdomimant regions are the two main regions on either side of the Dominant proper, and since the uncanny valleys around the Dominant are not as deep as those around the Tonic, there is more room for actual first derivative diatonic functionality, as well as other functionalities. Note that together with the Dominant itself, these functions are collectively called "'''Protodominants'''", as there are many systems where what would be the Dominant fails to get distinguished from either one or the other of the functions listed here.

'''Geminodominant''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents beyond [[22/15]] to about 3.5 cents short of the 3/2 perfect fifth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Dominant through detempering, occurring in [[5-limit]] diatonic environments, and acting as a sort of "fraternal twin" to the Dominant, hence its name. Specifically, as typified by intervals like [[40/27]], Geminodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution, though they also have a Preservient function. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the traditional weak harmonies- the Mediant and the Contramediant- which also has the benefits of strengthening interrupted cadences and creating the sense of impending movement, but outside of these usages and well supported chords, this kind of thing is best avoided since things repel the similar but not identical. Apart from diatonic contexts, Geminodominants only rise to prominence in systems where what might otherwise function as a Dominant is found just short of the sweet spot range near the standard issue 3/2.

'''Pseudodominant''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents to about 25 cents beyond the 3/2 perfect fifth in the scale's direction of construction. Because none of these intervals arise naturally in a [[5-limit]] diatonic scale, and because they fall within one of the Dominant's uncanny valleys, it is rare to see this functionality outside of systems where what might otherwise function as a Dominant is found just beyond the sweet spot range near the standard issue 3/2. However, these intervals naturally arise in the Circle of Thirteenth Harmonics, in which case they have a Predominant function.

=== Circumservient Regions ===

The circumservient regions are the two main regions on either side of the Servient proper, and since the uncanny valleys around the Servient are not as deep as those around the Tonic, there is more room for actual first derivative diatonic functionality, as well as other functionalities. Note that together with the Servient itself, these functions are collectively called "'''Protoservients'''", as there are many systems where what would be the Servient fails to get distinguished from either one or the other of the functions listed here.

'''Geminoservient''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents beyond the 4/3 perfect fourth to about 3.5 cents short of [[15/11]] in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminoservient is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Servient through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Servient, hence its name. As typified by intervals like [[27/20]], Geminoservients are dissonant intervals that often act as a sort of Predominant and or as the inverses of Geminodominants. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the Mediant and the Contramediant, but outside of these usages and well supported chords, this kind of thing is best avoided. Apart from diatonic contexts, Geminoservients only rise to prominence in systems where what might otherwise function as a Servient is found just beyond the sweet spot range near the standard issue 4/3.

'''Pseudoservient''' - This is a note that occurs roughly at intervals ranging from about 25 cents to about 3.5 cents short of the 4/3 perfect fourth in the scale's direction of construction. Because none of these intervals arise naturally in a [[5-limit]] diatonic scale, and because they fall within one of the Servient's uncanny valleys, it is rare to see this functionality outside of systems where what might otherwise function as a Servient is found just short the sweet spot range near the standard issue 4/3. However, these intervals naturally arise in the Circle of Thirteenth Harmonics, in which case they have a Preservient function.

=== Governing Rules ===

'''Commatic Repulsion''' - This rule is best summed up in Flora Canou's statement that things repel the similar but not identical. To illustrate this principle, let's take a look at the VImin chord as is present in [[meantone]] and contrast it to the situation outside of meantone. Believe it or not, the meantone VImin chord has no fewer than two separate functions relative to the Tonic. First of all, there's the Relative Minor functionality, which is 5/3–1/1–5/4 relative to the Tonic and can tonicize as a new Imin either temporarily, as in a deceptive cadence, or permanently as in a modulation. Secondly, there's the Tertiary Dominant functionality, which is 27/16-81/80-81/64 relative to the Tonic and, as the name suggests, acts as the Dominant of the Dominant's own Dominant. While both of these are fused together in meantone, these two functions are separated exactly by the syntonic comma in non-meantone environments, and the only way to use a VImin chord with both functions is to temper out the syntonic comma. As Flora Canou has stated:

<div style="font-style: italic; border: 1px solid silver; margin: 15px; padding: 15px;">

: If the comma is not tempered out, the progression does not hold. The idea to force it is absurd. Fitting one function will leave the other emergent function[s] misplaced by a comma, which is easily the most catastrophic scene [musically] – the uncanny valley of harmony.

</div>

However, in non-meantone settings, it is possible to have a chord which shares its root with the Tertiary Dominant and shares both its third and fifth with the Relative Minor. Since such an arrangement results in a wolf fifth, and since the wolf fifth is a dissonance requiring resolution, one could argue that such a chord has a function distinct from the more traditional options- specifically one which is involved in interrupted cadences, as well as in starting cadences that "wind down" such as VImin-IVmaj-Imaj or even VImin-Vmaj-Imaj. This particular function is what I call the ''Major Contramediant Tensive''.

'''Primary Adpositive Purity''' - This rule is that for every chord root located one step away from either the Tonic, Dominant or Servient along the Circle of Fifths, there is a demand for the fifth of the chord in question to be within 3.5 cents of a just 3/2, though in the case of a chord that has a root located at two steps away from the Tonic along the along the Circle of Fifths, the fifth of the chord can be deliberately subjected to the process of Displacement as mentioned before and not run afoul of this restriction. However, chord roots that are located three or more steps away from the Tonic along the circle of fifths are not subject to the aforementioned restriction due to their distance. This means that the Tonic, Dominant, Servient, Supertonic and Subtonic chords all demand a perfect fifth as the fifth of the chord, whether you are building the Tonality upwards or downwards, but wolf fifths can occur easily as the fifth of chords which are located three or more steps away from the Tonic.

== Beyond Diatonic and Chromatic Functional Harmony ==

In addition to all the aforementioned Diatonic and Chromatic functions, as well as the detemperings of diatonic functions, there is an additional set of categories for dealing with the notes in between the various Diatonic scale degrees.

=== History ===

I've been in the process of developing this since well before I officially joined the Microtonal community, in fact, it all started for me with my discovery of the nature of the eleventh harmonic as a quartertone, however, while it is only thanks to a YouTuber who goes by "Quartertone Harmony" <ref>[https://www.youtube.com/channel/UCeDhhWQYqGRPy5ES8gTtIAQ Quartertone Harmony - YouTube]</ref> that I've been able to fill in significant gaps in my theory, the reality is that the idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees, can trace at least some of its roots back to the work of [[Ivan Wyschnegradsky]].

Specifically, idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees- an idea that has at least some roots in Wyschnegradsky's concept of [[Wikipedia: Major fourth and minor fifth|"Major Fourth" and "Minor Fifth"]]. However, I wanted to use LCJI as a basis for defining these intervals and thus decided to take [[11/8]] as being the just version of Wyschnegradsky's "Major Fourth", and while I drew up sketches based loosely on [[24edo]] for early versions of this concept, I also realized that that two instances of [[33/32]] added up to an interval smaller than [[2187/2048]] but which had a similar function. Furthermore, since two instances of 11/8 resulted in an interval in the vicinity of a Major seventh, I decided to take stacks of 11/8 to form a second navigational axis which works together with the Diatonic Axis in order to define the microtonal functions positioned roughly halfway between the German and Viennese Diatonic functions, though there are a few other microtonal functions as well that are not immediately covered by this second axis.

[[File:Diatonic Function Map.png|thumb|Initial diagram of paradiatonic function locations I made around the time of officially joining the Xenharmonic community. Note that a number of the functions listed on this page are missing, while the Supercollocant, the Superabrogant, the Subabrogant, the Intersubiant, the Interregnant, the Superobstant and the Subobstant initially had different names.]]

Most traditional music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo. In [[Talk:159edo notation #My Second Idea for a Notation System|a conversation]] between myself and [[Kite Giedraitis]] about this topic, Kite mentioned that there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and the aforementioned 2187/2048- a chromatic semitone that is otherwise known as the Apotome- which, when added together, add up to a 9/8 whole tone. Furthermore, Kite also mentioned how in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]]. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields [[27/25]]- two additional diatonic semitones. On the other hand, subtracting 81/80 from the Apotome yields [[135/128]], and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.

Building on Kite's logic, I then decided to apply similar distinctions among quartertones, and thus make the argument that quartertones don't have to denote exactly one fourth of a whole tone in as of themselves, but rather, they only have to add up to a whole tone when paired up correctly. However, the catch was that for quartertones, there are sometimes multiple correct options, making things more complicated. I decided to define the musical functions of quartertones initially on an informal basis by drawing a distinction between the terms "'''Parachromatic'''" (from the prefix ''para-'' in both the senses of ''alongside'' and ''resembling''<ref>[[Wiktionary: para- #Etymology 1]]</ref>, and the word ''chromatic'') and "'''Paradiatonic'''" (from the same two senses of the prefix ''para-'' and the word ''diatonic'') for purposes of classifying quartertone intervals. This was easy, since I had found that two instances of 33/32 added up to [[1089/1024]] rather than 2187/2048, and, since I had informally added the "para-" prefix (in the same senses) to both "Major" and "Minor" to create the terms '''Paramajor''' and '''Paraminor''' to better describe how 11/8 and 16/11 related to 128/99 and 99/64 respectively in order to describe how, for instance, the notes at 99/64 and 16/11 above the Tonic relate to each other in much the same way as major and minor intervals do, except that this relationship occurs in a context where the note halfway between them is actually part of the base scale rather than the two notes in question, and there's a different interval between said two notes than the base scale's chroma.

The way I see it, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit inframinor seconds by default, while parachromatic quartertones are analogous to chromatic semitones in that they are denoted as primes, albiet as ultraprimes by default. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, it can be deduced that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, I ended up choosing the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. As a result of multiple factors, I ended up choosing the combination of three 33/32 parachromatic quartertones and one [[4096/3993]] paradiatonic quartertone as the JI basis for this in regards to both Diatonic theory and [[Alpharabian tuning]], and, through interactions with others in the Xenharmonic community, I was later influenced by others on Discord to take [[MOS]]-based structural considerations into account. This eventually resulted in the first formal definition of a "'''parachroma'''" (an interval that can be easily tempered to equal half of a MOS-chroma), and later, the "'''parastep'''" (the interval that remains after subtracting as many parachromas from a Major MOS-step as possible without resulting in a negative interval). Finally, drawing from the concept of "parachromas" as applied to MOS-based contexts, I was able to finally give a formal definition of terms like "paramajor" (the result of adding a parachroma to either a MOS generator or its period-inverse) and "paraminor" (the result of subtracting a parachroma from a MOS generator or its period-inverse), which I had previously come up with on an informal basis.

== The Realm of the Paradiatonic and the Parachormatic ==

In January of 2022, Quartertone Harmony posted a video in which he grouped together a series of functions he refered to in the video as the "shadow scale"<ref>[https://www.youtube.com/watch?v=P6WJryxB_0Y Quartertone Harmony - Harmonic Functions of Quartertones SD 480p]</ref>, which I will refer to here as a '''paradiatonic scale''' since there are technically two of these, and this in turn led to the separation of Paradiatonic and Parachromatic harmonic functions for me. This whole concept of a "shadow scale", in addition to everything else discuss in this section, paves the way for my idea of [[MOS-Shadow theory]], but, aside from how it applies to Diatonic-scale based functional harmony, MOS-Shadow theory is another whole discussion for another time.

=== The Paradiatonic Scales ===

The Paradiatonic scales from a given tonic acts as a sort of "second shelf" of that tonality. Note that scale degrees in parentheses are optional.

The '''Bright Paradiatonic Scale''' consists of the following scale degrees as analyzed relative to Viennese Theory's scale steps:

I, tII/dbIII, (dIII), tIV, dV, (dVI), tVI/dbVII, tVII

The '''Dark Paradiatonic Scale''' consists of the following scale degrees as analyzed relative to Viennese Theory's scale steps:

I, dbII tII/dbIII, (dIII), tIV, dV, (dVI), tVI/dbVII

While the Diatonic scale itself has seven notes, the two Paradiatonic scales each have eight notes, furthermore, the tunings of each note in each Paradiatonic scale not only depend upon the exact tuning of the Diatonic scale used as a basis, but also vary considerably when it comes to the notes of the Paradiatonic that occur between the main Diatonic interval category ranges.

=== Basic Paradiatonic Functions ===

Out of the various functions found in the Paradiatonic scales, four of them- specifically, the tII/dbIII, tIV, dV and tVI/dbVII- can be considered basic, while the other three are first derivatives. As with the three basic diatonic functions, the four most basic paradiatonic functions have their roots in LCJI. In the order listed, the tII/dbIII, tIV, dV and tVI/dbVII functions are the following...

'''Contravaricant''' - Named in contrast to the Varicant function, this is an interval that maps to both 1\5 and 5\24 in the scale's direction of construction, lying roughly in the middle of the 4/3 interval separating the Tonic and the Serviant above it. Intervals in the Contravaricant region often don't consistently act as either seconds or thirds, or even act as a cross between a second and a third, only with slight potential for crowding in chords depending on the exact tuning. In Bass-Up tonality, this functionality is first encountered in the form of [[8/7]], though [[7/6]] is another notable interval included in this range, with intervals in this range having Predominant, Preservient, and Dominant Parallel functions, as well as an overlap between Tonic Adjacent and Tonic Parallel functions. There are two different subtypes of this function that differ from each other based on how they are approached in interval stacking, the "'''Pseudobidominant'''", which arises from stacking two Pseudodominants, and the "'''Pseudotriservient'''", which arises from stacking three Pseudoservients.

'''Subient''' - This is an interval that maps to both 3\7 and 11\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[15/11]] to around 3.5 cents beyond [[11/8]]. Like both the Servient and the Sycophant, intervals in this region have a Predominant function, however, this predominant function is weaker than that of the surrounding interval regions since they neither act as a counterweight to the Dominant like a Servient nor do they completely tonicize the Dominant like a Sycophant- at least to those who are more familiar with quartertones. What is even less expected is that these same intervals also have Preservient and Mocktyrant functions. The name of this function comes from Latin "subeō"<ref>[[Wiktionary: subeo #Latin]]</ref>, specifically from the senses ranging from "approach" to "follow" and even "undergo", as all of these senses describe what the Subient actually does musically. Since the Subient is not found in the Tonic's quartertone field, there is a tendency for Subient chords to be preceded and or followed by other chords with roots in the same quartertone field, though one could also reasonably approach it from the Tonic, the Servient, some kind of Contramediant, some kind of Supervicinant, or, in some tuning systems, certain kinds of Mediant. In Bass-Up tonality, this functionality has its roots in the eleventh harmonic. This function used to be called the "Intersubiant" on an ad hoc basis, though that term has since been rendered obsolete.

'''Imponent''' - This is an interval that maps to both 4\7 and 13\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[16/11]] to around 3.5 cents beyond [[22/15]]. Accordingly, intervals in this region behave as a cross between a Tyrant Antitonic on one hand and a Dominant on the other in that they often contrast with the Tonic through some combination of harmonic connection and brute force contrast, though it unexpectedly has decent Predominant and Mocksycophant functions. Like the Subient, the Imponent is not found in the Tonic's quartertone field and is either very distant from the Tonic along the circle of fifths or is completely missed by said circle of fifths, thus, there is a tendency for Imponent chords to be preceded and or followed by other chords with roots in the same quartertone field. As if all that weren't enough, the Imponent can be used in the formation of chords, but because of [[delta-rational]] and other considerations, tertian chords framed with this type of interval usually have a markedly different structure imposed on them. The name of this function comes from Latin "impōnō"<ref>[[Wiktionary: impono #Latin]]</ref>, specifically from the senses ranging from "impose upon" to "inflict" and even "establish", as all of these senses describe what the Imponent actually does musically. Conspicuously, the distance between the Imponent and the Subient is only about a Minor Second- or, at the very most, a Supraminor Second- so as a result, the Imponent can be both a set-up and a follow-up to the Subient. In Bass-Up tonality, the Imponent functionality has its roots in the eleventh subharmonic. This function used to be called the "Interregnant" on an ad hoc basis, though that term has since been rendered obsolete.

'''Varicant''' - Just as a Mediant lies roughly in the middle of the 3/2 interval separating the Tonic and the Dominant above it, a Varicant lies roughly in the middle of the 4/3 interval separating the Dominant and the Tonic above it. Intervals in this region often don’t consistently act as either sixths or sevenths, or even act as a cross between a sixth and a seventh, only with slight potential for crowding in chords depending on the exact tuning- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō"<ref>[[Wiktionary: varico #Latin]]</ref>. This is an interval that maps to both 4\5 and 19\24 in the scale's direction of construction. In Bass-Up Tonality, this functionality is first encountered in the form of the [[7/4]] interval, though [[12/7]] is another notable interval included in this range. While many microtonalists think of 7/4 as being purely a type of seventh- and indeed, it most commonly acts as a sort of subminor seventh- I counterargue based on this same interval's relationships with 11/8 in particular that 7/4 is not merely a type of seventh, but rather, a type of a cross between a sixth and a seventh, with such a property explaining why [[14/11]] is generally considered to be a type of third. Furthermore, in contrast to the Subtonics of Bass-Up Tonality, Varicants are liable to act as Predominants, Preservients, and Serviant Parallels, as well as display an overlap between Tonic Adjacent and Tonic Parallel functions. There are two different subtypes of this function that differ from each other based on how they are approached in interval stacking, the "'''Pseudotridominant'''", which arises from stacking three Pseudodominants, and the "'''Pseudobiservient'''", which arises from stacking two Pseudoservients.

=== Derivative Paradiatonic Functions ===

'''Subgradient''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 25 cents below the octave reduplication of the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Subcollocant- however, there are functional differences between a Subgradient and a Subcollocant in Bass-Up Tonality. In Bass-Up Tonality, Subgradients are often more likely to be passing tones than Subcollocants, and, when they’re not merely passing non-chord tones, they are often harder to approach and or follow up without creating some kind of awkward tonal disconnect, with such a disconnect being especially noticeable for intervals like [[64/33]]. Furthermore, whereas a Subcollocant can resolve to the Tonic in part through a strong harmonic connection, a Subgradient is much more likely to do so through sheer brute force, and even these cases require a proper set-up, as otherwise, the awkward tonal disconnect between the Subgradient and the Tonic is likely to result in the Subgradient resolving back down to either the Subcollocant or the Subabrogant. As if that weren't enough, the Subgradient also has the Antsubient function. This function used to be called the "Subdietic", though that term has since been restricted to a related composite function (see below on Composite Functions).

'''Supergradient''' - This is a note that occurs at intervals between roughly 25 cents above the Tonic and 25/24 above the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Supercollocant. However, there are a few functional differences between a Supergradient and a Supercollocant in Bass-Up Tonality that are worth considering. For starters, Supergradients are often more likely to be passing tones than Supercollocants, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward through some sort of semitone-like motion to some form of Supercollocant or Superabrogant, as they are to resolve downwards toward the Tonic, a property which intervals like 33/32 in particular are apt to demonstrate. Furthermore, whereas a Supercollocant can resolve to the Tonic in part through a strong harmonic connection, a Supergradient is much more likely to do so through sheer brute force when such a resolution is noticeable. As if that weren't enough, the Subgradient also has the Antimponent function. This function used to be called the "Superdietic", though that term has since been restricted to a related composite function (see below on Composite Functions).

=== Parachromatic Functions ===

These are quartertone functions that are not on the Paradiatonic Scale. Of these, there are only two basic functions...

'''Superobstant''' - This is a note which, in Bass-Up tonality, is mapped to 9\24. This region is characterized by intervals that don’t consistently act as either thirds or fourths, or even act as a cross between a third and a fourth, as well as by intervals that act as parachromatic alterations of either the Mediant or the Serviant. As per the name, which comes from "super-" and "obstant", the latter of which comes from Latin "obstō"<ref>[[Wiktionary: obsto #Latin]]</ref>, intervals in this region are also generally more dissonant and have a tendency to disrupt both chords and melodies, leading to their avoidance in chords outside of deliberate dissonances. Chords of this type have Preservient, Presubient and Preimponent functions as well as Premediant functions.

'''Subobstant''' - This is a note which, in Bass-Up tonality, is mapped to 15\24. This region is characterized by intervals that don’t consistently act as either fifths or sixths, or even act as a cross between a fifth and a sixth, as well as by intervals that act as parachromatic alterations of either the Dominant or the Contramediant. As per the name, which comes from "sub-" and "obstant", which, as mentioned before, comes from Latin "obstō", intervals in this region are also generally more dissonant, leading to their avoidance in chords outside of deliberate dissonances. Chords of this type have Predominant, Presubient and Preimponent functions, as well as Precontramediant functions, and, perhaps very tellingly, tend to utilize Diminished Fourths instead of Major Thirds due to the functions of the Subgradient- which usually gets incorporated into these kinds of chords.

== Diatonic, Paradiatonic, and the Tonic's Proximal Pythagorean Aura ==

One of the things that Quartertone Harmony has found and mentioned<ref>[https://www.youtube.com/watch?v=3N_l5ciE14g Quartertone Harmony - The Truth About Quartertone Melodies]</ref> is that when dealing with quartertones, there seems to be something about a given 12-tone scale in 24edo which assures that notes in that same field will sound like they go together. However, in systems such as 159edo, you begin to see that things are slightly more complicated, as this distinctive atmosphere or quality turns out to be around 20 cents at widest and seems to surround and be generated by notes related to the Tonic by Pythagorean intervals that aren't all that far away from the Tonic, hence the term '''proximal Pythagorean aura''' to refer to it. Notably, the proximal Pythagorean aura is at its thickest at around six or seven steps away from the Tonic in either direction along the circle of fifths, and extends along the circle of fifths on either side of these areas from two steps away from the Tonic to eleven steps away from the Tonic. The presence of this aura explains things related to the uncanny valleys around the Dominant and Serviant- namely why these uncanny valleys are not as deep or as wide as those around the Tonic, and why simple intervals with single factors of prime 5 seem to be at least somewhat "in key" relative to the Tonic. It should be noted that the Tonic, the Dominant, and the Serviant work together with this aura to define safe regions for field shifts that are guaranteed to not come across as jarring. Intervals outside of this aura, such as most paradiatonic intervals, must abide by certain rules in order to not come across as jarring.

== Additional Composite Functions ==

Notes on the boundaries of functional regions have multiple functions due to occurring at the boundary between different functions, the process by which this happens is known as '''Compositing'''. As for the actual composite functions themselves, there are quite a few of them.

'''Subdietic''' - This function is a compositing of Subcommatic and Subgradient, and as Subcommatic is part of its nature, it is effectively repelled harmonically. It should be noted that this is one of only a handful of composite functions which are formed by the overlap between different functional regions rather than merely being located at a boundary.

'''Superdietic''' - This function is a compositing of Supercommatic and Supergradient, and as Supercommatic is part of its nature, it is effectively repelled harmonically. It should be noted that this is one of only a handful of composite functions which are formed by the overlap between different functional regions rather than merely being located at a boundary.

'''Subsanguinant''' - This function is one of two "bleeding tone" functions- hence its Latin-derived name- and is a compositing of Subgradient and Proximosubcollocant functions, and thus, has a more tense feel than a Proximosubcollocant alone. It should be noted that this function is formed by the overlap between different functional regions rather than merely being located at a boundary.

'''Supersanguinant''' - This function is one of two "bleeding tone" functions- hence its Latin-derived name- and is a compositing of Supergradient and Proximosupercollocant functions, and thus, has a more tense feel than a Proximosupercollocant alone. It should be noted that this function is formed by the overlap between different functional regions rather than merely being located at a boundary.

'''Submaculant''' - This function is a compositing of Subgradient and Subdislocant, and thus is like the chromatic version of the Subsanguinant.

'''Supermaculant''' - This function is a compositing of Supergradient and Superdislocant, and thus is like the chromatic version of the Supersanguinant.

'''Acuotyrant''' - This function is a compositing of Prototyrant and Imponent in Bass-Up Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless operates more on the side of brute force when it contrasts with the Tonic.

'''Gravosycophant''' - This function is a compositing of Protosycophant and Subient in Bass-Up Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless often runs a high risk of tonicizing the Geminodominant.

'''Gravotyrant''' this function is a compositing of Prototyrant and Imponent in Treble-Down Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless operates more on the side of brute force when it contrasts with the Tonic.

'''Acuosycophant''' - This function is a compositing of Protosycophant and Subient in Treble-Down Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless often runs a high risk of tonicizing the Geminodominant.

== Extra Functions of Prime Harmonics and Subharmonics ==

It should be noted that there are other layers of function besides those oriented around the [[3-limit]]. These are detailed here.

'''Paradominant''' - This function typically appears along prime axes other than that of the 3-limit in which they take Dominant-like function. Thus, for example, in Bass-Up Tonality, the note at 5/4 above the Tonic, being a direct prime harmonic of the Tonic, gets to serve this function in addition to its Mesodistomediant function as defined by the 3-limit, with the Paradominant function emerging mainly in a series of 5/4-based motions leading to the Tonic. When considered alongside the actual Dominant function of 3/2, however, Paradominants take on the function of Predominant owing to the fact that the Dominant function of 3/2 is actually stronger than that of any predominant due to being closer to the fundamental.

'''Paraservient''' - This function is essentially the inverse of the Paradominant function, and acts as a sort of counterweight to the Paradominant relative to the Tonic. Like the Paradominant function, it typically appears along prime axes other than that of the 3-limit in which they take Serviant-like function. Thus, for example, in Bass-Up tonality, the note at 8/5 above the Tonic, being a direct prime subharmonic of the Tonic, gets to serve this function in addition to its Mesodistocontramediant function as defined by the 3-limit. When considered alongside the actual Serviant function of 4/3, however, Paraservients take on the function of Preservient since the Serviant function of 4/3 is stronger.

== References ==
<references/>

[[Category:Terms]]
[[Category:Harmony]]
[[Category:Method]]
[[Category:Diatonic]]

User:Aura/Aura's Ideas on Functional Harmony (Part 1)

2026-03-02T19:33:01Z

Aura:

One construct from Western Classical music with potential implications for Microtonalists is '''[[Wikipedia:Function (music)|harmonic function]]'''- especially as it pertains to the [[5L 2s|diatonic]] MOS scale and its various relatives. In Mainstream Music Theory there were once two prevailing schools of thought in regards to diatonic functional harmony- '''[[Wikipedia:Function (music) #German functional theory|German Theory]]''' and '''[[Wikipedia:Function (music) #Viennese theory of the degrees|Viennese Theory]]'''- however, in a conversation with [[User:Mousemambo|Mousemambo]] on Discord, it has been revealed to me that in modern practice, the old ideas of functional harmony have largely disintegrated due firstly to the conviction that after around 1900 CE, art music took a turn away from Common Practice Period foundations and those old analyses just don't work anymore as originally formulated, and secondly due to a suspicion that they never really existed beyond the pareidolia of minds trying to see patterns in noise. Mousemambo has also pointed out to me that modern writers have moved away from the convoluted depths of the two Germanic schools, now more often simply identifying scale degrees and the chords for which they are the root as either Tonic; Dominant, which is basically anything leading to Tonic; and Predominant, which is basically anything leading to Dominant. However, upon listening to the ways in which Plagal cadences get used, and how the chords on the perfect 4th above the Tonic get used as a sort of "home away from home" in some tunings, it is obvious to me that the stance taken by modern writers is an oversimplification, and that there are more remnants of the ideas of the two schools in modern music than one would initially think. Furthermore, the genres of music I write call for a reconstruction of at least some of the ideals of the old Germanic schools from the ground up. Thus, ideas from both schools, as well as a number of other ideas, find a home in my microtonal theory and practice. If the reader will bear with me, I shall use narrative set-ups and character metaphors to describe how the various harmonic functions act in composition and the way they relate to one another, and, furthermore, I'll eventually be looking at ways to extend this reconstruction of functional harmony into the microtonal realm. However, before I get into that, I must answer a few questions about how function relates to different parts of the chord.

== Harmonic Information, Tonic, and Chord Structure ==

In modern theory, it is often contended that the third is the most important scale degree for determining harmonic information, followed by the root and the seventh while the fifth is the least important, however, I disagree with this assessment.

Instead, I contend that it's the root of a chord and the relationship between the chord root and the [[Tonic]] that dictates the bulk of the context for the function of the other notes in a given chord, with other bits of information being dictated by the relationship of other notes in the chord to both the Tonic and the actual chord root- do note that which note is considered to be the Tonic can in fact change based on additional context, such as the location of tritones- and, to a lesser extent, wolf fifths and wolf fourths- in a scale, as these, in combination with a tonality's direction of construction, can tonicize certain notes. From there, I think that only perfect fourths or perfect fifths that are either above or below a chord root can actually create stable frameworks for building chords, while dividing such intervals in two pieces without causing crowding creates the notes that impart character and color to chords. However, because perfect fifths are larger than perfect fourths, there's greater ease and a greater selection of options in dividing a perfect fifth without causing crowding than there is in doing the same with a perfect fourth.

I must also admit that I think additional harmonic information can be supplied by the likes of both otonal and utonal [[primodality]], albeit my approach is a bit more unusual. While primes other than 2 can form the basis of tonality, it should be mentioned that the higher the prime involved as common numerator and or common denominator, the weaker the tonicization effect. Furthermore, I'm of the opinion that if you want to add intervals from segments with higher-prime denominators such as /3 or /5 to an otherwise /2^n segment to help flesh out what is essentially a Bass-Up tonality, it will usually work out best if that /3 or /5 interval is also a 2^n/ type of interval- in this case, 4/3 or 8/5. Similarly, I'm of the opinion that if you want to add intervals from segments with higher-prime denominators such as /5 or /7 to an otherwise /3*2^n segment to help flesh out what is essentially a Bass-Up tonality, it will usually work out best if that /5 or /7 interval is also a 3*2^n/ type of interval- in this case, 6/5 or 12/7.

== Facets Derived from German Theory ==

Among the chief ideas that come from German Theory is that there are three basic, or primary functions, and that there are multiple operations that can be applied to these three basic functions in order to derive new functions. However, compared to those ideal functions, one of the three is in part original to my work.

[[File:Diatonic Function Map (Version 4).png|thumb|Diagram of diatonic and paradiatonic function locations as of an older edition of this article- note the use of the now obsolete terms "Interregnant" and "Intersubiant" for what are now the Imponent and Subient functions respectively. Apart from these, and the new spelling standardization of the term "Servient", there have only been minor changes, and a new diagram should be forthcoming.]]

=== Basic Diatonic Functions ===

The three basic functions have their roots in [[LCJI]]. The functions themselves are labeled as follows:

'''[[Tonic]]''' - This functionality has its roots in the fundamental at the root of both the harmonic and subharmonic series, which for all intents and purposes, can be thought of as [[1/1]], and, in [[octave equivalence|octave equivalent]] systems, [[2/1]]. To use a character metaphor for how the Tonic acts in functional harmony, the Tonic is the king of the Kingdom of Tonality- a very good king who not only exercises the highest authority in matters of governing the kingdom and does not tolerate challenges to his leadership, but also knows how be a top-notch confidante to his subjects both wherever and whenever possible. For more specifics on the functions of the Tonic, see the linked article.

'''Dominant''' - As per the name, and as noted on the [[Wikipedia:Dominant (music)|Wikipedia article]], the Dominant is the second most important after the Tonic, However, in contrast to what is stated about the Dominant in the article, there are several caveats which must be addressed in the realm of microtonality. Firstly, there's the matter of its origins- specifically, it is generated by the tonic as the first [[3-limit]] interval and indeed the first nontonic note in the "majoresque" direction. Secondly, there's the matter of just what it does, as it typically serves as one, or often more, of the following- a primary creator of instability in the "majoresque" direction that requires the Tonic for resolution, the second most important melodic and or harmonic anchor after the Tonic- a function that requires it to be tuned with a great deal of accuracy in order to blend well with the Tonic- as well as a generator of many of the "majoresque" notes in a [[5L 2s|diatonic MOS]], and or a discourager against the usage of other microtonally nearby pitches. Thirdly, one must take stock of the fact that, aside from the Unison and Octave, each octave-reduced harmonic and corresponding subharmonic interval come together to generate their own axis which has a preferred direction of travel<ref>[https://www.youtube.com/watch?v=HBdWxSxxe1M Quartertone Harmony - Beyond the Circle of Fifths SD 480p]</ref> which is determined by a Tonality's direction of construction. Finally, one must take stock of the fact that when you take the notes that occur before the Tonic on each of these axes when moving in the preferred direction of travel and place them in a sequence, one finds that a clear hierarchy of functional strength based on the closeness of harmonic and subharmonic connection to the Tonic becomes apparent, with the [[3/2]] Perfect 5th away from the Tonic in a tonality's direction of construction naturally emerging as the note with the strongest connection to the Tonic, though it should be noted that the relationships in this hierarchy are quite sensitive to detuning, and can even be scrambled by such detuning. Thus, the term "Dominant"- in its most basic form as referred to in this article, and specifically at the root level- is restricted to where it only refers to such notes that occur roughly at a 3/2 interval away from the Tonic in the scale's direction of construction, with acceptable detuning levels being at around 3.5 cents from JI on either side. On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that the Dominant does not see the Tonic occurring in the proximal tertian structure of its chord- that is, as a third or fifth. To use a character metaphor for how the Dominant acts in functional harmony, the Dominant is both the Head Steward of the Tonic's castle, and the one that executes the Tonic's directives as a Manager of Civil Service in the Kingdom of Tonality.

'''Servient''' - Compared to the term "''Subdominant''" from music theory of old, the term "Servient" (or, in older articles and manuscripts "Serviant"), specifically at the root level, is restricted to those notes that occur roughly at a [[4/3]] interval away from the Tonic in the scale's direction of construction since the Servient function is essentially the inverse of the Dominant function, and acts as a sort of counterweight to the Dominant relative to the Tonic. It typically serves as one, or often more, of the following— a primary creator of instability in the "minoresque" direction that can either be intensified with the dominant or resolved with the tonic, the third most important melodic and or harmonic anchor after the tonic and the dominant, a generator of many of the "minoresque" notes in a diatonic MOS, and or a discourager against the usage of other microtonally nearby pitches. To use a character metaphor for how the Servient acts in functional harmony, the Servient is a Servant who goes above and beyond the call of duty and acts as a confidante that observes things and reports back to the Dominant and Tonic about the way things are working both inside and outside the Tonic's castle due to its relationships to various Nontonic functions. Although one might think that the term "Subdominant" would be eligible for getting a similar treatment to the term "Dominant" here, the problems with such an option are threefold. Firstly, not all possible "Subdominant" harmonies have the same harmonic properties relative to the Tonic, as there is an extremely close connection between the Tonic and the 4/3 Perfect 4th. Secondly, in music built from the Treble downwards, the notes with these sorts of functions are actually located ''above'' the Dominant. Thirdly, in common parlance, "Subdominant" is often equated with "Predominant", however, while the Serviant does tend to resolve towards the Dominant, or else some other note that acts as a surrogate for the Dominant, the fact remains that it can also create plagal cadences and even semiplagal cadences, which break the modern Tonic-Dominant-Predominant paradigm and are better explained in part by the ideals of the two Germanic schools. On the chord level, not only is the root level definition of the Servient function at play, but it should also be noted that a Servient chord often sees the Tonic occurring in the proximal structure of its chord- that is, as either a third or fifth- which explains why Servient chords are weaker than their Dominant counterparts in both Bass-Up and Treble-Down Tonalities.

=== Basic Diatonic Function-Deriving Operations ===

The way I see it, there are seven known operations which can be used to derive additional diatonic functions from the three basic functions listed above.

'''Stacking''' - The notes that are arrived at through stacking multiple instances of either 3/2 or 4/3 (or their tempered counterparts) are dubbed according to the number of instances stacked, and the nature of the notes separated by the interval being stacked. Thus, stacking two instances of the Dominant or the Servient results in the creation of the "Bidominant" or "Biservient" respectively. This concept comes from the German language's way of referring to the chord built on the second scale degree of the Diatonic scale as the "Doppeldominante", which literally means "Double Dominant".

'''Parallelism''' - Notes located in the same primary [[tetrachord]] as either the Tonic, the Dominant, or the Servient but that don't crowd them tend to take on similar functions to said notes, with the caveat that functions derived from the Tonic in this fashion are still technically Nontonic functions. This process is one of two that create what in traditional music theory are referred to as [[Wikipedia:Parallel and counter parallel|"''parallels''" and "''counter parallels''"]]. It should be noted that the ability of an interval to relate to the Tonic through Parallelism, as well as the surrounding of more dissonant intervals by consonant intervals in the same region displaying such relationships to the Tonic, results in a tendency towards harmonic stagnation.

'''Adjacency''' - Notes within a suitable voice leading distance from either the Dominant or Servient tend to have the opposite function relative to the Tonic- this process even extends to the relationship between the Dominant and Servient themselves. On the other hand, notes within this same kind of distance from the Tonic often tend to have their functions colored more by their relationships to both the Dominant and Servient. This process is one of two that create what in traditional music theory are referred to as "''parallels''" and "''counter parallels''", however, unlike Parallelism proper, this process can establish these kinds of relationships outside the primary tetrachord.

'''Antipodism''' - Notes that are either opposite in pitch hue or nearly so due to being approximately half an octave away from the starting point are harmonically opposed to the starting point. Non-tonic notes related through this process tend to have the opposite function relative to the Tonic. For the notes related to the Tonic by this process, see Antitonic below.

'''Preparation''' - Notes that "prepare the way" for either a Dominant or a Servient through any of the above operations, or through some other mechanism, relate to said notes by this process. Functions which have this kind of role relative to some other function are denoted with a "pre-" prefix here.

'''Detempering''' - When the comma or subchroma that separates a note from one of the three primary functions is not tempered out, it results in the appearance of notes with either similar functions to the Dominant or Servient, or, in the case of this function being applied to the Tonic, a Nontonic function similar to one of the Chromatic functions. Often, though not always, these notes fall within the uncanny valleys of the three primary functions.

'''Neutralization''' - When one hybridizes Major and Minor Diatonic scale degrees of the same class or even tempers them together, one is using this process.

== Facets Derived from Viennese Theory ==

Among the chief ideas that come from Viennese Theory is the idea that each degree has its own function relative to the Tonic. However, while in Viennese Theory proper, the degrees are strictly defined only relative to the cycle of fifths, I, for the realm of Microtonality, not only take stacks of 3/2 to form a key navigational axis called the "'''Diatonic Axis'''", but also additionally take things like Bass-Up tonality (that is, tonal music built from the low pitches upwards) and Treble-Down tonality (that is, tonal music built from the high pitches downwards) into consideration. On top of that, I also contend that virtually all of the functions described by Viennese Theory find their roots in specific combinations of the different operations described above on the basic functions from German Theory.

=== Derivative Diatonic Functions ===

I should point out that all of the scale degree functions described in Viennese Theory, as well as a few additional functions listed on this page, can be classified as '''first derivative''' functions because only one instance of any given derivational process is needed to reach them. Do note that of all the original Viennese functions, only the Mediant remains virtually unchanged from the original theories to this reconstruction, as another has been renamed, while others have been grouped together under new functions.

'''Supervicinant''' - This is any note that maps to 1\7 above the Tonic, and in fact, intervals in this vicinity have both Tonic Adjacent and Serviant Parallel functions, beyond that, however, the precise behavior of notes in this range is determined by both mappings and concrete tuning ranges, see the Supervicinant and Subvicinant Subtypes section for more discussion of this function.

'''Mediant''' - This is the note that maps to 2\7 from the Tonic in the scale's direction of construction and is named due to being roughly halfway between the Tonic and the Dominant. This is the first of the two diatonic scale degrees that are located relatively far from the Tonic along the Circle of Third Harmonics, and, as a consequence, not only has the most possibilities for realization, but also a tendency towards stagnation, leading to the historical designation as a "weak harmony". Aside from these, the properties that are central to the Mediant function are all most easily derived through the Tonic Parallel function and the Servient Adjacent function, and in addition, Mediants have both Preservient and Predominant functions. Furthermore, since there's such a large range of Mediants, there are a multitude of subtypes, see the Mediant Subtypes section for more discussion of this function.

'''Antitonic''' - This is a special case, see the Antitonic Subtypes section for more discussion of this function.

'''Contramediant''' - Compared to the term "''Submediant''" from traditional music theory, the term "Contramediant" may have a slightly different frame of reference, as while a "Submediant" is halfway between the Tonic and a "Subdominant", the "Contramediant" is halfway between the Tonic and the Serviant. The Contramediant is the note that maps to 5\7 from the Tonic in the scale's direction of construction, and is the second of two that are located relatively far from the Tonic along the Circle of Third Harmonics, and, as a consequence, is not only tied with the Mediant for having the most possibilities for realization, but also has a tendency towards stagnation, leading to the historical designation as a "weak harmony". From a functional standpoint, the properties that are central to the Contramediant function are most easily derived through the Tonic Parallel function and the Dominant Adjacent function, and in addition, Contramediants have both Preservient and Predominant functions. Furthermore, since there's such a large range of Contramediants, there are a multitude of subtypes, see the Contramediant Subtypes section for more discussion of this function.

'''Subvicinant''' - This is any note that maps to 1\7 below the Tonic, and in fact, intervals in this vicinity have both Tonic Adjacent and Dominant Parallel functions, beyond that, however, the precise behavior of notes in this range is determined by both mappings and concrete tuning ranges, see the Supervicinant and Subvicinant Subtypes section for more discussion of this function.

== Supervicinant and Subvicinant Subtypes ==

The Supervicinant and Subvicinant functions each have two subtypes that are common to traditional music theory and one subtype that's not diatonic at all, although leading tones have been specified according to their position and have been given distinct functions.

=== Specific Types of Supervicinant ===

The four subtypes of Supervicinant are as follows...

'''Supercollocant''' - This subtype of Supervicinant, although not found in Viennese Theory proper, is an interval that usually maps to both 1\7 and 2\24 in Bass-Up tonality. Additionally, it has the Antidominant function as typified by the root of the [[Wikipedia:Neapolitan chord|Neapolitan chord]] in Bass-Up tonality and the Antiservient function in Treble-Down tonality. In actuality, two subtypes of Supercollocant exist- the "'''Proximosupercollocant'''", which always maps to both 1\7 and 2\24 in Bass-Up tonality, and the "'''Distosupercollocant'''", which maps to 1\7 and either 2\24 or 3\24 depending on a variety of factors, and is distinct from both Proximosupercollocant and Superabrogant only in finer tuning systems where it is usually more consonant, and thus, weaker. An example of a Proximosupercollocant is [[256/243]] while an example of a Distosupercollocant is [[16/15]].

'''Superabrogant''' - This subtype of Supervicinant is not a traditional diatonic function at all since it is an interval that maps to both 1\7 and 3\24 in Bass-Up tonality. As per this function's name, the intervals in this range are like Supercollocants in that they may cause listeners to forget the ending pitch's relationship to the starting pitch in voice-leading. However, they are noticeably too wide for them to convincingly pass off as being totally semitone-like since they feel as if they're rather disconnected from the Tonic, and furthermore, the occurrence of two successive instances of these sorts of intervals in the same melodic line in the same direction is liable to come across as jarring.

'''Superaequient''' - This subtype of Supervicinant is an interval that maps to 1\7 and either 3\24 or 4\24 depending on a variety of factors. It is so named for the [[equable heptatonic]] region above the Tonic, which comprises the bulk of its range, and for its capacity to straddle the line between the Supertonic and Superabrogant functions in voice-leading. Conspicuously, it is distinct from Supertonic and Superabrogant functions only in finer tuning systems, with the approximation ranges for [[11/10]] and [[10/9]] being located at its borders.

'''Supertonic''' - This subtype of Supervicinant is an interval that maps to both 1\7 and 4\24 in Bass-Up tonality and occurs above the Tonic as the second scale degree. However, it should be noted that the lower boundary of this function is situated at roughly 119/108 above the Tonic, since notes located at smaller distances from the Tonic are liable to cause listeners to forget the ending pitch's relationship to the starting pitch- something which Supertonics don't do under any circumstances. Conspicuously, one type of Supertonic is the "'''Bidominant'''", which is the function specifically of [[9/8]].

=== Specific Types of Subvicinant ===

The four subtypes of Subvicinant are as follows...

'''Subtonic''' - This subtype of Subvicinant is an interval that maps to both 6\7 and 20\24 in Bass-Up tonality and occurs above the Tonic as the seventh scale degree. However, it should be noted that the upper boundary of this function is situated at roughly 216/119 above the Tonic, since notes located at larger distances from the Tonic are liable to cause listeners to forget the ending pitch's relationship to the starting pitch- something which Subtonics don't do under any circumstances. Conspicuously, one type of Supertonic is the "'''Biservient'''", which is the function specifically of [[16/9]].

'''Subaequient''' - This subtype of Subvicinant is an interval that maps to 6\7 and either 20\24 or 21\24 depending on a variety of factors. It is so named for the equable heptatonic region below the Tonic, which comprises the bulk of its range, and for its capacity to straddle the line between the Subtonic and Subabrogant functions in voice-leading. Conspicuously, it is distinct from Subtonic and Subabrogant functions only in finer tuning systems, with the approximation ranges for [[9/5]] and [[20/11]] being located at its borders.

'''Subabrogant''' - This subtype of Subvicinant is not a traditional diatonic function at all since it is an interval that maps to both 6\7 and 21\24 in Bass-Up tonality. As per this function's name, the intervals in this range are like Subcollocants in that they may cause listeners to forget the ending pitch's relationship to the starting pitch in voice-leading. However, they, like Superabrogants, feel as if they're rather disconnected from the Tonic, and furthermore, the occurrence of two successive instances of these sorts of intervals in the same melodic line in the same direction is liable to come across as jarring.

'''Subcollocant''' - This subtype of Subvicinant is the note typically referred to when people say "the leading tone", and an interval that usually maps to both 6\7 and 22\24 in Bass-Up tonality. Additionally, it has the Antiserviant function in Bass-Up tonality and the Antidominant function in Treble-Down tonality. Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, my own analysis, while acknowledging the functional similarities between the Subcollocant and the Dominant in Bass-Up tonality, sees this interval as functionally distinct from the Dominant due to the Subcollocant also being potentially related to the Mediant in the same way that the Dominant is related to the Tonic- a key functionality that is often exploited in [[Wikipedia:Vi–ii–V–I|circle progression]]s. In actuality, two subtypes of Subcollocant exist- the "'''Proximosubcollocant'''", which always maps to both 6\7 and 22\24 in Bass-Up tonality, and the "'''Distosubcollocant'''", which maps to 6\7 and either 21\24 or 22\24 depending on a variety of factors, and is distinct from both Proximosubcollocant and Subabrogant only in finer tuning systems where it is usually more consonant, and thus, weaker. An example of a Proximosubcollocant is [[243/128]] while an example of a Distosubcollocant is [[15/8]].

== Mediant and Contramediant Subtypes ==

There are five basic types of Mediant, although only three of them are distinct in [[24edo]]- you'd need to go to [[41edo]] or [[53edo]] to see all five. Similarly, there are five basic types of Contramediant, with only three of them being distinct in [[24edo]], so you'd need to go to [[41edo]] or [[53edo]] to see all five.

=== Specific Types of Mediant ===

The five subtypes of Mediant are as follows...

'''Proximomediant''' - This type of Mediant is mapped to 2\7 and 6\24 and is so named due to being located on the end of the Mediant range closest to the Tonic. Conspicuously, it is one of the possible the types of Mediant seen in Minor keys and is only distinct from the Mesoproximomediant in finer tuning systems where it is usually more dissonant. One subtype of Proximomediant is the "'''Triservient'''", which is the function specifically of [[32/27]].

'''Mesoproximomediant''' - This type of Mediant is mapped to both 2\7 and either 6\24 or 7\24 depending on a variety of factors. It is so named due to being located between the proximal end of the Mediant range and the midline between the Tonic and the Dominant. Conspicuously, it is one of the possible the types of Mediant seen in Minor keys and is only distinct from the Proximomediant and Mesomediant in finer tuning systems where it's notable for containing the approximation range for [[6/5]] at its lower border and for being the [[fourth complement]] of the Superaequient, hence why it's not merely a simple compositing of the adjacent functions.

'''Mesomediant''' - This type of Mediant is mapped to both 2\7 and 7\24 and is so named due to being located along the midline between the Tonic and the Dominant. This type of Mediant is arguably the most dissonant, and is not a traditional diatonic function at all, and in fact, it doesn't serve well at phrase endings- rather, its Tonic Parallel function is only appropriate during the middle of musical phrases. Intervals with this type of function include [[11/9]] and [[27/22]].

'''Mesodistomediant''' - This type of Mediant is mapped to both 2\7 and either 7\24 or 8\24 depending on a variety of factors. It is so named due to being located between the distal end of the Mediant range and the midline between the Tonic and the Dominant. Conspicuously, it is one of the possible the types of Mediant seen in Major keys and is only distinct from the Mesomediant and Distomediant in finer tuning systems where it's notable for containing the approximation range for [[5/4]] at its upper border and for being the fourth complement of the Distosupercollocant, hence why it's not merely a simple compositing of the adjacent functions.

'''Distomediant''' - This type of Mediant is mapped to both 2\7 and 8\24 and is so named due to being located on the end of the Mediant range furthest from the Tonic. Conspicuously, it is one of the possible the types of Mediant seen in Major keys and is only distinct from the Mesodistomediant in finer tuning systems where it is usually more dissonant. One subtype of Distomediant is the "'''Quadridominant'''", which is the function specifically of [[81/64]].

=== Specific Types of Contramediant ===

The five subtypes of Contramediant are as follows...

'''Proximocontramediant''' - This type of Contramediant is mapped to 5\7 and 18\24 and is so named due to being located on the end of the Contramediant range closest to the Tonic. Conspicuously, it is one of the possible the types of Contraediant seen in Major keys and is only distinct from the Mesoproximocontramediant in finer tuning systems where it is usually more dissonant. One subtype of Proximocontramediant is the "'''Tridominant'''", which is the function specifically of [[27/16]].

'''Mesoproximocontramediant''' - This type of Contramediant is mapped to 5\7 and either 17\24 or 18\24 depending on a variety of factors. It is so named due to being located between the proximal end of the Contramediant range and the midline between the Tonic and the Servient. Conspicuously, it is one of the possible the types of Contramediant seen in Major keys and is only distinct from the Proximocontramediant and Mesocontramediant in finer tuning systems where it's notable for containing the approximation range for [[5/3]] at its upper border, hence why it's not merely a simple compositing of the adjacent functions.

'''Mesocontramediant''' - This type of Contramediant is mapped to both 5\7 and 17\24 and is so named due to being located along the midline between the Tonic and the Servient. This type of Contramediant is arguably the most dissonant, and is not a traditional diatonic function at all, and in fact, it doesn't serve well at phrase endings- rather, its Tonic Parallel function is only appropriate during the middle of musical phrases. Intervals with this type of function include [[18/11]] and [[44/27]].

'''Mesodistocontramediant''' - This type of Contramediant is mapped to both 5\7 and either 16\24 or 17\24 depending on a variety of factors. It is so named due to being located between the distal end of the Contramediant range and the midline between the Tonic and the Servient. Conspicuously, it is one of the possible the types of Contramediant seen in Minor keys and is only distinct from the Mesocontramediant and Distocontramediant in finer tuning systems where it's notable for containing the approximation range for [[8/5]] at its lower border, hence why it's not merely a simple compositing of the adjacent functions.

'''Distocontramediant''' - This type of Contramediant is mapped to both 5\7 and 16\24 and is so named due to being located on the end of the Contramediant range furthest from the Tonic. Conspicuously, it is one of the possible the types of Contramediant seen in Minor keys and is only distinct from the Mesodistocontramediant in finer tuning systems where it is usually more dissonant. One subtype of Distomediant is the "'''Quadriservient'''", which is the function specifically of [[128/81]].

== Antitonic Subtypes ==

Notes that occur around half an octave away from the Tonic, on account of harmonies built on notes in this area tending to oppose that of the Tonic, are referred to by the term "Antitonic" by myself and others. It should be noted that the Antitonic is basically a first derivative function as it is derived from the Tonic through either perfect or imperfect Antipodism. In addition, the term "Antitonic" acts as a generic term for any of a group of diatonic functions found in this region. While some microtonal theorists insist that the Antitonic functionality is more fundamental than perhaps even the Dominant or Servient, others, such as myself, disagree.

=== Specific Types of Antitonic ===

The exact outcome and specific function of any given Antitonic depends on whether or not the interval in question is an augmented fourth or a diminished fifth.

'''Sycophant''' - This type of Antitonic is mapped to both 3\7 and 12\24, and is named as such on account of it having a tendency to "kiss up to" and tonicize the Dominant- that is, to cause the Dominant to become a new Tonic- unless followed up by a different note such as some type of Mediant. A prototypical example of this type of Antitonic is [[45/32]].

'''Tyrant''' - This type of Antitonic is mapped to both 4\7 and 12\24, and it tends to contrast with the Tonic in a manner somewhat akin to that of a Dominant, but by sheer brute force and contrary harmonic nature, and indeed these brute force Dominant-esque tendencies are the source of the name "Tyrant". For example, if the Tonic harmony is Minor in nature, the Antitonic harmony will be Major- or more rarely, Supermajor- in nature. Furthermore, in scales such as the Locrian scale, any type of Serviant harmony tends to resolve towards some other type of substitute for a Dominant, often bypassing this type of Antitonic, though on rare occasions, a Tyrant will act as a leading tone to the Servient. A prototypical example of this type of Antitonic is [[64/45]].

== Chromatic Functions ==

These functions are either derived through Adjacency relative to some First Derivative Diatonic function, or else, are derived from the Primary Diatonic functions through a Chromatic function-deriving operation.

=== Primary Chromatic functions ===

The four basic Chromatic functions are as follows...

'''Superdislocant''' - This is a note that is to the Supercollocant what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone upwards and thus being displaced by a Nontonic function which leads away from the Tonic proper.

'''Subdislocant''' - This is a note that is to the Subcollocant what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone downwards and thus being displaced by a Nontonic function which leads away from the Tonic proper.

'''Protosycophant''' - This is a note which can tonicize the Dominant, but because its harmonies fail to completely oppose those of the Tonic, it fails to count as a true Sycophant. A prototypical example of this type of interval is [[7/5]].

'''Prototyrant''' - This is a note which can tonicize the Serviant, but because its harmonies fail to completely oppose those of the Tonic, it fails to count as a true Tyrant. A prototypical example of this type of interval is [[10/7]].

=== Basic Chromatic Function-Deriving Operations ===

The way I see it, there is at least one known operation which can be used to derive additional Chromatic functions from various Diatonic functions.

'''Displacement''' - This process results in a note that is near the Dominant or Serviant taking on similar functions to said notes rather than the opposite function, or, when applied to the Tonic, results in a Nontonic function that tends to want to lead away from the Tonic rather than towards it. This process is distinct from Detempering in that the notes created by this process are located at further distances from the note to which this process is applied.

== Wolf Fifths, Wolf Fourths and the Uncanny Valleys of Harmony ==

As per Flora Canou's analysis<ref>''[[User:FloraC/Analysis on the 13-limit just intonation space: episode ii #Chapter V. Things Repel the Similar but Not Identical| Analysis on the 13-limit just intonation space: episode ii]]''. Flora Canou. Xenharmonic Wiki.</ref>, there are uncanny valleys on either side of each of the three primary functions. In this analysis, she states the following:

<div style="font-style: italic; border: 1px solid silver; margin: 15px; padding: 15px;">

: From a historical perspective, meantone was discovered for a good reason. Letting it do its duty is of the greatest mercy. Not accounting for the discordance of wolf fifths or melodic bumps is equivalent to holding that meantone need not exist, and that most recent to modern theories are either failure or flukes based on wrong principles.

</div>

However, my contention is that while meantone doing its duty around the Tonic makes sense as [[81/80]] is really quite discordant, it seems that under certain circumstances, the average listener acquainted with [[12edo]] music will not notice wolf fourths or wolf fifths unless they are explicitly pointed out, thus not only are there other methods of accounting for the discordance of wolf fourths and wolf fifths besides shunning them, but there are additional functions hiding in the shadows of the three primary functions, along with rules governing their usage, which shall be covered in this section.

=== Circumtonic Regions ===

The Circumtonic regions are the two main regions on either side of the Tonic, outside the Tonic's "event horizon". These intervals are invariably inside the Tonic's uncanny valleys, and the uncanny valleys around the Tonic, unlike those around the other two functions, are very deep, and as a result, wolf intervals around the Tonic are tolerable in ornamentation but not melody or harmony. Thus, these intervals cannot be directly approached from the Tonic, even melodically, and so they're usually avoided outside of modulation.

'''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 35 cents above the Tonic. These intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance, or for a slightly less resolved version of a Unison or Octave.

'''Subcommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 35 cents below the Tonic. As with Supercommatic intervals, these intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance.

=== Circumdominant Regions ===

The circumdomimant regions are the two main regions on either side of the Dominant proper, and since the uncanny valleys around the Dominant are not as deep as those around the Tonic, there is more room for actual first derivative diatonic functionality, as well as other functionalities. Note that together with the Dominant itself, these functions are collectively called "'''Protodominants'''", as there are many systems where what would be the Dominant fails to get distinguished from either one or the other of the functions listed here.

'''Geminodominant''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents beyond [[22/15]] to about 3.5 cents short of the 3/2 perfect fifth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Dominant through detempering, occurring in [[5-limit]] diatonic environments, and acting as a sort of "fraternal twin" to the Dominant, hence its name. Specifically, as typified by intervals like [[40/27]], Geminodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution, though they also have a Preservient function. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the traditional weak harmonies- the Mediant and the Contramediant- which also has the benefits of strengthening interrupted cadences and creating the sense of impending movement, but outside of these usages and well supported chords, this kind of thing is best avoided since things repel the similar but not identical. Apart from diatonic contexts, Geminodominants only rise to prominence in systems where what might otherwise function as a Dominant is found just short of the sweet spot range near the standard issue 3/2.

'''Pseudodominant''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents to about 25 cents beyond the 3/2 perfect fifth in the scale's direction of construction. Because none of these intervals arise naturally in a [[5-limit]] diatonic scale, and because they fall within one of the Dominant's uncanny valleys, it is rare to see this functionality outside of systems where what might otherwise function as a Dominant is found just beyond the sweet spot range near the standard issue 3/2. However, these intervals naturally arise in the Circle of Thirteenth Harmonics, in which case they have a Predominant function.

=== Circumservient Regions ===

The circumservient regions are the two main regions on either side of the Servient proper, and since the uncanny valleys around the Servient are not as deep as those around the Tonic, there is more room for actual first derivative diatonic functionality, as well as other functionalities. Note that together with the Servient itself, these functions are collectively called "'''Protoservients'''", as there are many systems where what would be the Servient fails to get distinguished from either one or the other of the functions listed here.

'''Geminoservient''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents beyond the 4/3 perfect fourth to about 3.5 cents short of [[15/11]] in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminoservient is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Servient through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Servient, hence its name. As typified by intervals like [[27/20]], Geminoservients are dissonant intervals that often act as a sort of Predominant and or as the inverses of Geminodominants. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the Mediant and the Contramediant, but outside of these usages and well supported chords, this kind of thing is best avoided. Apart from diatonic contexts, Geminoservients only rise to prominence in systems where what might otherwise function as a Servient is found just beyond the sweet spot range near the standard issue 4/3.

'''Pseudoservient''' - This is a note that occurs roughly at intervals ranging from about 25 cents to about 3.5 cents short of the 4/3 perfect fourth in the scale's direction of construction. Because none of these intervals arise naturally in a [[5-limit]] diatonic scale, and because they fall within one of the Servient's uncanny valleys, it is rare to see this functionality outside of systems where what might otherwise function as a Servient is found just short the sweet spot range near the standard issue 4/3. However, these intervals naturally arise in the Circle of Thirteenth Harmonics, in which case they have a Preservient function.

=== Governing Rules ===

'''Commatic Repulsion''' - This rule is best summed up in Flora Canou's statement that things repel the similar but not identical. To illustrate this principle, let's take a look at the VImin chord as is present in [[meantone]] and contrast it to the situation outside of meantone. Believe it or not, the meantone VImin chord has no fewer than two separate functions relative to the Tonic. First of all, there's the Relative Minor functionality, which is 5/3–1/1–5/4 relative to the Tonic and can tonicize as a new Imin either temporarily, as in a deceptive cadence, or permanently as in a modulation. Secondly, there's the Tertiary Dominant functionality, which is 27/16-81/80-81/64 relative to the Tonic and, as the name suggests, acts as the Dominant of the Dominant's own Dominant. While both of these are fused together in meantone, these two functions are separated exactly by the syntonic comma in non-meantone environments, and the only way to use a VImin chord with both functions is to temper out the syntonic comma. As Flora Canou has stated:

<div style="font-style: italic; border: 1px solid silver; margin: 15px; padding: 15px;">

: If the comma is not tempered out, the progression does not hold. The idea to force it is absurd. Fitting one function will leave the other emergent function[s] misplaced by a comma, which is easily the most catastrophic scene [musically] – the uncanny valley of harmony.

</div>

However, in non-meantone settings, it is possible to have a chord which shares its root with the Tertiary Dominant and shares both its third and fifth with the Relative Minor. Since such an arrangement results in a wolf fifth, and since the wolf fifth is a dissonance requiring resolution, one could argue that such a chord has a function distinct from the more traditional options- specifically one which is involved in interrupted cadences, as well as in starting cadences that "wind down" such as VImin-IVmaj-Imaj or even VImin-Vmaj-Imaj. This particular function is what I call the ''Major Contramediant Tensive''.

'''Primary Adpositive Purity''' - This rule is that for every chord root located one step away from either the Tonic, Dominant or Servient along the Circle of Fifths, there is a demand for the fifth of the chord in question to be within 3.5 cents of a just 3/2, though in the case of a chord that has a root located at two steps away from the Tonic along the along the Circle of Fifths, the fifth of the chord can be deliberately subjected to the process of Displacement as mentioned before and not run afoul of this restriction. However, chord roots that are located three or more steps away from the Tonic along the circle of fifths are not subject to the aforementioned restriction due to their distance. This means that the Tonic, Dominant, Servient, Supertonic and Subtonic chords all demand a perfect fifth as the fifth of the chord, whether you are building the Tonality upwards or downwards, but wolf fifths can occur easily as the fifth of chords which are located three or more steps away from the Tonic.

== Beyond Diatonic and Chromatic Functional Harmony ==

In addition to all the aforementioned Diatonic and Chromatic functions, as well as the detemperings of diatonic functions, there is an additional set of categories for dealing with the notes in between the various Diatonic scale degrees.

=== History ===

I've been in the process of developing this since well before I officially joined the Microtonal community, in fact, it all started for me with my discovery of the nature of the eleventh harmonic as a quartertone, however, while it is only thanks to a YouTuber who goes by "Quartertone Harmony" <ref>[https://www.youtube.com/channel/UCeDhhWQYqGRPy5ES8gTtIAQ Quartertone Harmony - YouTube]</ref> that I've been able to fill in significant gaps in my theory, the reality is that the idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees, can trace at least some of its roots back to the work of [[Ivan Wyschnegradsky]].

Specifically, idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees- an idea that has at least some roots in Wyschnegradsky's concept of [[Wikipedia: Major fourth and minor fifth|"Major Fourth" and "Minor Fifth"]]. However, I wanted to use LCJI as a basis for defining these intervals and thus decided to take [[11/8]] as being the just version of Wyschnegradsky's "Major Fourth", and while I drew up sketches based loosely on [[24edo]] for early versions of this concept, I also realized that that two instances of [[33/32]] added up to an interval smaller than [[2187/2048]] but which had a similar function. Furthermore, since two instances of 11/8 resulted in an interval in the vicinity of a Major seventh, I decided to take stacks of 11/8 to form a second navigational axis which works together with the Diatonic Axis in order to define the microtonal functions positioned roughly halfway between the German and Viennese Diatonic functions, though there are a few other microtonal functions as well that are not immediately covered by this second axis.

[[File:Diatonic Function Map.png|thumb|Initial diagram of paradiatonic function locations I made around the time of officially joining the Xenharmonic community. Note that a number of the functions listed on this page are missing, while the Supercollocant, the Superabrogant, the Subabrogant, the Intersubiant, the Interregnant, the Superobstant and the Subobstant initially had different names.]]

Most traditional music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo. In [[Talk:159edo notation #My Second Idea for a Notation System|a conversation]] between myself and [[Kite Giedraitis]] about this topic, Kite mentioned that there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and the aforementioned 2187/2048- a chromatic semitone that is otherwise known as the Apotome- which, when added together, add up to a 9/8 whole tone. Furthermore, Kite also mentioned how in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]]. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields [[27/25]]- two additional diatonic semitones. On the other hand, subtracting 81/80 from the Apotome yields [[135/128]], and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.

Building on Kite's logic, I then decided to apply similar distinctions among quartertones, and thus make the argument that quartertones don't have to denote exactly one fourth of a whole tone in as of themselves, but rather, they only have to add up to a whole tone when paired up correctly. However, the catch was that for quartertones, there are sometimes multiple correct options, making things more complicated. I decided to define the musical functions of quartertones initially on an informal basis by drawing a distinction between the terms "'''Parachromatic'''" (from the prefix ''para-'' in both the senses of ''alongside'' and ''resembling''<ref>[[Wiktionary: para- #Etymology 1]]</ref>, and the word ''chromatic'') and "'''Paradiatonic'''" (from the same two senses of the prefix ''para-'' and the word ''diatonic'') for purposes of classifying quartertone intervals. This was easy, since I had found that two instances of 33/32 added up to [[1089/1024]] rather than 2187/2048, and, since I had informally added the "para-" prefix (in the same senses) to both "Major" and "Minor" to create the terms '''Paramajor''' and '''Paraminor''' to better describe how 11/8 and 16/11 related to 128/99 and 99/64 respectively in order to describe how, for instance, the notes at 99/64 and 16/11 above the Tonic relate to each other in much the same way as major and minor intervals do, except that this relationship occurs in a context where the note halfway between them is actually part of the base scale rather than the two notes in question, and there's a different interval between said two notes than the base scale's chroma.

The way I see it, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit inframinor seconds by default, while parachromatic quartertones are analogous to chromatic semitones in that they are denoted as primes, albiet as ultraprimes by default. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, it can be deduced that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, I ended up choosing the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. As a result of multiple factors, I ended up choosing the combination of three 33/32 parachromatic quartertones and one [[4096/3993]] paradiatonic quartertone as the JI basis for this in regards to both Diatonic theory and [[Alpharabian tuning]], and, through interactions with others in the Xenharmonic community, I was later influenced by others on Discord to take [[MOS]]-based structural considerations into account. This eventually resulted in the first formal definition of a "'''parachroma'''" (an interval that can be easily tempered to equal half of a MOS-chroma), and later, the "'''parastep'''" (the interval that remains after subtracting as many parachromas from a Major MOS-step as possible without resulting in a negative interval). Finally, drawing from the concept of "parachromas" as applied to MOS-based contexts, I was able to finally give a formal definition of terms like "paramajor" (the result of adding a parachroma to either a MOS generator or its period-inverse) and "paraminor" (the result of subtracting a parachroma from a MOS generator or its period-inverse), which I had previously come up with on an informal basis.

== The Realm of the Paradiatonic and the Parachormatic ==

In January of 2022, Quartertone Harmony posted a video in which he grouped together a series of functions he refered to in the video as the "shadow scale"<ref>[https://www.youtube.com/watch?v=P6WJryxB_0Y Quartertone Harmony - Harmonic Functions of Quartertones SD 480p]</ref>, which I will refer to here as a '''paradiatonic scale''' since there are technically two of these, and this in turn led to the separation of Paradiatonic and Parachromatic harmonic functions for me. This whole concept of a "shadow scale", in addition to everything else discuss in this section, paves the way for my idea of [[MOS-Shadow theory]], but, aside from how it applies to Diatonic-scale based functional harmony, MOS-Shadow theory is another whole discussion for another time.

=== The Paradiatonic Scales ===

The Paradiatonic scales from a given tonic acts as a sort of "second shelf" of that tonality. Note that scale degrees in parentheses are optional.

The '''Bright Paradiatonic Scale''' consists of the following scale degrees as analyzed relative to Viennese Theory's scale steps:

I, tII/dbIII, (dIII), tIV, dV, (dVI), tVI/dbVII, tVII

The '''Dark Paradiatonic Scale''' consists of the following scale degrees as analyzed relative to Viennese Theory's scale steps:

I, dbII tII/dbIII, (dIII), tIV, dV, (dVI), tVI/dbVII

While the Diatonic scale itself has seven notes, the two Paradiatonic scales each have eight notes, furthermore, the tunings of each note in each Paradiatonic scale not only depend upon the exact tuning of the Diatonic scale used as a basis, but also vary considerably when it comes to the notes of the Paradiatonic that occur between the main Diatonic interval category ranges.

=== Basic Paradiatonic Functions ===

Out of the various functions found in the Paradiatonic scales, four of them- specifically, the tII/dbIII, tIV, dV and tVI/dbVII- can be considered basic, while the other three are first derivatives. As with the three basic diatonic functions, the four most basic paradiatonic functions have their roots in LCJI. In the order listed, the tII/dbIII, tIV, dV and tVI/dbVII functions are the following...

'''Contravaricant''' - Named in contrast to the Varicant function, this is an interval that maps to both 1\5 and 5\24 in the scale's direction of construction, lying roughly in the middle of the 4/3 interval separating the Tonic and the Serviant above it. Intervals in the Contravaricant region often don't consistently act as either seconds or thirds, or even act as a cross between a second and a third, only with minor potential for crowding in chords depending on the exact tuning. In Bass-Up tonality, this functionality is first encountered in the form of [[8/7]], though [[7/6]] is another notable interval included in this range, with intervals in this range having Predominant, Preservient, and Dominant Parallel functions, as well as an overlap between Tonic Adjacent and Tonic Parallel functions. There are two different subtypes of this function that differ from each other based on how they are approached in interval stacking, the "'''Pseudobidominant'''", which arises from stacking two Pseudodominants, and the "'''Pseudotriservient'''", which arises from stacking three Pseudoservients.

'''Subient''' - This is an interval that maps to both 3\7 and 11\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[15/11]] to around 3.5 cents beyond [[11/8]]. Like both the Servient and the Sycophant, intervals in this region have a Predominant function, however, this predominant function is weaker than that of the surrounding interval regions since they neither act as a counterweight to the Dominant like a Servient nor do they completely tonicize the Dominant like a Sycophant- at least to those who are more familiar with quartertones. What is even less expected is that these same intervals also have Preservient and Mocktyrant functions. The name of this function comes from Latin "subeō"<ref>[[Wiktionary: subeo #Latin]]</ref>, specifically from the senses ranging from "approach" to "follow" and even "undergo", as all of these senses describe what the Subient actually does musically. Since the Subient is not found in the Tonic's quartertone field, there is a tendency for Subient chords to be preceded and or followed by other chords with roots in the same quartertone field, though one could also reasonably approach it from the Tonic, the Servient, some kind of Contramediant, some kind of Supervicinant, or, in some tuning systems, certain kinds of Mediant. In Bass-Up tonality, this functionality has its roots in the eleventh harmonic. This function used to be called the "Intersubiant" on an ad hoc basis, though that term has since been rendered obsolete.

'''Imponent''' - This is an interval that maps to both 4\7 and 13\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[16/11]] to around 3.5 cents beyond [[22/15]]. Accordingly, intervals in this region behave as a cross between a Tyrant Antitonic on one hand and a Dominant on the other in that they often contrast with the Tonic through some combination of harmonic connection and brute force contrast, though it unexpectedly has decent Predominant and Mocksycophant functions. Like the Subient, the Imponent is not found in the Tonic's quartertone field and is either very distant from the Tonic along the circle of fifths or is completely missed by said circle of fifths, thus, there is a tendency for Imponent chords to be preceded and or followed by other chords with roots in the same quartertone field. As if all that weren't enough, the Imponent can be used in the formation of chords, but because of [[delta-rational]] and other considerations, tertian chords framed with this type of interval usually have a markedly different structure imposed on them. The name of this function comes from Latin "impōnō"<ref>[[Wiktionary: impono #Latin]]</ref>, specifically from the senses ranging from "impose upon" to "inflict" and even "establish", as all of these senses describe what the Imponent actually does musically. Conspicuously, the distance between the Imponent and the Subient is only about a Minor Second- or, at the very most, a Supraminor Second- so as a result, the Imponent can be both a set-up and a follow-up to the Subient. In Bass-Up tonality, the Imponent functionality has its roots in the eleventh subharmonic. This function used to be called the "Interregnant" on an ad hoc basis, though that term has since been rendered obsolete.

'''Varicant''' - Just as a Mediant lies roughly in the middle of the 3/2 interval separating the Tonic and the Dominant above it, a Varicant lies roughly in the middle of the 4/3 interval separating the Dominant and the Tonic above it. Intervals in this region often don’t consistently act as either sixths or sevenths, or even act as a cross between a sixth and a seventh, only with minor potential for crowding in chords depending on the exact tuning- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō"<ref>[[Wiktionary: varico #Latin]]</ref>. This is an interval that maps to both 4\5 and 19\24 in the scale's direction of construction. In Bass-Up Tonality, this functionality is first encountered in the form of the [[7/4]] interval, though [[12/7]] is another notable interval included in this range. While many microtonalists think of 7/4 as being purely a type of seventh- and indeed, it most commonly acts as a sort of subminor seventh- I counterargue based on this same interval's relationships with 11/8 in particular that 7/4 is not merely a type of seventh, but rather, a type of a cross between a sixth and a seventh, with such a property explaining why [[14/11]] is generally considered to be a type of third. Furthermore, in contrast to the Subtonics of Bass-Up Tonality, Varicants are liable to act as Predominants, Preservients, and Serviant Parallels, as well as display an overlap between Tonic Adjacent and Tonic Parallel functions. There are two different subtypes of this function that differ from each other based on how they are approached in interval stacking, the "'''Pseudotridominant'''", which arises from stacking three Pseudodominants, and the "'''Pseudobiservient'''", which arises from stacking two Pseudoservients.

=== Derivative Paradiatonic Functions ===

'''Subgradient''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 25 cents below the octave reduplication of the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Subcollocant- however, there are functional differences between a Subgradient and a Subcollocant in Bass-Up Tonality. In Bass-Up Tonality, Subgradients are often more likely to be passing tones than Subcollocants, and, when they’re not merely passing non-chord tones, they are often harder to approach and or follow up without creating some kind of awkward tonal disconnect, with such a disconnect being especially noticeable for intervals like [[64/33]]. Furthermore, whereas a Subcollocant can resolve to the Tonic in part through a strong harmonic connection, a Subgradient is much more likely to do so through sheer brute force, and even these cases require a proper set-up, as otherwise, the awkward tonal disconnect between the Subgradient and the Tonic is likely to result in the Subgradient resolving back down to either the Subcollocant or the Subabrogant. As if that weren't enough, the Subgradient also has the Antsubient function. This function used to be called the "Subdietic", though that term has since been restricted to a related composite function (see below on Composite Functions).

'''Supergradient''' - This is a note that occurs at intervals between roughly 25 cents above the Tonic and 25/24 above the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Supercollocant. However, there are a few functional differences between a Supergradient and a Supercollocant in Bass-Up Tonality that are worth considering. For starters, Supergradients are often more likely to be passing tones than Supercollocants, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward through some sort of semitone-like motion to some form of Supercollocant or Superabrogant, as they are to resolve downwards toward the Tonic, a property which intervals like 33/32 in particular are apt to demonstrate. Furthermore, whereas a Supercollocant can resolve to the Tonic in part through a strong harmonic connection, a Supergradient is much more likely to do so through sheer brute force when such a resolution is noticeable. As if that weren't enough, the Subgradient also has the Antimponent function. This function used to be called the "Superdietic", though that term has since been restricted to a related composite function (see below on Composite Functions).

=== Parachromatic Functions ===

These are quartertone functions that are not on the Paradiatonic Scale. Of these, there are only two basic functions...

'''Superobstant''' - This is a note which, in Bass-Up tonality, is mapped to 9\24. This region is characterized by intervals that don’t consistently act as either thirds or fourths, or even act as a cross between a third and a fourth, as well as by intervals that act as parachromatic alterations of either the Mediant or the Serviant. As per the name, which comes from "super-" and "obstant", the latter of which comes from Latin "obstō"<ref>[[Wiktionary: obsto #Latin]]</ref>, intervals in this region are also generally more dissonant and have a tendency to disrupt both chords and melodies, leading to their avoidance in chords outside of deliberate dissonances. Chords of this type have Preservient, Presubient and Preimponent functions as well as Premediant functions.

'''Subobstant''' - This is a note which, in Bass-Up tonality, is mapped to 15\24. This region is characterized by intervals that don’t consistently act as either fifths or sixths, or even act as a cross between a fifth and a sixth, as well as by intervals that act as parachromatic alterations of either the Dominant or the Contramediant. As per the name, which comes from "sub-" and "obstant", which, as mentioned before, comes from Latin "obstō", intervals in this region are also generally more dissonant, leading to their avoidance in chords outside of deliberate dissonances. Chords of this type have Predominant, Presubient and Preimponent functions, as well as Precontramediant functions, and, perhaps very tellingly, tend to utilize Diminished Fourths instead of Major Thirds due to the functions of the Subgradient- which usually gets incorporated into these kinds of chords.

== Diatonic, Paradiatonic, and the Tonic's Proximal Pythagorean Aura ==

One of the things that Quartertone Harmony has found and mentioned<ref>[https://www.youtube.com/watch?v=3N_l5ciE14g Quartertone Harmony - The Truth About Quartertone Melodies]</ref> is that when dealing with quartertones, there seems to be something about a given 12-tone scale in 24edo which assures that notes in that same field will sound like they go together. However, in systems such as 159edo, you begin to see that things are slightly more complicated, as this distinctive atmosphere or quality turns out to be around 20 cents at widest and seems to surround and be generated by notes related to the Tonic by Pythagorean intervals that aren't all that far away from the Tonic, hence the term '''proximal Pythagorean aura''' to refer to it. Notably, the proximal Pythagorean aura is at its thickest at around six or seven steps away from the Tonic in either direction along the circle of fifths, and extends along the circle of fifths on either side of these areas from two steps away from the Tonic to eleven steps away from the Tonic. The presence of this aura explains things related to the uncanny valleys around the Dominant and Serviant- namely why these uncanny valleys are not as deep or as wide as those around the Tonic, and why simple intervals with single factors of prime 5 seem to be at least somewhat "in key" relative to the Tonic. It should be noted that the Tonic, the Dominant, and the Serviant work together with this aura to define safe regions for field shifts that are guaranteed to not come across as jarring. Intervals outside of this aura, such as most paradiatonic intervals, must abide by certain rules in order to not come across as jarring.

== Additional Composite Functions ==

Notes on the boundaries of functional regions have multiple functions due to occurring at the boundary between different functions, the process by which this happens is known as '''Compositing'''. As for the actual composite functions themselves, there are quite a few of them.

'''Subdietic''' - This function is a compositing of Subcommatic and Subgradient, and as Subcommatic is part of its nature, it is effectively repelled harmonically. It should be noted that this is one of only a handful of composite functions which are formed by the overlap between different functional regions rather than merely being located at a boundary.

'''Superdietic''' - This function is a compositing of Supercommatic and Supergradient, and as Supercommatic is part of its nature, it is effectively repelled harmonically. It should be noted that this is one of only a handful of composite functions which are formed by the overlap between different functional regions rather than merely being located at a boundary.

'''Subsanguinant''' - This function is one of two "bleeding tone" functions- hence its Latin-derived name- and is a compositing of Subgradient and Proximosubcollocant functions, and thus, has a more tense feel than a Proximosubcollocant alone. It should be noted that this function is formed by the overlap between different functional regions rather than merely being located at a boundary.

'''Supersanguinant''' - This function is one of two "bleeding tone" functions- hence its Latin-derived name- and is a compositing of Supergradient and Proximosupercollocant functions, and thus, has a more tense feel than a Proximosupercollocant alone. It should be noted that this function is formed by the overlap between different functional regions rather than merely being located at a boundary.

'''Submaculant''' - This function is a compositing of Subgradient and Subdislocant, and thus is like the chromatic version of the Subsanguinant.

'''Supermaculant''' - This function is a compositing of Supergradient and Superdislocant, and thus is like the chromatic version of the Supersanguinant.

'''Acuotyrant''' - This function is a compositing of Prototyrant and Imponent in Bass-Up Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless operates more on the side of brute force when it contrasts with the Tonic.

'''Gravosycophant''' - This function is a compositing of Protosycophant and Subient in Bass-Up Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless often runs a high risk of tonicizing the Geminodominant.

'''Gravotyrant''' this function is a compositing of Prototyrant and Imponent in Treble-Down Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless operates more on the side of brute force when it contrasts with the Tonic.

'''Acuosycophant''' - This function is a compositing of Protosycophant and Subient in Treble-Down Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless often runs a high risk of tonicizing the Geminodominant.

== Extra Functions of Prime Harmonics and Subharmonics ==

It should be noted that there are other layers of function besides those oriented around the [[3-limit]]. These are detailed here.

'''Paradominant''' - This function typically appears along prime axes other than that of the 3-limit in which they take Dominant-like function. Thus, for example, in Bass-Up Tonality, the note at 5/4 above the Tonic, being a direct prime harmonic of the Tonic, gets to serve this function in addition to its Mesodistomediant function as defined by the 3-limit, with the Paradominant function emerging mainly in a series of 5/4-based motions leading to the Tonic. When considered alongside the actual Dominant function of 3/2, however, Paradominants take on the function of Predominant owing to the fact that the Dominant function of 3/2 is actually stronger than that of any predominant due to being closer to the fundamental.

'''Paraservient''' - This function is essentially the inverse of the Paradominant function, and acts as a sort of counterweight to the Paradominant relative to the Tonic. Like the Paradominant function, it typically appears along prime axes other than that of the 3-limit in which they take Serviant-like function. Thus, for example, in Bass-Up tonality, the note at 8/5 above the Tonic, being a direct prime subharmonic of the Tonic, gets to serve this function in addition to its Mesodistocontramediant function as defined by the 3-limit. When considered alongside the actual Serviant function of 4/3, however, Paraservients take on the function of Preservient since the Serviant function of 4/3 is stronger.

== References ==
<references/>

[[Category:Terms]]
[[Category:Harmony]]
[[Category:Method]]
[[Category:Diatonic]]

User:Overthink/Table of 311edo intervals

2026-01-19T00:44:24Z

Aura: Corrected the step for the Pythagorean Minor Second based on the mappings.

{{Editable user page}}
Note: Second, sixth, and seventh names can be derived from the third names.
{| class="wikitable mw-collapsible right-1 right-2"
|+ Intervals of 311edo
|-
! rowspan="2" | Steps
! rowspan="2" | Cents
! rowspan="2" | Name
! colspan="3" | Approximate Ratios
|-
! 13-limit
! 23-limit
! 41-limit
|-
| 0
| 0.000
|Unison
|[[1/1]]
|
|
|-
| 1
| 3.859
|Keenanisma?
|
|
|
|-
| 2
| 7.717
|Kleisma
|
|
|
|-
| 3
| 11.576
|Semicomma
|
|
|
|-
| 4
| 15.434
|Subcomma
|
|
|
|-
| 5
| 19.293
|Quasicomma
|
|
|
|-
| 6
| 23.151
|Comma
|[[81/80]]
|
|
|-
| 7
| 27.010
|Superprime?
|[[64/63]]
|
|
|-
| 8
| 30.868
|
|
|
|
|-
| 9
| 34.727
|
|
|
|
|-
| 10
| 38.585
|
|
|
|
|-
| 11
| 42.444
|
|
|
|
|-
| 12
| 46.302
|
|
|
|
|-
| 13
| 50.161
|
|[[36/35]]
|
|
|-
| 14
| 54.019
|
|[[33/32]]
|
|
|-
| 15
| 57.878
|
|
|
|
|-
| 16
| 61.736
|Subminor 2nd
|[[28/27]]
|
|
|-
| 17
| 65.595
|
|[[27/26]]
|
|
|-
| 18
| 69.453
|
|[[25/24]], [[26/25]]
|
|
|-
| 19
| 73.312
|
|
|
|
|-
| 20
| 77.170
|
|
|
|
|-
| 21
| 81.029
|
|[[22/21]]
|
|
|-
| 22
| 84.887
|
|[[21/20]]
|
|
|-
| 23
| 88.746
|Pythagorean minor 2nd
|
|
|
|-
| 24
| 92.605
|
|
|
|
|-
| 25
| 96.463
|
|
|
|
|-
| 26
| 100.322
|
|
|
|
|-
| 27
| 104.180
|
|
|
|
|-
| 28
| 108.039
|
|
|
|
|-
| 29
| 111.897
|Classical minor 2nd
|[[16/15]]
|
|
|-
| 30
| 115.756
|
|''[[2187/2048]]''
|
|[[31/29]]
|-
| 31
| 119.614
|
|[[15/14]]
|
|
|-
| 32
| 123.473
|
|
|
|
|-
| 33
| 127.331
|
|[[14/13]]
|
|
|-
| 34
| 131.190
|
|
|
|
|-
| 35
| 135.048
|
|
|
|
|-
| 36
| 138.907
|
|[[13/12]]
|
|
|-
| 37
| 142.765
|
|
|
|
|-
| 38
| 146.624
|
|
|
|
|-
| 39
| 150.482
|
|[[12/11]]
|
|
|-
| 40
| 154.341
|
|
|
|
|-
| 41
| 158.199
|
|
|
|
|-
| 42
| 162.058
|
|
|
|
|-
| 43
| 165.916
|
|[[11/10]]
|
|
|-
| 44
| 169.775
|
|
|
|
|-
| 45
| 173.633
|
|
|
|
|-
| 46
| 177.492
|
|
|
|
|-
| 47
| 181.350
|Classical major 2nd
|[[10/9]]
|
|
|-
| 48
| 185.209
|
|
|
|
|-
| 49
| 189.068
|
|
|
|[[29/26]]
|-
| 50
| 192.926
|Mean major 2nd
|
|[[19/17]]
|
|-
| 51
| 196.785
|
|
|[[28/25]]
|
|-
| 52
| 200.643
|
|
|
|
|-
| 53
| 204.502
|Pythagorean major 2nd
|[[9/8]]
|
|
|-
| 54
| 208.360
|
|
|
|
|-
| 55
| 212.219
|
|
|
|
|-
| 56
| 216.077
|
|
|
|
|-
| 57
| 219.936
|
|
|
|
|-
| 58
| 223.794
|
|
|
|
|-
| 59
| 227.653
|
|
|
|
|-
| 60
| 231.511
|Supermajor 2nd
|[[8/7]]
|
|
|-
| 61
| 235.370
|
|
|
|
|-
| 62
| 239.228
|
|
|
|
|-
| 63
| 243.087
|
|
|
|
|-
| 64
| 246.945
|
|[[15/13]]
|
|
|-
| 65
| 250.804
|
|
|
|
|-
| 66
| 254.662
|
|
|
|
|-
| 67
| 258.521
|
|
|
|
|-
| 68
| 262.379
|
|
|
|
|-
| 69
| 266.238
|Subminor 3rd
|[[7/6]]
|
|
|-
| 70
| 270.096
|
|
|
|
|-
| 71
| 273.955
|
|
|
|
|-
| 72
| 277.814
|
|
|
|
|-
| 73
| 281.672
|Diatismic minor 3rd
|
|
|
|-
| 74
| 285.531
|Pentacircle minor 3rd
|[[33/28]]
|
|
|-
| 75
| 289.389
|Major minthmic minor 3rd
|[[13/11]]
|
|
|-
| 76
| 293.248
|Pythagorean minor 3rd
|[[32/27]]
|
|
|-
| 77
| 297.106
|Boethius minor 3rd
|
|
|
|-
| 78
| 300.965
|Quasi-tempered minor 3rd?
|[[25/21]]
|
|
|-
| 79
| 304.823
|
|
|
|
|-
| 80
| 308.682
|
|
|
|
|-
| 81
| 312.540
|
|
|
|
|-
| 82
| 316.399
|Classical minor 3rd
|[[6/5]]
|
|
|-
| 83
| 320.257
|Keenanismic minor 3rd
|
|
|
|-
| 84
| 324.116
|Marvelous minor 3rd
|
|
|
|-
| 85
| 327.974
|Undetricesimal supraminor 3rd
|
|
|29/24
|-
| 86
| 331.833
|
|
|
|
|-
| 87
| 335.691
|
|
|
|
|-
| 88
| 339.550
|
|
|
|
|-
| 89
| 343.408
|Tridecimal subneutral 3rd
|[[39/32]]
|
|
|-
| 90
| 347.267
|Undecimal artoneutral 3rd
|[[11/9]]
|
|
|-
| 91
| 351.125
|Neutral 3rd
|
|
|
|-
| 92
| 354.984
|Undecimal tendoneutral 3rd
|[[27/22]]
|
|
|-
| 93
| 358.842
|Tridecimal supraneutral 3rd
|[[16/13]]
|
|
|-
| 94
| 362.701
|
|
|
|
|-
| 95
| 366.559
|
|
|
|
|-
| 96
| 370.418
|
|
|
|
|-
| 97
| 374.277
|Undetricesimal submajor 3rd
|
|
|36/29
|-
| 98
| 378.135
|
|
|
|
|-
| 99
| 381.994
|
|
|
|
|-
| 100
| 385.852
|Classical major 3rd
|[[5/4]]
|
|
|-
| 101
| 389.711
|
|
|
|
|-
| 102
| 393.569
|
|
|
|
|-
| 103
| 397.428
|
|
|
|
|-
| 104
| 401.286
|
|
|
|
|-
| 105
| 405.145
|Boethius major 3rd
|
|
|
|-
| 106
| 409.003
|Pythagorean major 3rd
|
|
|
|-
| 107
| 412.862
|Major minthmic major 3rd
|
|
|
|-
| 108
| 416.720
|Pentacircle major 3rd
|[[14/11]]
|
|
|-
| 109
| 420.579
|Diatismic major 3rd
|
|
|
|-
| 110
| 424.437
|Shrubmajor 3rd
|
|[[23/18]]
|
|-
| 111
| 428.296
|
|
|
|
|-
| 112
| 432.154
|Swetismic supermajor 3rd
|
|
|
|-
| 113
| 436.013
|Septimal major 3rd
|[[9/7]]
|
|
|-
| 114
| 439.871
|
|
|
|
|-
| 115
| 443.730
|
|
|
|
|-
| 116
| 447.588
|
|
|
|
|-
| 117
| 451.447
|
|
|
|
|-
| 118
| 455.305
|
|
|
|
|-
| 119
| 459.164
|
|
|
|
|-
| 120
| 463.023
|
|
|
|
|-
| 121
| 466.881
|
|
|
|
|-
| 122
| 470.740
|
|
|
|
|-
| 123
| 474.598
|
|
|
|
|-
| 124
| 478.457
|
|
|
|
|-
| 125
| 482.315
|
|
|
|
|-
| 126
| 486.174
|
|
|
|
|-
| 127
| 490.032
|
|
|
|
|-
| 128
| 493.891
|
|
|
|
|-
| 129
| 497.749
|Perfect fourth
|[[4/3]]
|
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| 130
| 501.608
|
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| 131
| 505.466
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| 132
| 509.325
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| 513.183
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| 517.042
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| 135
| 520.900
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| 524.759
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| 528.617
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| 532.476
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| 139
| 536.334
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| 540.193
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| 544.051
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| 547.910
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| 551.768
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| 555.627
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| 559.486
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| 563.344
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| 567.203
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| 571.061
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| 149
| 574.920
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| 150
| 578.778
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| 582.637
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| 586.495
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| 590.354
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| 594.212
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| 598.071
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| 601.929
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| 605.788
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| 609.646
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| 613.505
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| 617.363
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| 621.222
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| 625.080
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| 632.797
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| 636.656
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| 640.514
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| 644.373
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| 648.232
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| 652.090
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| 655.949
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| 671.383
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| 675.241
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| 679.100
|Grave fifth
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| 694.534
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| 698.392
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| 702.251
|Perfect fifth
|[[3/2]]
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| 1080.386
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| 1095.820
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| 1099.678
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| 1103.537
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| 1107.395
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| 1111.254
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| 1115.113
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| 1118.971
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| 1122.830
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| 1138.264
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| 1142.122
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| 1145.981
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| 1149.839
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| 1153.698
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| 1157.556
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| 1161.415
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| 1165.273
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| 1169.132
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| 1172.990
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| 1176.849
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| 1180.707
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| 1184.566
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| 1188.424
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| 1192.283
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| 1196.141
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| 1200.000
|Octave
|[[2/1]]
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|}

User talk:Overthink/Table of 311edo intervals

2026-01-19T00:33:28Z

Aura:

== Superprime ==

I don't know about you, but I'd label 64/63 as the superprime. As for 1\311, I think we need a better name... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 23:47, 18 January 2026 (UTC)

Ooh! How about the Keenanisma for 1\311? That's about how big that comma is anyways... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 23:48, 18 January 2026 (UTC)

: Oh yeah, the keenanisma is quite important, and seperates many important intervals. 64/63 as the superprime makes sense as well, due to 8/7 being a supermajor 2nd and 9/8 being a major 2nd.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 00:00, 19 January 2026 (UTC)

:: Oh, and 33/32 would be the Ultraprime, and you can derive ultramajor and inframinor from ~33/32 relative to Pythagorean intervals, such as 8192/8019 being the inframinor second and 297/256 being the ultramajor second. There's also the paramajor fourth (11/8), and the paraminor fourth (128/99) and the paramajor fifth (99/64) and the paraminor fifth (16/11). --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 00:32, 19 January 2026 (UTC)

User talk:Overthink/Table of 311edo intervals

2026-01-19T00:32:53Z

Aura:

== Superprime ==

I don't know about you, but I'd label 64/63 as the superprime. As for 1\311, I think we need a better name... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 23:47, 18 January 2026 (UTC)

Ooh! How about the Keenanisma for 1\311? That's about how big that comma is anyways... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 23:48, 18 January 2026 (UTC)

: Oh yeah, the keenanisma is quite important, and seperates many important intervals. 64/63 as the superprime makes sense as well, due to 8/7 being a supermajor 2nd and 9/8 being a major 2nd.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 00:00, 19 January 2026 (UTC)

: Oh, and 33/32 would be the Ultraprime, and you can derive ultramajor and inframinor from ~33/32 relative to Pythagorean intervals, such as 8192/8019 being the inframinor second and 297/256 being the ultramajor second. There's also the paramajor fourth (11/8), and the paraminor fourth (128/99) and the paramajor fifth (99/64) and the paraminor fifth (16/11). --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 00:32, 19 January 2026 (UTC)

User talk:Overthink/Table of 311edo intervals

2026-01-18T23:48:30Z

Aura:

User talk:Overthink/Table of 311edo intervals

2026-01-18T23:47:05Z

Aura: /* Superprime */ new section

2.3.5.11 subgroup

2026-01-05T17:19:13Z

Aura:

The '''2.3.5.11 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 5, and 11 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 11. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[11/8]], [[11/9]], [[27/22]], and so on.

In can be thought as either an extension of [[Alpharabian tuning]] with the familiar 5-limit chords and stuctures, or a retraction of the 11-limit by removing prime 7. It can be similar to the [[2.3.5.13 subgroup]], specially considering neutral interval pairs such as 39/32 ~ 11/9 and 16/13 ~ 27/22, which are connected by the small comma of [[352/351]].

== Regular temperaments ==
=== Rank-1 temperaments (edos) ===
It is relatively well approximated by the following edos (bold ones do particularly well in this subgroup): [[7edo|7]], [[15edo|15]], [[22edo|22]], [[24edo|'''24''']], [[31edo|31]], [[38edo|38]], [[41edo|41]], [[46edo|46]], [[65edo|'''65''']], [[72edo|'''72''']], [[80edo|80]], [[87edo|'''87''']], [[94edo|94]], [[96edo|96]], [[118edo|'''118''']], [[130edo|130]], [[137edo|137]], [[159edo|'''159''']], [[183edo|183]], [[217edo|217]], [[224edo|224]], [[270edo|'''270''']], [[311edo|311]], …

=== Rank-2 temperaments ===
[[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[cassandra]] mapping, and another by finding 11/8 at the quadruple-diminished seventh (-30 fifths) through the [[helenus]] mapping. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction.

[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. [[65edo]] is the intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.

=== Rank-3 temperaments ===
[[Vishdel]] provides a low-complexity, accurate temperament, but for those searching a much higher accuracy system, [[tritomere]] is among the best rank-3 temperaments for this case, having tremendous accuracy with manageable complexity, tempering out the difference between three [[243/242|rastmas]] and one [[81/80|syntonic comma]] (0.08 cents). Its boundary of usability begins at [[152edo|152]] and [[159edo]], the latter inheriting the marvelous fifths from 53edo, one that [[Aura]] has shown great interest in. Bigger edos that support this excellent temperament include [[342edo]], [[494edo]], [[677edo]], [[1171edo]], among others.

User:Eufalesio/Telicity

2025-12-29T17:20:03Z

Aura:

{{Editable user page}}This is a major rewrite of the "[[Telicity]]" article, as it was incredibly mathed up and very hard to parse and understand what is meant to be explained. So I stepped in. (Thanks to [[Aura]] for suggestions and help)

There's still math, but ''much'' less math. And also continued fractions are important.

Feel free to change anything after '''ARTICLE START'''. I left out loads of cool bits, but I can't be writing articles all day now can I?

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

== '''ARTICLE START''' ==
'''Telicity''' is a property of both [[equal temperament]]s and [[comma]]s and how they relate to each other. An edo is p-2 '''telic''' when it tempers a comma in 2.p subgroup for a prime p, and that comma is smaller than half an edostep.

Commas and equal temperaments that demonstrate this property are referred to as as being '''telic'''. When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be '''multitelic'''.

== Telicity and Continued fractions ==

In order to understand how and why telicity is useful, one must first look at continued fractions to see how telicity can be derived.

n-m telicity in any equal division of n satisfies the following:

* The equal division of m is a denominator appearing in the continued fraction of logm(n).
* The comma that arises from stacking m<sup>numerator</sup>/n<sup>denominator</sup> of the convergent is smaller than half an ed-m-step.

Mathematically, this is satisfied with the following:

<math>d_{enominator}\left(C\right)\left|d_{enominator}\left(C\right)\log_{m}\left(n\right)-n_{umerator}\left(C\right)\right|<\frac{1}{2}</math>

When m is equal to 2, the result is an n-2 telic edo, if it's not equal to 2, it's an [[edonoi]].

=== Multitelicity ===
If said produced comma is also smaller than k halves of an ed-m-step, then the edm is k-strong m-n telic, which means that the comma is smaller than not only half of an ed-m-step, but also half/2 (a quarter), or half/3 (a sixth)... etc. Essentially not only the (denominator)ed-m is convergent, but also its multiples. This makes it '''multitelic'''.

Mathematically, this is expressed as the following:

<math>k\cdot d_{enominator}\left(C\right)\left|d_{enominator}\left(C\right)\log_{m}\left(n\right)-n_{umerator}\left(C\right)\right|<\frac{1}{2}</math>

Multitelicity is not the same as having many telicities. For example, 12edo is 3-2 telic and 5-2 telic, but only multitelic in the 3-2.

=== Examples ===
Here is the series of convergents for log2(3):

* 1/[[1edo|1]], 2/1, 3/[[2edo|2]], 8/[[5edo|5]], 19/[[12edo|12]], 65/[[41edo|41]], 84/[[53edo|53]], 485/[[306edo|306]], 1054/[[665edo|665]], 24727/[[15601edo|15601]], 50508/[[31867edo|31867]], 125743/[[79335edo|79335]], 176251/[[111202edo|111202]]...

The commas that arise from these edos are the following, with the corresponding :

* 1edo: [[4/3]] [<nowiki/>[[Bixby|bixby, degenerate case]]]
* 2edo: [[9/8]] [<nowiki/>[[Very low accuracy temperaments#Antitonic|antitonic, degenerate case]]]
* 5edo: [[256/243]] [<nowiki/>[[blackwood]]]
* 12edo: [[Pythagorean comma|531441/524288]] [<nowiki/>[[compton]]]
* ''41edo:'' [[41-comma|[65 -41⟩]] [<nowiki/>[[Countercomp family|countercomp]]]
* 53edo: [[Mercator's comma|[-84 53⟩]] [<nowiki/>[[mercator]]]
* 306edo: [[Qian's small comma|[485 -306⟩]] [sasktel?]
* 665edo: [[Satanic comma|[-1054 665⟩]] [satanic?]

Of those, 41edo is not telic because its comma, the countercomp comma, is larger than half an edostep. (19.845*2 > 29.268). The next non-telic convergent is [[111202edo]].

Of those, 12, 53, 665 are multitelic, because they have a k-strength value greater than one; being 2, 3, and 11 respectively, which means that [[24edo|24]], [[106edo|106]], [[159edo|159]], [[1330edo|1330]], [[1995edo|1995]], [[2660edo|2660]], [[3325edo|3325]], [[3990edo|3990]], [[4655edo|4655]], [[5320edo|5320]], [[5985edo|5985]], [[6650edo|6650]], and [[7315edo|7315]] are also 3-2 telic.

== Applications ==

=== Prime approximations ===

3-2 telic edos have record-breakingly accurate [[3/2|perfect fifths]]. As well as 5-2 telic edos having record-breakingly accurate [[5/4|ptolemaic major thirds]], and so on. These telic edos can stack their optimized intervals extremely well with minimal error, being perfect for incredibly intricate modulations, and overall because they offer astoundingly great approximations of intervals within their telic subgroups.

Non-telic edos with convergent fifths, or semiconvergent fifths (which are never telic), are also incredibly good and offer comparably great intervals, specially if the edo is big enough. Edos with semiconvergent fifths include [[7edo|7]], [[17edo|17]], [[29edo|29]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[359edo|359]], [[971edo|971]]...

=== MOS ===
MOS scales generated by a pure prime interval have [[strictly proper]] scales, with the softest hardness when they have a scale size that corresponds with a telic edo. For Pythagorean tuning these include [[1L 1s|1L 1s (monowood)]], [[2L 3s|2L 3s (pentic)]], [[5L 7s|5L 7s (p-chromatic)]], [[41L 12s]].

Scale sizes for edos that have semiconvergent or non-telic convergent generators may generate proper but never strictly proper, or improper scales ''(I hypothesize this)''

=== Table of P-2 telic edos ===
{| class="wikitable"
|+
|3-2 telic
| rowspan="4" |1
|2
|5
|12
2-strong
|53
3-strong
|306
|665
11-strong
|15601
|31867
|79335
|190537
28-strong
|-
|5-2 telic
|3
4-strong
|28
|59
|146
2-strong
|643
3-strong
|4004
|8651
|12655
|21306
2-strong
|97879
9-strong
|-
|7-2 telic
|5
2-strong
|26
2-strong
|109
2-strong
|571
2-strong
|2694
15-strong
|91313
2-strong
|453601
4-strong
|
|
|
|-
|11-2 telic
|2
3-strong
|13
|37
13-strong
|986
|1935
|4856
|16503
12-strong
|
|
|
|}
WIP

== See also ==
* [[Consistent circle]]
[[Category:EDO theory pages]]
[[Category:Terms]]

User:Eufalesio/Telicity

2025-12-29T17:18:47Z

Aura:

{{Editable user page}}This is a major rewrite of the "[[Telicity]]" article, as it was incredibly mathed up and very hard to parse and understand what is meant to be explained. So I stepped in. (Thanks to [[Aura]] for suggestions and help)

There's still math, but ''much'' less math. And also continued fractions are important.

Feel free to change anything after '''ARTICLE START'''. I left out loads of cool bits, but I can't be writing articles all day now can I?

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

== '''ARTICLE START''' ==
'''Telicity''' is a property of both [[equal temperament]]s and [[comma]]s and how they relate to each other. An edo is p-2 '''telic''' when it tempers a comma in 2.p subgroup for a prime p, and that comma is smaller than half an edostep.

Commas and equal temperaments that demonstrate this property are referred to as as being '''telic'''. When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be '''multitelic'''.

== Telicity and Continued fractions ==

In order to understand how and why telicity is useful, one must first look at continued fractions to see how telicity can be derived.

n-m telicity in any equal division of n satisfies the following:

* The equal division of m is a denominator appearing in the continued fraction of logm(n).
* The comma that arises from stacking m<sup>numerator</sup>/n<sup>denominator</sup> of the convergent is smaller than half an ed-m-step.

Mathematically, this is satisfied with the following:

<math>d_{enominator}\left(C\right)\left|d_{enominator}\left(C\right)\log_{m}\left(n\right)-n_{umerator}\left(C\right)\right|<\frac{1}{2}</math>

When m is equal to 2, the result is an n-2 telic edo, if it's not equal to 2, it's an [[edonoi]].

=== Multitelicity ===
If said produced comma is also smaller than k halves of an ed-m-step, then the edm is k-strong m-n telic, which means that the comma is smaller than not only half of an ed-m-step, but also half/2 (a quarter), or half/3 (a sixth)... etc. Essentially not only the (denominator)ed-m is convergent, but also its multiples. This makes it '''multitelic'''.

Mathematically, this is expressed as the following:

<math>k\cdot d_{enominator}\left(C\right)\left|d_{enominator}\left(C\right)\log_{m}\left(n\right)-n_{umerator}\left(C\right)\right|<\frac{1}{2}</math>

Multitelicity is not the same as having many telicities. For example, 12edo is 3-2 telic and 5-2 telic, but only multitelic in the 3-2.

=== Examples ===
Here is the series of convergents for log2(3):

* 1/[[1edo|1]], 2/1, 3/[[2edo|2]], 8/[[5edo|5]], 19/[[12edo|12]], 65/[[41edo|41]], 84/[[53edo|53]], 485/[[306edo|306]], 1054/[[665edo|665]], 24727/[[15601edo|15601]], 50508/[[31867edo|31867]], 125743/[[79335edo|79335]], 176251/[[111202edo|111202]]...

The commas that arise from these edos are the following, with the corresponding :

* 1edo: [[4/3]] [<nowiki/>[[Bixby|bixby, degenerate case]]]
* 2edo: [[9/8]] [<nowiki/>[[Very low accuracy temperaments#Antitonic|antitonic, degenerate case]]]
* 5edo: [[256/243]] [<nowiki/>[[blackwood]]]
* 12edo: [[Pythagorean comma|531441/524288]] [<nowiki/>[[compton]]]
* ''41edo: [[41-comma|[65 -41⟩]] [<nowiki/>[[Countercomp family|countercomp]]]
* 53edo: [[Mercator's comma|[-84 53⟩]] [<nowiki/>[[mercator]]]
* 306edo: [[Qian's small comma|[485 -306⟩]] [sasktel?]
* 665edo: [[Satanic comma|[-1054 665⟩]] [satanic?]

Of those, 41edo is not telic because its comma, the countercomp comma, is larger than half an edostep. (19.845*2 > 29.268). The next non-telic convergent is [[111202edo]].

Of those, 12, 53, 665 are multitelic, because they have a k-strength value greater than one; being 2, 3, and 11 respectively, which means that [[24edo|24]], [[106edo|106]], [[159edo|159]], [[1330edo|1330]], [[1995edo|1995]], [[2660edo|2660]], [[3325edo|3325]], [[3990edo|3990]], [[4655edo|4655]], [[5320edo|5320]], [[5985edo|5985]], [[6650edo|6650]], and [[7315edo|7315]] are also 3-2 telic.

== Applications ==

=== Prime approximations ===

3-2 telic edos have record-breakingly accurate [[3/2|perfect fifths]]. As well as 5-2 telic edos having record-breakingly accurate [[5/4|ptolemaic major thirds]], and so on. These telic edos can stack their optimized intervals extremely well with minimal error, being perfect for incredibly intricate modulations, and overall because they offer astoundingly great approximations of intervals within their telic subgroups.

Non-telic edos with convergent fifths, or semiconvergent fifths (which are never telic), are also incredibly good and offer comparably great intervals, specially if the edo is big enough. Edos with semiconvergent fifths include [[7edo|7]], [[17edo|17]], [[29edo|29]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[359edo|359]], [[971edo|971]]...

=== MOS ===
MOS scales generated by a pure prime interval have [[strictly proper]] scales, with the softest hardness when they have a scale size that corresponds with a telic edo. For Pythagorean tuning these include [[1L 1s|1L 1s (monowood)]], [[2L 3s|2L 3s (pentic)]], [[5L 7s|5L 7s (p-chromatic)]], [[41L 12s]].

Scale sizes for edos that have semiconvergent or non-telic convergent generators may generate proper but never strictly proper, or improper scales ''(I hypothesize this)''

=== Table of P-2 telic edos ===
{| class="wikitable"
|+
|3-2 telic
| rowspan="4" |1
|2
|5
|12
2-strong
|53
3-strong
|306
|665
11-strong
|15601
|31867
|79335
|190537
28-strong
|-
|5-2 telic
|3
4-strong
|28
|59
|146
2-strong
|643
3-strong
|4004
|8651
|12655
|21306
2-strong
|97879
9-strong
|-
|7-2 telic
|5
2-strong
|26
2-strong
|109
2-strong
|571
2-strong
|2694
15-strong
|91313
2-strong
|453601
4-strong
|
|
|
|-
|11-2 telic
|2
3-strong
|13
|37
13-strong
|986
|1935
|4856
|16503
12-strong
|
|
|
|}
WIP

== See also ==
* [[Consistent circle]]
[[Category:EDO theory pages]]
[[Category:Terms]]

441/440

2025-12-14T17:22:08Z

Aura:

{{Infobox Interval
| Name = werckisma
| Color name = 1uzzg2, luzozogu 2nd,<br>Luzozogu comma
| Comma = yes
}}
'''441/440''', the '''werckisma''', also known as ''Werckmeister's undecimal septenarian schisma'', is a [[small comma|small]] [[superparticular]] [[11-limit]] [[comma]] with a size of roughly 3.93 [[cent]]s. It is the difference between [[21/20]] and [[22/21]], and between [[80/63]] ( = ([[10/9]])⋅([[8/7]])) and [[14/11]], as well as the amount by which a stack of two [[21/16]] septimal subfourths exceeds a single [[55/32]] supermajor sixth. It also arises as the following comma differences: ([[49/48]])/([[55/54]]), ([[56/55]])/([[64/63]]), ([[81/80]])/([[99/98]]), ([[126/125]])/([[176/175]]), and ([[243/242]])/([[540/539]]).

== Temperaments ==
Tempering it out splits [[11/10]] into two even halves by equating 21/20 and 22/21, splits [[55/32]] into two even halves by equating 21/16 with [[55/42]], and makes [[werckismic chords]] possible. See [[Rank-4 temperament #Werckismic (441/440)]] for some technical data. See [[Werckismic temperaments]] for a collection of rank-3 temperaments where it is tempered out.

== See also ==
* [[List of superparticular intervals]]

[[Category:Werckismic]]
[[Category:Commas named after composers]]
[[Category:Commas named after music theorists]]

User:Aura/Aura's introduction to 159edo

2025-11-29T17:54:51Z

Aura:

While large EDOs like [[159edo]] have a bit of a learning curve compared to smaller systems, the fact that this system sports not only consistency to the [[17-odd-limit]], but also both additional options for imitating the pitch-hue palettes of smaller tuning systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, and, the ability to perform sophisticated harmonic and melodic maneuvers that involve substituting intervals that are one or two steps away from the equivalents of traditional musical intervals in such a way as to fly under the radar of many listeners, all while having a step size that is above the average listener's melodic [[JND]], makes this system well worth the price of admission- particularly for those who can use or make synthesizers and other digital-system-based instruments.

== Intervals and Notation ==

159edo contains all the intervals of [[53edo]], however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. Furthermore, just as with 24edo can be thought of as essentially having two fields of 12edo separated by a quartertone, 159edo can be thought of as having three fields of 53edo, each separated from the others by a third of a 53edo step on either side. This even lends to 159edo having its own variation on the [[Dinner Party Rules]]—represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals
|-
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! [[SKULO interval names|SKULO]]-based interval names
! [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic]]-based interval names
! [[Syntonic-rastmic subchroma notation|SRS notation]]
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Prime
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superprime
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superprime
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superprime, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superprime
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraprime
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraprime, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Prime
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Prime
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Prime
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Prime
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Prime
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Prime
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Prime
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Prime
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Prime
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Prime
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Prime
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Prime
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Prime
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== 5-limit diatonic music ==

Although 159edo inherits 53edo's close approximations of both the [[5-limit]] Zarlino scale and the [[3-limit]] [[diatonic]] MOS, these are not the only scales that one can use for fixed-pitch diatonic music even in that system. In fact, both of them are less than optimal for many facets of traditional Western classical music, seeing as for the most part, each of the traditional diatonic modes has its own optimized scale in the 5-limit. The reason for this is that in non-meantone 5-limit systems, one inevitably has to deal with the ~[[40/27]] wolf fifth and the ~[[27/20]] wolf fourth, and there's only two ideal positions in the scale for those intervals to be situated in fixed-pitch diatonic music. For major modes, the ~27/20 wolf fourth is ideally situated between the ~[[5/4]] major third and the ~[[27/16]] major sixth, while for the minor modes and the diatonic blighted mode Locrian, the ~40/27 wolf fifth is ideally situated between the ~[[6/5]] minor third and the ~[[16/9]] minor seventh. Not only that, but it also pays to distinguish the Pythagorean major and minor thirds from their Ptolemaic counterparts as both types of major and minor third have distinct roles to play in the 5-limit diatonic music.

=== Scales and Harmony ===

For purposes of this section, we shall assume the tonic to be D-natural as in the interval chart above. Note that the following trines are available in 5-limit diatonic harmony.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo diatonic trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Supradusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|}

==== Ionian and Major ====

This mode- along the corresponding tonality- is optimized for 5-limit using a variation on the Didymian diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Ionian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| ↓VII
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| I
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| III
| 10:12:15:18
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Major with Ptolemaic Major Seventh
| D, F#, A, C#↓
| 0, 54, 93, 144
| IV
| 64:81:96:120
| This dissonant tetrad is an unconventional servient (subdominant) seventh for 5-limit music, and is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Dominant Seventh
| D, F#↓, A, C
| 0, 51, 93, 132
| V
| 36:45:54:64
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~3/2 below its root
|-
| Supradusthumic Pythagorean Minor Seventh
| D, F, A↓, C
| 0, 39, 93, 132
| VI
| 54:32:40:48
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Half-Diminished
| D, F↑, Ab↑, C↑
| 0, 42, 81, 135
| VII
| 45:54:64:81
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Ionian in 159edo is considerably more tonally stable than its more familiar counterpart from 12edo and 24edo. It is also one of only two traditional diatonic modes in which one is able to perform a complete [[Wikipedia: Vi–ii–V–I|circle progression]].

==== Dorian ====

Unlike what is expected, the form of Dorian mode optimized for 5-limit is not symmetrical, owing to the need to maintain [[Rothenberg propriety]].

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Dorian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV, bVII
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II, V
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| VI
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Pythagorean Major Seventh
| D, F#, A, C#
| 0, 54, 93, 147
| bVII
| 128:162:192:243
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II, V
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Supradusthumic Ptolemaic Major Seventh
| D, F#↓, A↓, C#↓
| 0, 51, 90, 144
| b↑III
| 240:256:320:405
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Parallel Dominant Seventh
| D, F#, A, C↑
| 0, 54, 93, 135
| IV
| 1/(45:54:64:81)
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~9/8 above its root
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| VI
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Dorian in 159edo has a few noteworthy tricks- especially once one considers suspensions, which will have to be covered later in this document.

==== Phrygian ====

This mode is optimized for 5-limit using a variation on the left-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Phrygian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Ptolemaic Minor Second
| Eb↑
| 15
| Distosupercollocant
| [[16/15]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Ptolemaic Minor Sixth
| Bb↑
| 108
| Mesodistocontramediant
| [[8/5]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| b↑II, b↑VI
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I, IV, bVII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| V
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑II
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Supradusthumic Ptolemaic Dominant Seventh
| D, F#↓, A↓, C
| 0, 51, 90, 132
| b↑III
| 1/(45:54:64:80)
| This dissonant dominant tetrad has a slightly different function than its more traditional counterpart
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| IV, bVII
| 10:12:15:18
|
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| V
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑VI
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|}

With the above scale steps and chords, this variation of Phrygian in 159edo actually presents unexpected possibilities compared to its more traditional counterpart.

==== Lydian ====

This mode is optimized for 5-limit using a variation on the right-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Lydian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G#↓
| 78
| Sycophant
| [[45/32]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Lydian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, II, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III, ↓VII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| #↓IV
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

[[Category:Method]]
[[Category:Approaches to tuning systems]]

== Notes ==
<references group="note" />

User:Aura/Aura's introduction to 159edo

2025-11-29T17:54:33Z

Aura:

While large EDOs like [[159edo]] have a bit of a learning curve compared to smaller systems, the fact that this system sports not only consistency to the [[17-odd-limit]], but also both additional options for imitating the pitch-hue palettes of smaller tuning systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, and, the ability to perform sophisticated harmonic and melodic maneuvers that involve substituting intervals that are one or two steps away from the equivalents of traditional musical intervals in such a way as to fly under the radar of many listeners, all while having a step size that is above the average listener's melodic [[JND]], makes this system well worth the price of admission- particularly for those who can use or make synthesizers and other digital-system-based instruments.

== Intervals and Notation ==

159edo contains all the intervals of [[53edo]], however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. Furthermore, just as with 24edo can be thought of as essentially having two fields of 12edo separated by a quartertone, 159edo can be thought of as having three fields of 53edo, each separated from the others by a third of a 53edo step on either side. This even lends to 159edo having its own variation on the [[Dinner Party Rules]]—represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals
|-
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! [[SKULO interval names|SKULO]]-based interval names
! [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic]]-based interval names
! [[Syntonic-rastmic subchroma notation|SRS notation]]
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Prime
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superprime
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superprime
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superprime, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superprime
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraprime
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraprime, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Prime
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Prime
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Prime
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Prime
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Prime
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Prime
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Prime
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Prime
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Prime
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Prime
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Prime
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Prime
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Prime
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== 5-limit diatonic music ==

Although 159edo inherits 53edo's close approximations of both the [[5-limit]] Zarlino scale and the [[3-limit]] [[diatonic]] MOS, these are not the only scales that one can use for fixed-pitch diatonic music even in that system. In fact, both of them are less than optimal for many facets of traditional Western classical music, seeing as for the most part, each of the traditional diatonic modes has its own optimized scale in the 5-limit. The reason for this is that in non-meantone 5-limit systems, one inevitably has to deal with the ~[[40/27]] wolf fifth and the ~[[27/20]] wolf fourth, and there's only two ideal positions in the scale for those intervals to be situated in fixed-pitch diatonic music. For major modes, the ~27/20 wolf fourth is ideally situated between the ~[[5/4]] major third and the ~[[27/16]] major sixth, while for the minor modes and the diatonic blighted mode Locrian, the ~40/27 wolf fifth is ideally situated between the ~[[6/5]] minor third and the ~[[16/9]] minor seventh. Not only that, but it also pays to distinguish the Pythagorean major and minor thirds from their Ptolemaic counterparts as both types of major and minor third have distinct roles to play in the 5-limit diatonic music.

=== Scales and Harmony ===

For purposes of this section, we shall assume the tonic to be D-natural as in the interval chart above. Note that the following trines are available in 5-limit diatonic harmony.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo diatonic trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Supradusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|}

==== Ionian and Major ====

This mode- along the corresponding tonality- is optimized for 5-limit using a variation on the Didymian diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Ionian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| ↓VII
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| I
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| III
| 10:12:15:18
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Major with Ptolemaic Major Seventh
| D, F#, A, C#↓
| 0, 54, 93, 144
| IV
| 64:81:96:120
| This dissonant tetrad is an unconventional servient (subdominant) seventh for 5-limit music, and is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Dominant Seventh
| D, F#↓, A, C
| 0, 51, 93, 132
| V
| 36:45:54:64
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~3/2 below its root
|-
| Supradusthumic Pythagorean Minor Seventh
| D, F, A↓, C
| 0, 39, 93, 132
| VI
| 54:32:40:48
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Half-Diminished
| D, F↑, Ab↑, C↑
| 0, 42, 81, 135
| VII
| 45:54:64:81
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Ionian in 159edo is considerably more tonally stable than its more familiar counterpart from 12edo and 24edo. It is also one of only two traditional diatonic modes in which one is able to perform a complete [[Wikipedia: Vi–ii–V–I|circle progression]].

==== Dorian ====

Unlike what is expected, the form of Dorian mode optimized for 5-limit is not symmetrical, owing to the need to maintain [[Rothenberg propriety]].

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Dorian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV, bVII
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II, V
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| VI
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Pythagorean Major Seventh
| D, F#, A, C#
| 0, 54, 93, 147
| bVII
| 128:162:192:243
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II, V
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Supradusthumic Ptolemaic Major Seventh
| D, F#↓, A↓, C#↓
| 0, 51, 90, 144
| b↑III
| 240:256:320:405
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Parallel Dominant Seventh
| D, F#, A, C↑
| 0, 54, 93, 135
| IV
| 1/(45:54:64:81)
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~9/8 above its root
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| VI
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Dorian in 159edo has a few noteworthy tricks- especially once one considers suspensions, which will have to be covered later in this document.

==== Phrygian ====

This mode is optimized for 5-limit using a variation on the left-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Phrygian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Ptolemaic Minor Second
| Eb↑
| 15
| Distosupercollocant
| [[16/15]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Ptolemaic Minor Sixth
| Bb↑
| 108
| Mesodistocontramediant
| [[8/5]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| b↑II, b↑VI
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I, IV, bVII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| V
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑II
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Supradusthumic Ptolemaic Dominant Seventh
| D, F#↓, A↓, C
| 0, 51, 90, 132
| b↑III
| 1/(45:54:64:80)
| This dissonant dominant tetrad has a slightly different function than its more traditional counterpart
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| IV, bVII
| 10:12:15:18
|
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| V
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑VI
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|}

With the above scale steps and chords, this variation of Phrygian in 159edo actually presents unexpected possibilities compared to its more traditional counterpart.

==== Lydian ====

This mode is optimized for 5-limit using a variation on the right-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Lydian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G#↓
| 78
| Sycophant
| [[45/32]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic LYdian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, II, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III, ↓VII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| #↓IV
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

[[Category:Method]]
[[Category:Approaches to tuning systems]]

== Notes ==
<references group="note" />

User:Aura/Aura's introduction to 159edo

2025-11-29T17:54:06Z

Aura:

While large EDOs like [[159edo]] have a bit of a learning curve compared to smaller systems, the fact that this system sports not only consistency to the [[17-odd-limit]], but also both additional options for imitating the pitch-hue palettes of smaller tuning systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, and, the ability to perform sophisticated harmonic and melodic maneuvers that involve substituting intervals that are one or two steps away from the equivalents of traditional musical intervals in such a way as to fly under the radar of many listeners, all while having a step size that is above the average listener's melodic [[JND]], makes this system well worth the price of admission- particularly for those who can use or make synthesizers and other digital-system-based instruments.

== Intervals and Notation ==

159edo contains all the intervals of [[53edo]], however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. Furthermore, just as with 24edo can be thought of as essentially having two fields of 12edo separated by a quartertone, 159edo can be thought of as having three fields of 53edo, each separated from the others by a third of a 53edo step on either side. This even lends to 159edo having its own variation on the [[Dinner Party Rules]]—represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals
|-
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! [[SKULO interval names|SKULO]]-based interval names
! [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic]]-based interval names
! [[Syntonic-rastmic subchroma notation|SRS notation]]
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Prime
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superprime
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superprime
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superprime, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superprime
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraprime
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraprime, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Prime
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Prime
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Prime
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Prime
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Prime
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Prime
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Prime
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Prime
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Prime
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Prime
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Prime
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Prime
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Prime
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== 5-limit diatonic music ==

Although 159edo inherits 53edo's close approximations of both the [[5-limit]] Zarlino scale and the [[3-limit]] [[diatonic]] MOS, these are not the only scales that one can use for fixed-pitch diatonic music even in that system. In fact, both of them are less than optimal for many facets of traditional Western classical music, seeing as for the most part, each of the traditional diatonic modes has its own optimized scale in the 5-limit. The reason for this is that in non-meantone 5-limit systems, one inevitably has to deal with the ~[[40/27]] wolf fifth and the ~[[27/20]] wolf fourth, and there's only two ideal positions in the scale for those intervals to be situated in fixed-pitch diatonic music. For major modes, the ~27/20 wolf fourth is ideally situated between the ~[[5/4]] major third and the ~[[27/16]] major sixth, while for the minor modes and the diatonic blighted mode Locrian, the ~40/27 wolf fifth is ideally situated between the ~[[6/5]] minor third and the ~[[16/9]] minor seventh. Not only that, but it also pays to distinguish the Pythagorean major and minor thirds from their Ptolemaic counterparts as both types of major and minor third have distinct roles to play in the 5-limit diatonic music.

=== Scales and Harmony ===

For purposes of this section, we shall assume the tonic to be D-natural as in the interval chart above. Note that the following trines are available in 5-limit diatonic harmony.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo diatonic trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Supradusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|}

==== Ionian and Major ====

This mode- along the corresponding tonality- is optimized for 5-limit using a variation on the Didymian diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Ionian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| ↓VII
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| I
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| III
| 10:12:15:18
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Major with Ptolemaic Major Seventh
| D, F#, A, C#↓
| 0, 54, 93, 144
| IV
| 64:81:96:120
| This dissonant tetrad is an unconventional servient (subdominant) seventh for 5-limit music, and is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Dominant Seventh
| D, F#↓, A, C
| 0, 51, 93, 132
| V
| 36:45:54:64
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~3/2 below its root
|-
| Supradusthumic Pythagorean Minor Seventh
| D, F, A↓, C
| 0, 39, 93, 132
| VI
| 54:32:40:48
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Half-Diminished
| D, F↑, Ab↑, C↑
| 0, 42, 81, 135
| VII
| 45:54:64:81
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Ionian in 159edo is considerably more tonally stable than its more familiar counterpart from 12edo and 24edo. It is also one of only two traditional diatonic modes in which one is able to perform a complete [[Wikipedia: Vi–ii–V–I|circle progression]].

==== Dorian ====

Unlike what is expected, the form of Dorian mode optimized for 5-limit is not symmetrical, owing to the need to maintain [[Rothenberg propriety]].

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Dorian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV, bVII
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II, V
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| VI
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Pythagorean Major Seventh
| D, F#, A, C#
| 0, 54, 93, 147
| bVII
| 128:162:192:243
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II, V
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Supradusthumic Ptolemaic Major Seventh
| D, F#↓, A↓, C#↓
| 0, 51, 90, 144
| b↑III
| 240:256:320:405
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Parallel Dominant Seventh
| D, F#, A, C↑
| 0, 54, 93, 135
| IV
| 1/(45:54:64:81)
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~9/8 above its root
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| VI
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Dorian in 159edo has a few noteworthy tricks- especially once one considers suspensions, which will have to be covered later in this document.

==== Phrygian ====

This mode is optimized for 5-limit using a variation on the left-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Phrygian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Ptolemaic Minor Second
| Eb↑
| 15
| Distosupercollocant
| [[16/15]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Ptolemaic Minor Sixth
| Bb↑
| 108
| Mesodistocontramediant
| [[8/5]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| b↑II, b↑VI
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I, IV, bVII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| V
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑II
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Supradusthumic Ptolemaic Dominant Seventh
| D, F#↓, A↓, C
| 0, 51, 90, 132
| b↑III
| 1/(45:54:64:80)
| This dissonant dominant tetrad has a slightly different function than its more traditional counterpart
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| IV, bVII
| 10:12:15:18
|
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| V
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑VI
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|}

With the above scale steps and chords, this variation of Phrygian in 159edo actually presents unexpected possibilities compared to its more traditional counterpart.

==== Lydian ====

This mode is optimized for 5-limit using a variation on the right-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Lydian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G#↓
| 78
| Sycophant
| [[45/32]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, II, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III, ↓VII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| #↓IV
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

[[Category:Method]]
[[Category:Approaches to tuning systems]]

== Notes ==
<references group="note" />

User:Aura/Aura's introduction to 159edo

2025-11-29T17:36:50Z

Aura: Fixed the name of a table

While large EDOs like [[159edo]] have a bit of a learning curve compared to smaller systems, the fact that this system sports not only consistency to the [[17-odd-limit]], but also both additional options for imitating the pitch-hue palettes of smaller tuning systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, and, the ability to perform sophisticated harmonic and melodic maneuvers that involve substituting intervals that are one or two steps away from the equivalents of traditional musical intervals in such a way as to fly under the radar of many listeners, all while having a step size that is above the average listener's melodic [[JND]], makes this system well worth the price of admission- particularly for those who can use or make synthesizers and other digital-system-based instruments.

== Intervals and Notation ==

159edo contains all the intervals of [[53edo]], however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. Furthermore, just as with 24edo can be thought of as essentially having two fields of 12edo separated by a quartertone, 159edo can be thought of as having three fields of 53edo, each separated from the others by a third of a 53edo step on either side. This even lends to 159edo having its own variation on the [[Dinner Party Rules]]—represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals
|-
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! [[SKULO interval names|SKULO]]-based interval names
! [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic]]-based interval names
! [[Syntonic-rastmic subchroma notation|SRS notation]]
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Prime
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superprime
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superprime
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superprime, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superprime
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraprime
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraprime, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Prime
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Prime
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Prime
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Prime
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Prime
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Prime
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Prime
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Prime
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Prime
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Prime
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Prime
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Prime
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Prime
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== 5-limit diatonic music ==

Although 159edo inherits 53edo's close approximations of both the [[5-limit]] Zarlino scale and the [[3-limit]] [[diatonic]] MOS, these are not the only scales that one can use for fixed-pitch diatonic music even in that system. In fact, both of them are less than optimal for many facets of traditional Western classical music, seeing as for the most part, each of the traditional diatonic modes has its own optimized scale in the 5-limit. The reason for this is that in non-meantone 5-limit systems, one inevitably has to deal with the ~[[40/27]] wolf fifth and the ~[[27/20]] wolf fourth, and there's only two ideal positions in the scale for those intervals to be situated in fixed-pitch diatonic music. For major modes, the ~27/20 wolf fourth is ideally situated between the ~[[5/4]] major third and the ~[[27/16]] major sixth, while for the minor modes and the diatonic blighted mode Locrian, the ~40/27 wolf fifth is ideally situated between the ~[[6/5]] minor third and the ~[[16/9]] minor seventh. Not only that, but it also pays to distinguish the Pythagorean major and minor thirds from their Ptolemaic counterparts as both types of major and minor third have distinct roles to play in the 5-limit diatonic music.

=== Scales and Harmony ===

For purposes of this section, we shall assume the tonic to be D-natural as in the interval chart above. Note that the following trines are available in 5-limit diatonic harmony.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo diatonic trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Supradusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|}

==== Ionian and Major ====

This mode- along the corresponding tonality- is optimized for 5-limit using a variation on the Didymian diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Ionian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| ↓VII
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| I
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| III
| 10:12:15:18
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Major with Ptolemaic Major Seventh
| D, F#, A, C#↓
| 0, 54, 93, 144
| IV
| 64:81:96:120
| This dissonant tetrad is an unconventional servient (subdominant) seventh for 5-limit music, and is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Dominant Seventh
| D, F#↓, A, C
| 0, 51, 93, 132
| V
| 36:45:54:64
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~3/2 below its root
|-
| Supradusthumic Pythagorean Minor Seventh
| D, F, A↓, C
| 0, 39, 93, 132
| VI
| 54:32:40:48
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Half-Diminished
| D, F↑, Ab↑, C↑
| 0, 42, 81, 135
| VII
| 45:54:64:81
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Ionian in 159edo is considerably more tonally stable than its more familiar counterpart from 12edo and 24edo. It is also one of only two traditional diatonic modes in which one is able to perform a complete [[Wikipedia: Vi–ii–V–I|circle progression]].

==== Dorian ====

Unlike what is expected, the form of Dorian mode optimized for 5-limit is not symmetrical, owing to the need to maintain [[Rothenberg propriety]].

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Dorian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV, bVII
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II, V
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| VI
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Pythagorean Major Seventh
| D, F#, A, C#
| 0, 54, 93, 147
| bVII
| 128:162:192:243
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II, V
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Supradusthumic Ptolemaic Major Seventh
| D, F#↓, A↓, C#↓
| 0, 51, 90, 144
| b↑III
| 240:256:320:405
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Parallel Dominant Seventh
| D, F#, A, C↑
| 0, 54, 93, 135
| IV
| 1/(45:54:64:81)
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~9/8 above its root
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| VI
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Dorian in 159edo has a few noteworthy tricks- especially once one considers suspensions, which will have to be covered later in this document.

==== Phrygian ====

This mode is optimized for 5-limit using a variation on the left-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Phrygian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Ptolemaic Minor Second
| Eb↑
| 15
| Distosupercollocant
| [[16/15]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Ptolemaic Minor Sixth
| Bb↑
| 108
| Mesodistocontramediant
| [[8/5]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| b↑II, b↑VI
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I, IV, bVII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| V
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑II
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Supradusthumic Ptolemaic Dominant Seventh
| D, F#↓, A↓, C
| 0, 51, 90, 132
| b↑III
| 1/(45:54:64:80)
| This dissonant dominant tetrad has a slightly different function than its more traditional counterpart
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| IV, bVII
| 10:12:15:18
|
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| V
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑VI
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|}

With the above scale steps and chords, this variation of Phrygian in 159edo actually presents unexpected possibilities compared to its more traditional counterpart.

==== Lydian ====

This mode is optimized for 5-limit using a variation on the right-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Lydian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G#↓
| 78
| Antitonic (Sycophant)
| [[45/32]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

[[Category:Method]]
[[Category:Approaches to tuning systems]]

== Notes ==
<references group="note" />

User:Aura/Aura's introduction to 159edo

2025-11-29T17:36:23Z

Aura: Will eventually add more.

While large EDOs like [[159edo]] have a bit of a learning curve compared to smaller systems, the fact that this system sports not only consistency to the [[17-odd-limit]], but also both additional options for imitating the pitch-hue palettes of smaller tuning systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, and, the ability to perform sophisticated harmonic and melodic maneuvers that involve substituting intervals that are one or two steps away from the equivalents of traditional musical intervals in such a way as to fly under the radar of many listeners, all while having a step size that is above the average listener's melodic [[JND]], makes this system well worth the price of admission- particularly for those who can use or make synthesizers and other digital-system-based instruments.

== Intervals and Notation ==

159edo contains all the intervals of [[53edo]], however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. Furthermore, just as with 24edo can be thought of as essentially having two fields of 12edo separated by a quartertone, 159edo can be thought of as having three fields of 53edo, each separated from the others by a third of a 53edo step on either side. This even lends to 159edo having its own variation on the [[Dinner Party Rules]]—represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals
|-
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! [[SKULO interval names|SKULO]]-based interval names
! [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic]]-based interval names
! [[Syntonic-rastmic subchroma notation|SRS notation]]
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Prime
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superprime
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superprime
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superprime, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superprime
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraprime
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraprime, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Prime
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Prime
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Prime
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Prime
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Prime
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Prime
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Prime
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Prime
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Prime
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Prime
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Prime
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Prime
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Prime
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== 5-limit diatonic music ==

Although 159edo inherits 53edo's close approximations of both the [[5-limit]] Zarlino scale and the [[3-limit]] [[diatonic]] MOS, these are not the only scales that one can use for fixed-pitch diatonic music even in that system. In fact, both of them are less than optimal for many facets of traditional Western classical music, seeing as for the most part, each of the traditional diatonic modes has its own optimized scale in the 5-limit. The reason for this is that in non-meantone 5-limit systems, one inevitably has to deal with the ~[[40/27]] wolf fifth and the ~[[27/20]] wolf fourth, and there's only two ideal positions in the scale for those intervals to be situated in fixed-pitch diatonic music. For major modes, the ~27/20 wolf fourth is ideally situated between the ~[[5/4]] major third and the ~[[27/16]] major sixth, while for the minor modes and the diatonic blighted mode Locrian, the ~40/27 wolf fifth is ideally situated between the ~[[6/5]] minor third and the ~[[16/9]] minor seventh. Not only that, but it also pays to distinguish the Pythagorean major and minor thirds from their Ptolemaic counterparts as both types of major and minor third have distinct roles to play in the 5-limit diatonic music.

=== Scales and Harmony ===

For purposes of this section, we shall assume the tonic to be D-natural as in the interval chart above. Note that the following trines are available in 5-limit diatonic harmony.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo diatonic trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Supradusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|}

==== Ionian and Major ====

This mode- along the corresponding tonality- is optimized for 5-limit using a variation on the Didymian diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Ionian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| ↓VII
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| I
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| III
| 10:12:15:18
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Major with Ptolemaic Major Seventh
| D, F#, A, C#↓
| 0, 54, 93, 144
| IV
| 64:81:96:120
| This dissonant tetrad is an unconventional servient (subdominant) seventh for 5-limit music, and is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Dominant Seventh
| D, F#↓, A, C
| 0, 51, 93, 132
| V
| 36:45:54:64
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~3/2 below its root
|-
| Supradusthumic Pythagorean Minor Seventh
| D, F, A↓, C
| 0, 39, 93, 132
| VI
| 54:32:40:48
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Half-Diminished
| D, F↑, Ab↑, C↑
| 0, 42, 81, 135
| VII
| 45:54:64:81
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Ionian in 159edo is considerably more tonally stable than its more familiar counterpart from 12edo and 24edo. It is also one of only two traditional diatonic modes in which one is able to perform a complete [[Wikipedia: Vi–ii–V–I|circle progression]].

==== Dorian ====

Unlike what is expected, the form of Dorian mode optimized for 5-limit is not symmetrical, owing to the need to maintain [[Rothenberg propriety]].

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Dorian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV, bVII
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II, V
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| VI
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Pythagorean Major Seventh
| D, F#, A, C#
| 0, 54, 93, 147
| bVII
| 128:162:192:243
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II, V
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Supradusthumic Ptolemaic Major Seventh
| D, F#↓, A↓, C#↓
| 0, 51, 90, 144
| b↑III
| 240:256:320:405
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Parallel Dominant Seventh
| D, F#, A, C↑
| 0, 54, 93, 135
| IV
| 1/(45:54:64:81)
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~9/8 above its root
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| VI
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Dorian in 159edo has a few noteworthy tricks- especially once one considers suspensions, which will have to be covered later in this document.

==== Phrygian ====

This mode is optimized for 5-limit using a variation on the left-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Phrygian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Ptolemaic Minor Second
| Eb↑
| 15
| Distosupercollocant
| [[16/15]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Ptolemaic Minor Sixth
| Bb↑
| 108
| Mesodistocontramediant
| [[8/5]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| b↑II, b↑VI
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I, IV, bVII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| V
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑II
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Supradusthumic Ptolemaic Dominant Seventh
| D, F#↓, A↓, C
| 0, 51, 90, 132
| b↑III
| 1/(45:54:64:80)
| This dissonant dominant tetrad has a slightly different function than its more traditional counterpart
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| IV, bVII
| 10:12:15:18
|
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| V
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑VI
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|}

With the above scale steps and chords, this variation of Phrygian in 159edo actually presents unexpected possibilities compared to its more traditional counterpart.

==== Lydian ====

This mode is optimized for 5-limit using a variation on the right-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Ionian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G#↓
| 78
| Antitonic (Sycophant)
| [[45/32]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

[[Category:Method]]
[[Category:Approaches to tuning systems]]

== Notes ==
<references group="note" />

8/3

2025-11-29T16:28:20Z

Aura: Found this out a few weeks ago

{{Infobox Interval
| Name = perfect eleventh
| Color name = w11, wa 11th
| Sound = jid_8_3_pluck_adu_dr220.mp3
}}

'''8/3''', the '''perfect eleventh''', is the ratio between the 3rd and 8th [[harmonic]]s; one octave above [[4/3]]. See also [[ed8/3]].

== Chord construction ==
Notably, 8/3 can be used as a framework for chords, but the usage of 8/3 as a framework for chords is intimately connected with the use of [[perfect fifth]]s in the same capacity- at least in [[Octave #Octave equivalence|octave-equivalent]] systems- due to the same pitch classes being involved in both 4:5:6 and 3:5:8 where 5 is kept as the same note, thus rendering the two chords as different voicings of the same underlying harmonic unit.

[[Category:Tritave-reduced harmonics]]

Talk:Voicings of 4:5:6

2025-11-25T10:38:40Z

Aura: /* Max number of voices */ new section

== Max number of voices ==

I'm thinking that the max number of voices we cover for chords should be seven, reason is that that number of voices can make for rich chord voicings. Besides which, I look at 1:2:3:4:5:6:8 as a voicing of 4:5:6, and indeed it's the most common voicing for that chord I use. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 10:38, 25 November 2025 (UTC)

User talk:Overthink

2025-11-16T07:01:08Z

Aura: /* 23-limit in 159edo */Fixed formatting

== Hi ==
Welcome to my talk page! -- [[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:40, 21 September 2025 (UTC)

== Overly large EDO pages ==
I see that you've done some pretty good work on updating temperament pages and infoboxes and the like. However, we have already been dealing with a glut of stubs - i.e. pages with very minimal content beyond what is automatically generated - created for large EDOs or impractical intervals. You have failed to demonstrate notability for these EDOs, which is a very high burden when dealing with systems in the thousands or even hundreds of thousands of notes.

One reason you might be making these pages is that you just want to look at the harmonics table. That's reasonable, and I do that on [[User:Lériendil/ET harmonic testing page|a subpage of my userpage]] for the sake of minimizing disruption to the wiki, and I advise that you do similar in the future. You seem to want to make useful contributions in general, so take this advice so we can continue to mutually assume good faith.

Thanks,
- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 07:23, 22 September 2025 (UTC)

: To add to this, the standard format for prime/odd harmonic tables is columns = 11. That shouldn't be exceeded in the majority of cases, especially where you're continuing the harmonics into several tables. (Columns = 9 is another standard.) -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 09:47, 22 September 2025 (UTC)

Do I move some or all of pages of EDOs [[7200edo|7200]], [[7474edo|7474]], [[17100edo|17100]], [[74740edo|74740]], and [[747400edo|747400]] to user pages? Maybe even some of {{edos|322, 484, 486, 528, and 699}}? --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 21:53, 22 September 2025 (UTC)

Sorry, I made a big mess while trying to move 747400edo to User:Overthink/747400edo. Please have someone clean it up. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 22:46, 22 September 2025 (UTC)

: Think what you did is sufficient for these purposes. Cleaned up the 747400 mess. -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 23:16, 22 September 2025 (UTC)

== Should I take more breaks ==

On my article [[User:Overthink/The circle of relative error|The circle of relative error]], I wrote 11000 characters and about 2000 words in a single day. Perhaps I should spend a bit less time on this wiki and focus on other things. Just look at the first sentence of my user page.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:15, 29 September 2025 (UTC)

== 23-limit in 159edo ==

Hello! I see you've been working on a 159edo well temperament. I should mention that when I work in that tuning system and I use the 23-limit, I choose either the no-17's or the no-19's form, with the former being viable up to the 29-odd-limit, and the latter being viable up to the 27-odd-limit. I think you'd do well to take stock of the 2.3.11 subgroup in 159edo as that is the skeleton of what I work with, though I will add the 5-prime and I'm admittedly trying to learn how to add the 7-prime and the 13-prime without stacking them multiple times. I've also literally invented a microtonal cadence in the 2.3.5.11 subgroup that might be of interest. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 05:37, 19 October 2025 (UTC)

: I believe a 159-note mos of [[tribilo]] (2.3.11 nexus) temperament may be useful, and it could contain better approximations of intervals outside of 2.3.11 as well. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 06:10, 24 October 2025 (UTC)

:: You'd be right about that, but you'd also be right if you decided on a 159-note MOS of [[frameshift]]. Fortunately, 159edo itself tempers out both the nexus comma and the frameshift comma. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:37, 25 October 2025 (UTC)

: It would be nice if the 2.3.5.11 subgroup was more connected. Prime 5 is equated to -8 fifths octave reduced, which is relatively close, but three 11/8s reach the original chain of fifths +28/-25 fifths away, which is almost exactly the opposite side of the circle. Three 11/10's return only a single fifth down, but unfortunately intervals with 3 factors of 5 are (just barely) inconsistent. There doesn't seem to be a very simple way to connect 2.3.11 to prime 5, but think I found a solution: Prime 17! The 4 intervals closest to the 12edo semitone in 159edo are 256/243, '''18/17''', '''17/16''', and 2187/2048. Prime 17 could be quite useful in classical music, such as in the 17:20:24 diminished chord. Prime 5 is found much more easily and consistently, as simply 3/2 minus 3 17/16s, and nothing of this simplicity is in 2.3.5.11. I believe it would be a good idea to devise a system of 2.3.17 analogous to alpharabian tuning, and you could easily just add back prime 11. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:24, 16 November 2025 (UTC)

: I believe a 12edo-based classification of intervals based on 2.3.5.17 [[Schismatic family#Term|Term temperament]] may be good, and for prime 11 use an 24edo-based classification from [[Schismatic family#Hemiterm|hemiterm]]. Note that 159edo supports both.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:07, 16 November 2025 (UTC)

: Or maybe just start with simpler intervals of 2.3.17.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:17, 16 November 2025 (UTC)

:: One way I've noticed for connecting primes 11 and 5, or rather combinations of 3 and 11 with combinations of 3 and 5, is to equate three instances of 243/242 with 81/80. Prime 17 is also a good addition, and it simplifies certain 11-based gestures in 159edo by 17/16 being equated with two instances of 33/32. I should note that 159edo supports both of these options. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:12, 16 November 2025 (UTC)

:: One more thing I just remembered about connecting combinations of 3, 5 and 11 is to stack three instances of 8192/8019 to get 16/15. I don't know what you make of that, but it's a gesture supported by both 65edo and 159edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:26, 16 November 2025 (UTC)

:: A third option I just found for connecting primes 3, 5 and 11 is to equate two instances of 81/80 with 4096/3993. You are right that none of these are particularly simple, but I still find these methods to be highly valuable- especially with prime 17 coming in to simplify things. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 07:00, 16 November 2025 (UTC)

User talk:Overthink

2025-11-16T07:00:24Z

Aura: /* 23-limit in 159edo */More information

== Hi ==
Welcome to my talk page! -- [[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:40, 21 September 2025 (UTC)

== Overly large EDO pages ==
I see that you've done some pretty good work on updating temperament pages and infoboxes and the like. However, we have already been dealing with a glut of stubs - i.e. pages with very minimal content beyond what is automatically generated - created for large EDOs or impractical intervals. You have failed to demonstrate notability for these EDOs, which is a very high burden when dealing with systems in the thousands or even hundreds of thousands of notes.

One reason you might be making these pages is that you just want to look at the harmonics table. That's reasonable, and I do that on [[User:Lériendil/ET harmonic testing page|a subpage of my userpage]] for the sake of minimizing disruption to the wiki, and I advise that you do similar in the future. You seem to want to make useful contributions in general, so take this advice so we can continue to mutually assume good faith.

Thanks,
- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 07:23, 22 September 2025 (UTC)

: To add to this, the standard format for prime/odd harmonic tables is columns = 11. That shouldn't be exceeded in the majority of cases, especially where you're continuing the harmonics into several tables. (Columns = 9 is another standard.) -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 09:47, 22 September 2025 (UTC)

Do I move some or all of pages of EDOs [[7200edo|7200]], [[7474edo|7474]], [[17100edo|17100]], [[74740edo|74740]], and [[747400edo|747400]] to user pages? Maybe even some of {{edos|322, 484, 486, 528, and 699}}? --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 21:53, 22 September 2025 (UTC)

Sorry, I made a big mess while trying to move 747400edo to User:Overthink/747400edo. Please have someone clean it up. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 22:46, 22 September 2025 (UTC)

: Think what you did is sufficient for these purposes. Cleaned up the 747400 mess. -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 23:16, 22 September 2025 (UTC)

== Should I take more breaks ==

On my article [[User:Overthink/The circle of relative error|The circle of relative error]], I wrote 11000 characters and about 2000 words in a single day. Perhaps I should spend a bit less time on this wiki and focus on other things. Just look at the first sentence of my user page.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:15, 29 September 2025 (UTC)

== 23-limit in 159edo ==

Hello! I see you've been working on a 159edo well temperament. I should mention that when I work in that tuning system and I use the 23-limit, I choose either the no-17's or the no-19's form, with the former being viable up to the 29-odd-limit, and the latter being viable up to the 27-odd-limit. I think you'd do well to take stock of the 2.3.11 subgroup in 159edo as that is the skeleton of what I work with, though I will add the 5-prime and I'm admittedly trying to learn how to add the 7-prime and the 13-prime without stacking them multiple times. I've also literally invented a microtonal cadence in the 2.3.5.11 subgroup that might be of interest. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 05:37, 19 October 2025 (UTC)

: I believe a 159-note mos of [[tribilo]] (2.3.11 nexus) temperament may be useful, and it could contain better approximations of intervals outside of 2.3.11 as well. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 06:10, 24 October 2025 (UTC)

:: You'd be right about that, but you'd also be right if you decided on a 159-note MOS of [[frameshift]]. Fortunately, 159edo itself tempers out both the nexus comma and the frameshift comma. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:37, 25 October 2025 (UTC)

: It would be nice if the 2.3.5.11 subgroup was more connected. Prime 5 is equated to -8 fifths octave reduced, which is relatively close, but three 11/8s reach the original chain of fifths +28/-25 fifths away, which is almost exactly the opposite side of the circle. Three 11/10's return only a single fifth down, but unfortunately intervals with 3 factors of 5 are (just barely) inconsistent. There doesn't seem to be a very simple way to connect 2.3.11 to prime 5, but think I found a solution: Prime 17! The 4 intervals closest to the 12edo semitone in 159edo are 256/243, '''18/17''', '''17/16''', and 2187/2048. Prime 17 could be quite useful in classical music, such as in the 17:20:24 diminished chord. Prime 5 is found much more easily and consistently, as simply 3/2 minus 3 17/16s, and nothing of this simplicity is in 2.3.5.11. I believe it would be a good idea to devise a system of 2.3.17 analogous to alpharabian tuning, and you could easily just add back prime 11. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:24, 16 November 2025 (UTC)

: I believe a 12edo-based classification of intervals based on 2.3.5.17 [[Schismatic family#Term|Term temperament]] may be good, and for prime 11 use an 24edo-based classification from [[Schismatic family#Hemiterm|hemiterm]]. Note that 159edo supports both.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:07, 16 November 2025 (UTC)

: Or maybe just start with simpler intervals of 2.3.17.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:17, 16 November 2025 (UTC)

:: One way I've noticed for connecting primes 11 and 5, or rather combinations of 3 and 11 with combinations of 3 and 5, is to equate three instances of 243/242 with 81/80. Prime 17 is also a good addition, and it simplifies certain 11-based gestures in 159edo by 17/16 being equated with two instances of 33/32. I should note that 159edo supports both of these options. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:12, 16 November 2025 (UTC)

:: One more thing I just remembered about connecting combinations of 3, 5 and 11 is to stack three instances of 8192/8019 to get 16/15. I don't know what you make of that, but it's a gesture supported by both 65edo and 159edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:26, 16 November 2025 (UTC)

A third option I just found for connecting primes 3, 5 and 11 is to equate two instances of 81/80 with 4096/3993. You are right that none of these are particularly simple, but I still find these methods to be highly valuable- especially with prime 17 coming in to simplify things. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 07:00, 16 November 2025 (UTC)

User talk:Overthink

2025-11-16T06:38:17Z

Aura: /* 23-limit in 159edo */Second correction

User talk:Overthink

2025-11-16T06:26:14Z

Aura: /* 23-limit in 159edo */Correction and addition

== Hi ==
Welcome to my talk page! -- [[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:40, 21 September 2025 (UTC)

== Overly large EDO pages ==
I see that you've done some pretty good work on updating temperament pages and infoboxes and the like. However, we have already been dealing with a glut of stubs - i.e. pages with very minimal content beyond what is automatically generated - created for large EDOs or impractical intervals. You have failed to demonstrate notability for these EDOs, which is a very high burden when dealing with systems in the thousands or even hundreds of thousands of notes.

One reason you might be making these pages is that you just want to look at the harmonics table. That's reasonable, and I do that on [[User:Lériendil/ET harmonic testing page|a subpage of my userpage]] for the sake of minimizing disruption to the wiki, and I advise that you do similar in the future. You seem to want to make useful contributions in general, so take this advice so we can continue to mutually assume good faith.

Thanks,
- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 07:23, 22 September 2025 (UTC)

: To add to this, the standard format for prime/odd harmonic tables is columns = 11. That shouldn't be exceeded in the majority of cases, especially where you're continuing the harmonics into several tables. (Columns = 9 is another standard.) -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 09:47, 22 September 2025 (UTC)

Do I move some or all of pages of EDOs [[7200edo|7200]], [[7474edo|7474]], [[17100edo|17100]], [[74740edo|74740]], and [[747400edo|747400]] to user pages? Maybe even some of {{edos|322, 484, 486, 528, and 699}}? --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 21:53, 22 September 2025 (UTC)

Sorry, I made a big mess while trying to move 747400edo to User:Overthink/747400edo. Please have someone clean it up. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 22:46, 22 September 2025 (UTC)

: Think what you did is sufficient for these purposes. Cleaned up the 747400 mess. -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 23:16, 22 September 2025 (UTC)

== Should I take more breaks ==

On my article [[User:Overthink/The circle of relative error|The circle of relative error]], I wrote 11000 characters and about 2000 words in a single day. Perhaps I should spend a bit less time on this wiki and focus on other things. Just look at the first sentence of my user page.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:15, 29 September 2025 (UTC)

== 23-limit in 159edo ==

Hello! I see you've been working on a 159edo well temperament. I should mention that when I work in that tuning system and I use the 23-limit, I choose either the no-17's or the no-19's form, with the former being viable up to the 29-odd-limit, and the latter being viable up to the 27-odd-limit. I think you'd do well to take stock of the 2.3.11 subgroup in 159edo as that is the skeleton of what I work with, though I will add the 5-prime and I'm admittedly trying to learn how to add the 7-prime and the 13-prime without stacking them multiple times. I've also literally invented a microtonal cadence in the 2.3.5.11 subgroup that might be of interest. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 05:37, 19 October 2025 (UTC)

: I believe a 159-note mos of [[tribilo]] (2.3.11 nexus) temperament may be useful, and it could contain better approximations of intervals outside of 2.3.11 as well. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 06:10, 24 October 2025 (UTC)

:: You'd be right about that, but you'd also be right if you decided on a 159-note MOS of [[frameshift]]. Fortunately, 159edo itself tempers out both the nexus comma and the frameshift comma. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:37, 25 October 2025 (UTC)

: It would be nice if the 2.3.5.11 subgroup was more connected. Prime 5 is equated to -8 fifths octave reduced, which is relatively close, but three 11/8s reach the original chain of fifths +28/-25 fifths away, which is almost exactly the opposite side of the circle. Three 11/10's return only a single fifth down, but unfortunately intervals with 3 factors of 5 are (just barely) inconsistent. There doesn't seem to be a very simple way to connect 2.3.11 to prime 5, but think I found a solution: Prime 17! The 4 intervals closest to the 12edo semitone in 159edo are 256/243, '''18/17''', '''17/16''', and 2187/2048. Prime 17 could be quite useful in classical music, such as in the 17:20:24 diminished chord. Prime 5 is found much more easily and consistently, as simply 3/2 minus 3 17/16s, and nothing of this simplicity is in 2.3.5.11. I believe it would be a good idea to devise a system of 2.3.17 analogous to alpharabian tuning, and you could easily just add back prime 11. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:24, 16 November 2025 (UTC)

: I believe a 12edo-based classification of intervals based on 2.3.5.17 [[Schismatic family#Term|Term temperament]] may be good, and for prime 11 use an 24edo-based classification from [[Schismatic family#Hemiterm|hemiterm]]. Note that 159edo supports both.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:07, 16 November 2025 (UTC)

: Or maybe just start with simpler intervals of 2.3.17.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:17, 16 November 2025 (UTC)

:: One way I've noticed for connecting primes 11 and 5, or rather combinations of 3 and 11 with combinations of 3 and 5, is to equate three instances of 243/242 with 81/80. Prime 17 is also a good addition, and it simplifies certain 11-based gestures in 159edo by 17/16 being equated with two instances of 33/32. I should note that 159edo supports both of these options. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:12, 16 November 2025 (UTC)

:: One more thing I just remembered about connecting combinations of 3 and 5 is to stack three instances of 8192/8019 to get 16/15. I don't know what you make of that, but it's a gesture supported by both 65edo and 159edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:26, 16 November 2025 (UTC)

User talk:Overthink

2025-11-16T06:12:25Z

Aura: /* 23-limit in 159edo */

== Hi ==
Welcome to my talk page! -- [[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:40, 21 September 2025 (UTC)

== Overly large EDO pages ==
I see that you've done some pretty good work on updating temperament pages and infoboxes and the like. However, we have already been dealing with a glut of stubs - i.e. pages with very minimal content beyond what is automatically generated - created for large EDOs or impractical intervals. You have failed to demonstrate notability for these EDOs, which is a very high burden when dealing with systems in the thousands or even hundreds of thousands of notes.

One reason you might be making these pages is that you just want to look at the harmonics table. That's reasonable, and I do that on [[User:Lériendil/ET harmonic testing page|a subpage of my userpage]] for the sake of minimizing disruption to the wiki, and I advise that you do similar in the future. You seem to want to make useful contributions in general, so take this advice so we can continue to mutually assume good faith.

Thanks,
- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 07:23, 22 September 2025 (UTC)

: To add to this, the standard format for prime/odd harmonic tables is columns = 11. That shouldn't be exceeded in the majority of cases, especially where you're continuing the harmonics into several tables. (Columns = 9 is another standard.) -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 09:47, 22 September 2025 (UTC)

Do I move some or all of pages of EDOs [[7200edo|7200]], [[7474edo|7474]], [[17100edo|17100]], [[74740edo|74740]], and [[747400edo|747400]] to user pages? Maybe even some of {{edos|322, 484, 486, 528, and 699}}? --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 21:53, 22 September 2025 (UTC)

Sorry, I made a big mess while trying to move 747400edo to User:Overthink/747400edo. Please have someone clean it up. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 22:46, 22 September 2025 (UTC)

: Think what you did is sufficient for these purposes. Cleaned up the 747400 mess. -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 23:16, 22 September 2025 (UTC)

== Should I take more breaks ==

On my article [[User:Overthink/The circle of relative error|The circle of relative error]], I wrote 11000 characters and about 2000 words in a single day. Perhaps I should spend a bit less time on this wiki and focus on other things. Just look at the first sentence of my user page.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:15, 29 September 2025 (UTC)

== 23-limit in 159edo ==

Hello! I see you've been working on a 159edo well temperament. I should mention that when I work in that tuning system and I use the 23-limit, I choose either the no-17's or the no-19's form, with the former being viable up to the 29-odd-limit, and the latter being viable up to the 27-odd-limit. I think you'd do well to take stock of the 2.3.11 subgroup in 159edo as that is the skeleton of what I work with, though I will add the 5-prime and I'm admittedly trying to learn how to add the 7-prime and the 13-prime without stacking them multiple times. I've also literally invented a microtonal cadence in the 2.3.5.11 subgroup that might be of interest. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 05:37, 19 October 2025 (UTC)

: I believe a 159-note mos of [[tribilo]] (2.3.11 nexus) temperament may be useful, and it could contain better approximations of intervals outside of 2.3.11 as well. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 06:10, 24 October 2025 (UTC)

:: You'd be right about that, but you'd also be right if you decided on a 159-note MOS of [[frameshift]]. Fortunately, 159edo itself tempers out both the nexus comma and the frameshift comma. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:37, 25 October 2025 (UTC)

: It would be nice if the 2.3.5.11 subgroup was more connected. Prime 5 is equated to -8 fifths octave reduced, which is relatively close, but three 11/8s reach the original chain of fifths +28/-25 fifths away, which is almost exactly the opposite side of the circle. Three 11/10's return only a single fifth down, but unfortunately intervals with 3 factors of 5 are (just barely) inconsistent. There doesn't seem to be a very simple way to connect 2.3.11 to prime 5, but think I found a solution: Prime 17! The 4 intervals closest to the 12edo semitone in 159edo are 256/243, '''18/17''', '''17/16''', and 2187/2048. Prime 17 could be quite useful in classical music, such as in the 17:20:24 diminished chord. Prime 5 is found much more easily and consistently, as simply 3/2 minus 3 17/16s, and nothing of this simplicity is in 2.3.5.11. I believe it would be a good idea to devise a system of 2.3.17 analogous to alpharabian tuning, and you could easily just add back prime 11. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:24, 16 November 2025 (UTC)

: I believe a 12edo-based classification of intervals based on 2.3.5.17 [[Schismatic family#Term|Term temperament]] may be good, and for prime 11 use an 24edo-based classification from [[Schismatic family#Hemiterm|hemiterm]]. Note that 159edo supports both.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:07, 16 November 2025 (UTC)

: Or maybe just start with simpler intervals of 2.3.17.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:17, 16 November 2025 (UTC)

:: One way I've noticed for connecting primes 11 and 5, or rather combinations of 3 and 11 with five, is to equate three instances of 243/242 with 81/80. Prime 17 is also a good addition, and it simplifies certain 11-based gestures in 159edo by 17/16 being equated with two instances of 33/32. I should note that 159edo supports both of these options. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:12, 16 November 2025 (UTC)

Talk:159edo

2025-11-15T06:16:09Z

Aura: /* How much consistency matters */Re

== Approximate errors ==

Okay... I have a list of the approximate errors in cents for 159edo's approximations of certain prime intervals:

*3: -0.068
*5: -1.408
*7: -2.788
*11: -0.374
*13: -2.792
*17: +0.705
*19: -3.173
*23: -1.859
*29: -3.162
*31: +2.134

I'm hoping that someone can make tables for Just Approximation like the ones found on the page for [[94edo]]... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 07:18, 7 September 2020 (UTC)

: Done. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:02, 7 September 2020 (UTC)

:: Thanks! Once we do a lot more exploring of 159edo, I hope to put our findings here. After all, there's no way I'm just letting an EDO as useful as this just languish anymore. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 13:08, 7 September 2020 (UTC)

:: I have 159edo's patent val for primes up to the 19 limit- {{val|159 252 369 446 550 588 650 675}}. How consistent is this EDO when it comes to this group of primes? (preceding unsigned comment by [[User:Aura|Aura]] ([[User talk:Aura|talk]]))

:::According to Scala it's only consistent up to 17-odd limit. It might still be consistent when we add some higher odd numbers, though. [[User:IlL|IlL]] ([[User talk:IlL|talk]]) 15:36, 7 September 2020 (UTC)

:::: Let's check it out then... let's try 19, 21, 23, 25, 27, 29 and 31... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 15:39, 7 September 2020 (UTC)

::: Easy to speculate with an understanding of [[relative error]]. It's consistent in 17-limit or no-17 29-limit. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:41, 7 September 2020 (UTC)

:::: Wait... why specifically a no-17 29-limit? Is it consistent in 19-limit or 23-limit? Perhaps I ought to reveal one final patent val for 159edo- that of the 23-prime limit... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 15:54, 7 September 2020 (UTC)

:::: Okay, so, if 159edo is extended to the 23-prime, 159edo has the patent val of {{val|159 252 369 446 550 588 650 675 719}}... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 16:01, 7 September 2020 (UTC)

:::: I must admit that the main reason I'm interested in whether or not 159edo is consistent up to the 23-limit is because I'm currently compiling a list of Just Intervals corresponding to the various steps in 159edo, and 23 is the highest prime I've had to use so far... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 16:08, 7 September 2020 (UTC)

:::: Okay, I've managed to confirm that 159edo is ''not'' consistent in the 19-odd limit as the difference between the best 17/16 and the best 19/16 is 25 steps, while the best 19/17 is 26 steps... Not good at all... Looks like I need to search for several new values for step sizes --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 16:32, 7 September 2020 (UTC)

::::: 159edo has two intervals in 29-limit with >50% relative error —- 19/17 and 29/17. That's why you have to decide, full 17-limit or no-17 29-limit. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 03:03, 8 September 2020 (UTC)

::::: Afaik no edo between 94 and 282 is fully consistent in 23-limit. There's 111, 149 and 217 fully consistent in 19-limit. 94 is special consistency-wise but it's not superior in accuracy, so not all edos above 94 need to directly compare with it, especially when there's nothing to relate them. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 02:59, 8 September 2020 (UTC)

:::::: I have used 94edo in the past, and the article on 94edo states that it is "a remarkable all-around utility temperament", while 159edo has other strengths, so I figured a comparison was at least somewhat warranted in this case. However, if such a comparison is not really warranted here, I'll remove the comparison altogether. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 03:35, 8 September 2020 (UTC)

== Reverting factually wrong additions ==
Facts:
# There's basically no relationship between contorsion and inconsistency.
# There's basically no relationship between comma size and inconsistency.
# There's only one reasonable mapping for 5 and 7 and it's consistent.
[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 17:18, 7 January 2021 (UTC)

: How then do you judge inconsistency? I note that [[128/125]], when approached by way of a chain of [[5/4]] intervals doesn't match the step that best fits [[128/125]] directly in terms of absolute error, and I have the same problem with [[49/32]]. I also noted that Mercator's comma is less than half the size of a single step in 159edo, so why is what I said about that entirely wrong? Please do tell. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:26, 7 January 2021 (UTC)

: Also, I wasn't talking about odd-limit here, I was talking about prime limit. I agree that there's only one reasonable mapping for 5/4 and 7/4, but once you get beyond the 17-odd-limit, that's where we start to have issues. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:33, 7 January 2021 (UTC)

:: In your way every edo would be "inconsistent in the 3-limit" because the 3-limit contains an infinity of different intervals and there're always some intervals with error over 50% of step size. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 17:39, 7 January 2021 (UTC)

::: Ah. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:40, 7 January 2021 (UTC)

::: I don't know about you, but to me, that high error rate does affect how the interval in question is actually used from a musical standpoint. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:42, 7 January 2021 (UTC)

:::: The error "rate" of a specific prime is the same as its relative error, I suppose. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 17:50, 7 January 2021 (UTC)

::::: I would say that's a reasonable conclusion, but only in part. I'm saying that the end of the usable portion of the harmonic lattice for a given prime as represented in a given EDO is marked by the relative error being less than 50%- or at least that's my policy on the matter. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 17:58, 7 January 2021 (UTC)

: I, too, am somewhat unsure about this issue. Is it correct that [[consistent|inconsistency/consistency]] is only defined in relation to a specific odd limit? Otherwise it would not be in the Boolean domain. I wished we had another measure for consistency, something that does not depend on an odd limit, but tells how many nodes of a (p-1)-dimensional lattice could be (somehow) reached from the unison. (But unfortunately my mathematical skills are not sufficient to comprehend this "somehow".) --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 17:53, 7 January 2021 (UTC)

:: I don't know how well my response to Flora manages to solve the problem you just stated, but here's to hoping... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:00, 7 January 2021 (UTC)

:: Is it me, or can it be said that "Boolean Consistency" means being able to go from the unison through a set of nodes in one p-limit to connect with an interval of a lower p-limit without the relative error reaching above the 50% marker? If so, then "Boolean Consistency" for the 3-limit means being able to connect with the pitch class used as the [[unison]] and [[octave]] a second time after going around a complete set of nodes without the relative error reaching above the 50% marker. If my speculation is correct, then we're talking about a different type of "consistency" than the kind that Flora's talking about. It's like comparing apples and oranges in a way- apples and oranges are both fruit but have a lot of differences between them. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:11, 7 January 2021 (UTC)

:: The consistency is defined on "an interval set S". There's not a rule against prime limit but that doesn't make sense since it simply can't be consistent. I remember reading about an "n-consistent" somewhere, in which 53edo is hundreds-consistent in the 3-limit as you can stack hundreds of 3's without relative error reaching over 50%. That might be what you look for. Somebody in the FB group also proposed another "n-consistent", in which the n is something substituting 50%, similar to relative error. Another fascinating idea is the ''pepper ambiguity'' (forgive me for saving links in talk pages) – its definition is not completely clear to me and I hope to work on it soon. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 18:31, 7 January 2021 (UTC)

::: It looks like there are multiple types of n-consistency being proposed even within the Facebook group, so yes, we need a discussion on this. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 19:01, 7 January 2021 (UTC)

::: I must point out that the degree of n-consistency that I look for on "an interval set S" in the 3-limit has everything to do with whether or not you can go around a complete circle of fifths in a given EDO without accumulating a relative error of 50% or more. That's the specific type of n-consistency that I think I can regard as "complete". --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 19:07, 7 January 2021 (UTC)

:::: This kind of consistency ("complete circle of fifths") seems problematic to me: How will you generalize these rings to other prime intervals? Also, aren't you interested in combinations of multiple prime dimensions (besides 2, of course)? --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:22, 7 January 2021 (UTC)

::::: The kind of n-consistency I'm alluding to involves being able to go from the unison through a set of nodes in one p-limit to connect with an interval of a lower p-limit without the relative error reaching or exceeding the 50% marker. Since the 3-prime can only connect with the 2-prime in this fashion, and since the 2-prime simply results in manifestations of the unison at different registers- meaning that the unison is the only available target- that means that the 3-prime requires a complete circle of fifths without accumulating 50% relative error or more to achieve a form of "complete consistency". However, higher primes have more options for a form of "complete consistency". For instance, the 11-prime in 159edo connects with the 3-prime easily without breaching the 50% relative error marker by means of tempering out the nexus comma, and similarly, the 5-prime connects with the 3-prime by means of tempering out the schisma. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:41, 7 January 2021 (UTC)

::::: As to combinations of multiple prime dimensions, I find these to be largely of secondary importance, but to be fair, they are subject to the same constraints- they must be able to connect to a p-limit lower than the lowest p-limit that is directly involved in the combination in question. For example, stacks of 14/13 trienthirds connect with the 5-prime by means of tempering out the cantonisma- the factor of 2 in 14 is trivial for this since the 2-prime simply results in manifestations of the unison at different registers. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:36, 7 January 2021 (UTC)

::::: Now this is fascinating... According to my calculations, subtracting the [[symbiosma]] from [[7/4]] results in an interval with the prime factorization of (3^9)/(2^10*11), so it looks like the symbiosma bridges the 7-prime and a combination of 3 and 11. Perhaps I should fix my definition of "complete consistency" by adding the following condition- if one is able to go from the unison through a set of nodes in one p-limit to connect with an interval made purely from a combination of two other primes, complete consistency is only achieved when the highest prime directly involved in the combination in question connects to the lowest prime in that same combination without breaching the 50% relative error marker once octave equivalence is accounted for. This would mean that in 159edo, the connection between the 7-prime on one hand and a combination of 11 and 3 on the other can only be regarded as "complete consistency" because the 11-prime connects to the 3-prime without breaching the 50% relative error marker on account of the nexus comma being tempered out. I still need to work out the details regarding more complicated combinations, but other than that, do you have any thoughts on this idea, Xenwolf? --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 20:20, 17 January 2021 (UTC)

:::::: I can't say anything about that. Considering the precision of 3.7 cents with which any interval is hit in 159edo and the generally accepted detuning degree of 13.7 cents of the major third in 12edo, considerations regarding consistency seem rather remote to me. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:47, 17 January 2021 (UTC)

::::::: Actually, if you think about it, the generally accepted detuning of the major third in [[12edo]] still follows the same rules that I'm laying down, as the Syntonic comma ([[81/80]]), which is responsible for that detuning, is smaller than half a step in 12edo, and it's still smaller than half a step in [[24edo]]. In fact, the [[Pythagorean comma]] is also less than half of a step in 24edo, and thus, the 3-prime and the 5-prime can both be regarded as having "complete consistency" in 24edo as well as in 12edo. However, when you start looking at [[36edo]], [[48edo]] and [[72edo]], suddenly, things don't turn out as good on this front, as the relative error percentage in these EDOs- especially for the Pythagorean comma- exceeds 50%. This is why I moved on from the larger 12-based EDOs and was finally open to detwelvulating. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:41, 17 January 2021 (UTC)

::::::: If you apply this same type of thinking to smaller EDOs, you see that 2edo is the first to have complete consistency in the 3-limit, but 3edo and 4edo both fail this test as the commas produced by their respective circles of fifths are larger than half of their respective step size. After that, the next EDO to have complete consistency in the 3-limit is 5edo, which accomplishes a completely consistent representation of the 3-prime as 256/243, the interval produced from a single circle of fifths in 5edo, is smaller than half of a step in 5edo. After that, the next EDO to have complete consistency in the 3-prime is 12edo itself, as 6edo, 7edo, 8edo, 9edo, 10edo, and 11edo all fail the test- of course, as I said, 24edo, which is related to 12edo, also passes this test. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:50, 17 January 2021 (UTC)

::::::: So, what about the EDOs between 12edo and 24edo? Well, according to my calculations, literally none of the EDOs from 13edo to 23edo demonstrate complete consistency in the 3-limit. Even the well known [[22edo]] fails this test- looks like I've found one of that EDO's significant weaknesses, and a good enough reason for me not to use it. Anyhow, I'll continue my calculations to see what other EDOs demonstrate the kind of complete 3-prime consistency, and I'll let y'all know about the first dozen or so members of the sequence that emerges from this. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 23:12, 17 January 2021 (UTC)

::::::: I just got to thinking, and, the term "complete consistency" seems like a misleading term for the type of consistency I'm after- perhaps "telic consistency" or even "telicity" are a better terms for this, since this type of consistency means that stacking intervals of one prime will eventually reach an interval of a lower prime without reaching or exceeding 50% relative error, and "telic" is related to "telos" meaning "end" or "goal". Since "telicity" is the noun used to refer to the property of being "telic", I think I'll use the term "telicity" for this type of n-consistency from now on. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 03:30, 18 January 2021 (UTC)

== Linking 159edo Songs to This Page ==

Hey, Xenwolf, since I've written like three songs in 159edo now, I'm wondering how to link these songs of mine to this page. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:37, 26 February 2021 (UTC)

: I started the [[159edo #Music|''Music'']] section, please feel free to add what you like there. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 19:04, 26 February 2021 (UTC)

== Gentle comma (364/363) and region, also tempering out 352/351 ==

Please note that while this article correctly notes that 159-ed2 tempers out364/363, the gentle region and grntle temperament also involves tempering out 352/351. In other words, -3 fifths represents 13/11 or 33/28; and +4 fifths represents 14/11 or 33/28. Thus there sre two genle commas: 159-ed2 tempers out 364/363, but not 352/351; compare 38\159 for 13/11 or 33/28 with 39\159 (-3 fifths) for 32/27. In gentle temperament as I described it in 2002, 32;27 and 13/11 or 33/28 map to -3 fifths.

[[User:Mschulter1325|Mschulter1325]] 01:18, 11 November 2022 (UTC)

: Would you say the gentle comma should refer to either 352/351 or 364/363? And that gentle temperament is the 13-limit temperament tempering out both 352/351 and 364/363? In that case we'll need to come up with another name for 364/363 cuz right now it's known specifically as ''the'' gentle comma. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:14, 11 November 2022 (UTC)

I eould say it's important that any change or updating of terms be graceful and as backward-compatible as possible. Maybe the larger minthma/gentle comma for 352/351 (old minthma) and smaller minthma/gentle comma for 364/363 (old gentle comma). I know that people have relied on the old names, and developed temperaments that, unlike my gentle but just as validly, temper out one but not the other. So this kind of collegiality and consultation is very helpful in seeking out, if you'll forgive the pun, the kindest and most gentle solution. [[User:Mschulter1325|Mschulter1325]] 02:46, 13 November 2022 (UTC)

== How much consistency matters ==

Of course you would want an interval like 6/5 or 11/7 to be consistent, as these are simple ratios that have their consonance affected by mistuning. However, there should be less significance in getting intervals like 35/32 and 49/32 consistent, as the consonance of intervals like these is not as obvious , even if still plausible. Also, telicity seems an interesting topic, but its importance seems limited. For example, 41edo is not 3-2 telic because 617673396283947/562949953421312 (monzo: [-49 31>) is inconsistent, but who would memorize the size of pythagorean intervals that complex? I hardly care about consistency of pythagorean intervals more complex than 2187/2048, and at most 531441/524288, so telicity seems to be redundant when we have more than even 12 notes. Telicity involves higher-limit intervals being on the 2-3 chain as well, but how much does it matter that it also lines up with the really complex pythagorean interval? There are temperaments like Garibaldi that use a long chain of fifths, but telicity seems far from essential in general. All that being said, 159edo still seems like an interesting system, though its quite complex and I will try systems like 17edo and 19edo first.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:34, 27 September 2025 (UTC)

: Now that I think about it, this section doesn't make sense. It's not about having the intervals be convincing as consonances, but having the system act similarly to JI, and intervals that are complex in terms of their ratios are often used.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 21:04, 16 October 2025 (UTC)

:: Telicity matters more for navigational and modulatory purposes than anything, and indeed, having complex 3-limit intervals act similarly to JI while the circle of fifths still closes on the octave is very useful for that reason. Also, telicity is a concept that was devised before consistent circles, and it actually affects mappings of higher primes relative to lower primes in general, so it's not just stuff along the Pythagorean chain. For instance, because three instances of 11/8 octave-reduced add up to something that's really close to a stack of four Pythagorean Chromatic Semitones, that helps make the 2.11 chain more navigable. Unfortunately, with EDOs, sometimes you're forced to pick some kinds of telicity over others for practical reasons. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:56, 16 October 2025 (UTC)

: I think I get why it matters now. Someone who is accustomed to 12edo will find notes up to ~50 cents apart, like 7/4 and 12/7, to sound similar, while notes further apart, like 5/4 and 6/5, sound completely different. If someone is accustomed to 159edo, and can distinguish all of its intervals, then this threshold drops to half a step of 159edo, around 3.7 cents so inconsistency is ''always'' a problem. The circle of fifths is crucial to western music theory, so 3-2 telicity is important in that regard. The article mentions intervals like 49/32 and 35/32 being inconsistent, but fortunately there aren't many chords involving these with concordant structures, unlike 55/32 which appears in chords like 32:40:48:55, which ''is'' mapped consistently. The condition of telicity quickly becomes restrictive in larger EDOs, though some like 159edo manage to slip through. Its very lucky that prime 11 specifically is tuned well in 159edo, and I personally find it to sound cool and perfectly consonant, while prime 7 doesn't feel as cool (though still consonant). Also, 159edo is distinctly consistent to the 17-odd-limit, which is at the limit for an EDO this size, with only 149edo before it also being distinctly consistent to the 17-odd-limit. 159edo fails 19-odd-limit consistency because 19/17 is mapped to 25 steps while it is closer to 26, though even in the no-17 19-limit it equates intervals like 19/15 and 24/19, which doesn't occur in lower odd limits, showing that you can't expect an EDO of this size to do so well past the 17-limit. An EDO with 2-3-11 telicity and similar or better properties doesn't occur for a while, by which point you might as well use JI. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:06, 15 November 2025 (UTC)

:: Your assessment is almost on the nose. Inconsistency is always a problem, so telicity offers a way to prune off the inconsistent intervals. You are right about 3-2 telicity being crucial to Western music theory, and thus, the kinds of music I write, but 5-3 telicity is also important for this kind of music, although to a lesser extent, and of course, 11-3 telicity is needed for good quartertones. The only real quibble I have with your assessment is that it should be noted both that even a system like 159edo is vastly simplified in terms of interval arithmetic compared to JI, and that the just-noticeable difference between pitches, which is situated at about 3.5 cents, is another significant practical restriction against large EDOs as well as large prime limits. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:16, 15 November 2025 (UTC)

User:Aura/Aura's introduction to 159edo

2025-10-30T23:17:08Z

Aura:

While large EDOs like [[159edo]] have a bit of a learning curve compared to smaller systems, the fact that this system sports not only consistency to the [[17-odd-limit]], but also both additional options for imitating the pitch-hue palettes of smaller tuning systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, and, the ability to perform sophisticated harmonic and melodic maneuvers that involve substituting intervals that are one or two steps away from the equivalents of traditional musical intervals in such a way as to fly under the radar of many listeners, all while having a step size that is above the average listener's melodic [[JND]], makes this system well worth the price of admission- particularly for those who can use or make synthesizers and other digital-system-based instruments.

== Intervals and Notation ==

159edo contains all the intervals of [[53edo]], however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. Furthermore, just as with 24edo can be thought of as essentially having two fields of 12edo separated by a quartertone, 159edo can be thought of as having three fields of 53edo, each separated from the others by a third of a 53edo step on either side. This even lends to 159edo having its own variation on the [[Dinner Party Rules]]—represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals
|-
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! [[SKULO interval names|SKULO]]-based interval names
! [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic]]-based interval names
! [[Syntonic-rastmic subchroma notation|SRS notation]]
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Prime
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superprime
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superprime
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superprime, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superprime
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraprime
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraprime, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Prime
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Prime
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Prime
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Prime
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Prime
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Prime
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Prime
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Prime
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Prime
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Prime
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Prime
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Prime
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Prime
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== 5-limit diatonic music ==

Although 159edo inherits 53edo's close approximations of both the [[5-limit]] Zarlino scale and the [[3-limit]] [[diatonic]] MOS, these are not the only scales that one can use for fixed-pitch diatonic music even in that system. In fact, both of them are less than optimal for many facets of traditional Western classical music, seeing as for the most part, each of the traditional diatonic modes has its own optimized scale in the 5-limit. The reason for this is that in non-meantone 5-limit systems, one inevitably has to deal with the ~[[40/27]] wolf fifth and the ~[[27/20]] wolf fourth, and there's only two ideal positions in the scale for those intervals to be situated in fixed-pitch diatonic music. For major modes, the ~27/20 wolf fourth is ideally situated between the ~[[5/4]] major third and the ~[[27/16]] major sixth, while for the minor modes and the diatonic blighted mode Locrian, the ~40/27 wolf fifth is ideally situated between the ~[[6/5]] minor third and the ~[[16/9]] minor seventh. Not only that, but it also pays to distinguish the Pythagorean major and minor thirds from their Ptolemaic counterparts as both types of major and minor third have distinct roles to play in the 5-limit diatonic music.

=== Scales and Harmony ===

For purposes of this section, we shall assume the tonic to be D-natural as in the interval chart above. Note that the following trines are available in 5-limit diatonic harmony.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo diatonic trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Supradusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|}

==== Ionian and Major ====

This mode- along the corresponding tonality- is optimized for 5-limit using a variation on the Didymian diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Ionian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| ↓VII
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| I
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| III
| 10:12:15:18
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Major with Ptolemaic Major Seventh
| D, F#, A, C#↓
| 0, 54, 93, 144
| IV
| 64:81:96:120
| This dissonant tetrad is an unconventional servient (subdominant) seventh for 5-limit music, and is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Dominant Seventh
| D, F#↓, A, C
| 0, 51, 93, 132
| V
| 36:45:54:64
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~3/2 below its root
|-
| Supradusthumic Pythagorean Minor Seventh
| D, F, A↓, C
| 0, 39, 93, 132
| VI
| 54:32:40:48
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Half-Diminished
| D, F↑, Ab↑, C↑
| 0, 42, 81, 135
| VII
| 45:54:64:81
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Ionian in 159edo is considerably more tonally stable than its more familiar counterpart from 12edo and 24edo. It is also one of only two traditional diatonic modes in which one is able to perform a complete [[Wikipedia: Vi–ii–V–I|circle progression]].

==== Dorian ====

Unlike what is expected, the form of Dorian mode optimized for 5-limit is not symmetrical, owing to the need to maintain [[Rothenberg propriety]].

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Dorian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV, bVII
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II, V
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| VI
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Pythagorean Major Seventh
| D, F#, A, C#
| 0, 54, 93, 147
| bVII
| 128:162:192:243
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II, V
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Supradusthumic Ptolemaic Major Seventh
| D, F#↓, A↓, C#↓
| 0, 51, 90, 144
| b↑III
| 240:256:320:405
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Parallel Dominant Seventh
| D, F#, A, C↑
| 0, 54, 93, 135
| IV
| 1/(45:54:64:81)
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~9/8 above its root
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| VI
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Dorian in 159edo has a few noteworthy tricks- especially once one considers suspensions, which will have to be covered later in this document.

==== Phrygian ====

This mode is optimized for 5-limit using a variation on the left-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Phrygian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Ptolemaic Minor Second
| Eb↑
| 15
| Distosupercollocant
| [[16/15]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Ptolemaic Minor Sixth
| Bb↑
| 108
| Mesodistocontramediant
| [[8/5]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| b↑II, b↑VI
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I, IV, bVII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| V
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑II
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Supradusthumic Ptolemaic Dominant Seventh
| D, F#↓, A↓, C
| 0, 51, 90, 132
| b↑III
| 1/(45:54:64:80)
| This dissonant dominant tetrad has a slightly different function than its more traditional counterpart
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| IV, bVII
| 10:12:15:18
|
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| V
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑VI
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|}

[[Category:Method]]
[[Category:Approaches to tuning systems]]

== Notes ==
<references group="note" />

User:Aura/Aura's introduction to 159edo

2025-10-30T23:12:55Z

Aura:

While large EDOs like [[159edo]] have a bit of a learning curve compared to smaller systems, the fact that this system sports not only consistency to the [[17-odd-limit]], but also both additional options for imitating the pitch-hue palettes of smaller tuning systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, and, the ability to perform sophisticated harmonic and melodic maneuvers that involve substituting intervals that are one or two steps away from the equivalents of traditional musical intervals in such a way as to fly under the radar of many listeners, all while having a step size that is above the average listener's melodic [[JND]], makes this system well worth the price of admission- particularly for those who can use or make synthesizers and other digital-system-based instruments.

== Intervals and Notation ==

159edo contains all the intervals of [[53edo]], however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. Furthermore, just as with 24edo can be thought of as essentially having two fields of 12edo separated by a quartertone, 159edo can be thought of as having three fields of 53edo, each separated from the others by a third of a 53edo step on either side. This even lends to 159edo having its own variation on the [[Dinner Party Rules]]—represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals
|-
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! [[SKULO interval names|SKULO]]-based interval names
! [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic]]-based interval names
! [[Syntonic-rastmic subchroma notation|SRS notation]]
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Prime
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superprime
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superprime
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superprime, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superprime
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraprime
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraprime, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Prime
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Prime
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Prime
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Prime
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Prime
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Prime
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Prime
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Prime
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Prime
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Prime
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Prime
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Prime
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Prime
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== 5-limit diatonic music ==

Although 159edo inherits 53edo's close approximations of both the [[5-limit]] Zarlino scale and the [[3-limit]] [[diatonic]] MOS, these are not the only scales that one can use for fixed-pitch diatonic music even in that system. In fact, both of them are less than optimal for many facets of traditional Western classical music, seeing as for the most part, each of the traditional diatonic modes has its own optimized scale in the 5-limit. The reason for this is that in non-meantone 5-limit systems, one inevitably has to deal with the ~[[40/27]] wolf fifth and the ~[[27/20]] wolf fourth, and there's only two ideal positions in the scale for those intervals to be situated in fixed-pitch diatonic music. For major modes, the ~27/20 wolf fourth is ideally situated between the ~[[5/4]] major third and the ~[[27/16]] major sixth, while for the minor modes and the diatonic blighted mode Locrian, the ~40/27 wolf fifth is ideally situated between the ~[[6/5]] minor third and the ~[[16/9]] minor seventh. Not only that, but it also pays to distinguish the Pythagorean major and minor thirds from their Ptolemaic counterparts as both types of major and minor third have distinct roles to play in the 5-limit diatonic music.

=== Scales and Harmony ===

For purposes of this section, we shall assume the tonic to be D-natural as in the interval chart above. Note that the following trines are available in 5-limit diatonic harmony.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo diatonic trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Supradusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|}

==== Ionian and Major ====

This mode- along the corresponding tonality- is optimized for 5-limit using a variation on the Didymian diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Ionian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Major Third
| F#↓
| 51
| Mesodistomediant
| [[5/4]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Ptolemaic Major Seventh
| C#↓
| 144
| Distosubcollocant
| [[15/8]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| I, V
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| ↓III
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Pythagorean Minor
| D, F, A↓
| 0, 39, 90
| VI
| 27:32:40
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Greater Ptolemaic Diminished
| D, F↑, Ab↑
| 0, 42, 81
| ↓VII
| 45:54:64
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Ionian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| I
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| III
| 10:12:15:18
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Major with Ptolemaic Major Seventh
| D, F#, A, C#↓
| 0, 54, 93, 144
| IV
| 64:81:96:120
| This dissonant tetrad is an unconventional servient (subdominant) seventh for 5-limit music, and is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Dominant Seventh
| D, F#↓, A, C
| 0, 51, 93, 132
| V
| 36:45:54:64
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~3/2 below its root
|-
| Supradusthumic Pythagorean Minor Seventh
| D, F, A↓, C
| 0, 39, 93, 132
| VI
| 54:32:40:48
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Half-Diminished
| D, F↑, Ab↑, C↑
| 0, 42, 81, 135
| VII
| 45:54:64:81
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Ionian in 159edo is considerably more tonally stable than its more familiar counterpart from 12edo and 24edo. It is also one of only two traditional diatonic modes in which one is able to perform a complete [[Wikipedia: Vi–ii–V–I|circle progression]].

==== Dorian ====

Unlike what is expected, the form of Dorian mode optimized for 5-limit is not symmetrical, owing to the need to maintain [[Rothenberg propriety]].

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Dorian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Pythagorean Major Second
| E
| 27
| Supertonic (Bidominant)
| [[9/8]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Pythagorean Major Sixth
| B
| 120
| Proximocontramediant (Tridominant)
| [[27/16]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Pythagorean Major
| D, F#, A
| 0, 54, 93
| IV, bVII
| 1/(54:64:81)
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Pythagorean Minor
| D, F, A
| 0, 39, 93
| II, V
| 54:64:81
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| VI
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Dorian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Pythagorean Major Seventh
| D, F#, A, C#
| 0, 54, 93, 147
| bVII
| 128:162:192:243
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Pythagorean Minor Seventh
| D, F, A, C
| 0, 39, 93, 132
| II, V
| 54:64:81:96
| This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
|-
| Supradusthumic Ptolemaic Major Seventh
| D, F#↓, A↓, C#↓
| 0, 51, 90, 144
| b↑III
| 240:256:320:405
| This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
|-
| Ptolemaic Parallel Dominant Seventh
| D, F#, A, C↑
| 0, 54, 93, 135
| IV
| 1/(45:54:64:81)
| This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~9/8 above its root
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| VI
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

With the above scale steps and chords, this variation of Dorian in 159edo has a few noteworthy tricks- especially once one considers suspensions, which will have to be covered later in this document.

==== Phrygian ====

This mode is optimized for 5-limit using a variation on the left-handed Zarlino diatonic scale.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of Phrygian notes and intervals
|-
! Interval Name
! Notation (from D)
! Steps from Tonic
! Function
! Corresponding JI
|-
| Perfect Unison
| D
| 0
| Tonic
| [[1/1]]
|-
| Ptolemaic Minor Second
| Eb↑
| 15
| Distosupercollocant
| [[16/15]]
|-
| Ptolemaic Minor Third
| F↑
| 42
| Mesoproximomediant
| [[6/5]]
|-
| Perfect Fourth
| G
| 66
| Servient (Subdominant)
| [[4/3]]
|-
| Perfect Fifth
| A
| 93
| Dominant
| [[3/2]]
|-
| Ptolemaic Minor Sixth
| Bb↑
| 108
| Mesodistocontramediant
| [[8/5]]
|-
| Pythagorean Minor Seventh
| C
| 132
| Subtonic (Biservient)
| [[16/9]]
|-
| Perfect Octave
| D
| 159
| Tonic
| [[2/1]]
|}

As a consequence of this particular scale structure, you have the following basic chords...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian triads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| b↑II, b↑VI
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| I, IV, bVII
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
|-
| Supradusthumic Ptolemaic Major
| D, F#↓, A↓
| 0, 51, 90
| b↑III
| 1/(27:32:40)
| This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
|-
| Lesser Ptolemaic Diminished
| D, F, Ab↑
| 0, 39, 81
| V
| 1/(45:54:64)
| This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of basic Phrygian tetrads
|-
! Name
! Notation (from D)
! Steps
! Occur(s) on Scale Degree(s)
! Approximate JI
! Notes
|-
| Ptolemaic Minor with Pythagorean Minor Seventh
| D, F↑, A, C
| 0, 42, 93, 132
| I
| 90:108:135:160
|
|-
| Ptolemaic Major Seventh
| D, F#↓, A, C#↓
| 0, 51, 93, 144
| b↑II
| 8:10:12:15
| This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
|-
| Supradusthumic Ptolemaic Dominant Seventh
| D, F#↓, A↓, C
| 0, 51, 90, 132
| b↑III
| 1/(45:54:64:80)
| This dissonant dominant tetrad has a slightly different function than its more traditional counterpart
|-
| Ptolemaic Minor Seventh
| D, F↑, A, C↑
| 0, 42, 93, 135
| IV, bVII
| 10:12:15:18
|
|-
| Ptolemaic Parallel Half-Diminished
| D, F, Ab↑, C
| 0, 39, 81, 132
| V
| 1/(36:45:54:64)
| This dissonant tetrad is an option for imperfect half cadences in the 5-limit
|}

[[Category:Method]]
[[Category:Approaches to tuning systems]]

== Notes ==
<references group="note" />

Talk:Amity family

2025-10-30T02:51:02Z

Aura:

== Proposal to rename catamite ==
We should rename "catamite" temperament: [[Amity family#Catamite]].

''Wikipedia'': "A catamite (Latin: catamītus) was a pubescent boy who was the intimate companion of an older male, usually in a pederastic relationship."

While catamites were not always children, they often were, and we should not have a temperament named after child abuse.

This name contravenes ''[[Temperament naming]]'': "the name must not be offensive, where "offensive" is defined as referring to topics that quite a few would deem controversial and/or where the name is NSFW (not-safe-for-work)."

I propose the name "stalagmite", since it's associated with caves so it preserves the meaning of "cata-" ("down").

Please reply and let me know:
# If you think "catamite" should be renamed to something else.
# If you would be okay with the name "stalagmite".
# If you have any other names you would like to suggest?

If all or the large majority of replies are in favour, then after I receive 3 replies or after 24 hours pass, whichever happens first, I will go ahead and make the change.

--[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 01:10, 20 October 2025 (UTC)
: I agree that catamite is not an appropriate name, but stalagmite sounds kind of weird as a temperament name. I propose the name "inframity", as "infra" means "below".--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 22:02, 29 October 2025 (UTC)

: I've been thinking this temperament needs to be renamed also, but I wasn't sure if anyone else besides me really thought this was the case until now. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:51, 30 October 2025 (UTC)

Talk:4:5:6

2025-10-30T02:49:02Z

Aura:

== Scale information off-topic? Why? ==

I recently tried to expand the article by mentioning scale information because I think it's good to know how certain microtonal chords can be used in composition. I fail to see how such information is off-topic. If there's guidelines for what sorts of information to add to these chord pages that I don't know about, then would someone mind explaining that to me? --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:48, 30 October 2025 (UTC)

Talk:4:5:6

2025-10-30T02:48:41Z

Aura: /* Scale information off-topic? Why? */ new section

== Scale information off-topic? Why? ==

I recently tried to expand the article by mentioning scale information because I think it's good to know how certain microtonal chords can be used in composition. I fail to see how such information is off-topic. If there's guidelines for what sorts of information to add to these chord pages that I don't know about, then would someone mind explaining that to me. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:48, 30 October 2025 (UTC)

Xenharmonic Wiki:Elections

2025-10-28T02:06:47Z

Aura:

This '''Elections''' page is where the xen community decides who will become [[administrators]] or [[bureaucrats]] of the xenwiki. Voting is open to all xenwiki editors.

To vote, simply put your comments and signature under the relevant section:

<code><nowiki># '''Support'''. I think this is a very good candidate! ~~~~</nowiki></code>

You can also add comments or questions for the candidate under 'comments'.

==Fredg999==

===Nomination===

I would like to nominate [[User:Fredg999]] for [[Xenharmonic_Wiki:Bureaucrats|bureaucrat]] (he is already an admin currently). The wiki currently has very few active bureaucrats and I think it's a good idea to have at least one more.

I'll keep it brief. Fredg999 has a track record of good edits, mostly focused on improving the general quality and usability of the wiki. More importantly though, he has shown himself to be a very level-headed person who is good at conflict resolution.

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 18:45, 21 October 2025 (UTC)

===Support===
# '''Support'''. As a nominator. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 18:45, 21 October 2025 (UTC)
# '''Support'''. Absolutely. – [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 22:10, 21 October 2025 (UTC)
# '''Support'''. I was hoping someone would propose this actually, I think he'd be great for the role. - [[Budjarn Lambeth]], 22 October 2025 (UTC+10)
# '''Support'''. Can't think of a better person for the job. – [[User:Nick Vuci|Nick Vuci]] ([[User talk:Nick Vuci|talk]]) 23:42, 21 October 2025 (UTC)
# '''Support'''. he is pretty chill i approve of this – [[User:DotuXil|dotuXil]] <span style="color:#f50c7d">(i like microtones)</span> 23:44, 21 October 2025 (UTC)
# '''Support'''. Fredg would be perfect for this. [[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 02:25, 22 October 2025 (UTC)
# '''Support'''. I accept the nomination. – [[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 04:37, 22 October 2025 (UTC)
# '''Support'''. I just agree with the arguments the others already wrote. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 20:44, 22 October 2025 (UTC)
# '''Support'''. Fredg999 has always been very reasonable and drives toward good compromises. [[User:Bcmills|Bcmills]] ([[User talk:Bcmills|talk]]) 17:26, 24 October 2025 (UTC)
# '''Support'''. Fredg seems level-headed. I also trust the judgment of his supporters. [[Gordon Wery]] 4:40, 25 October 2025 (UTC)
# '''Support'''. I think Fred will prove good at this job based on my own experiences with him. [[User:Aura|Aura]] ([[User talk:Aura|talk]]) 05:29, 27 October 2025 (UTC)

===Oppose===

===Neutral===
# '''Neutral.''' Meh. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]])

===Comments===
The one and only problem I see so far is that if we don't have enough bureaucrats, Fred could be overloaded- especially since he seems to be so busy as is- so we need to nominate at least one to two more bureaucrats. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 05:29, 27 October 2025 (UTC)

: Hi Aura,

: This is a fair concern; I think promoting Fred is the best first step for addressing it.

: Once Fred is promoted, there will be 3 active bureaucrats instead of 2: Tyler, FloraC and Fred.

: If someone else were to become a bureaucrat, it would probably make sense for them to be a regular admin for a year or so first (as was the case with Flora and Fred).

: Here is the list of current admins: [[Xenharmonic Wiki:Administrators#Current administrators]]. On that list, aside from Fred, I don't see anyone who is both active enough and non-controversial enough to be an ideal candidate for bureaucrat.

: As such, I think it might make some sense to nominate more new admins; and maybe a year or two down the line if things are going well, they might be nominated as full bureaucrats.

: I think that any of:
* Ganaram Inukshuk
* Bcmills
* Sintel
* Nick Vuci
* Gordon Wery
* Xenoindex
: <small>''(alphabetical order by surname)''</small>

: would be good choices (only if they themselves were okay with being nominated though of course; we'd need to ask them individually first).

: Are there any of those suggestions you agree or disagree with? Any other names you think it might be a good idea to nominate?

: --[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 01:08, 28 October 2025 (UTC)

:: For some reason, I thought that Admin was above Bureaucrat, but it seems I was mistaken. Anyhow, of that list you propose, the two that really come to mind are Xenoindex and Bcmills. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 01:59, 28 October 2025 (UTC)

:: In addition, I would think that Ganaram Inukshuk and Sintel would make good admins. I do disagree with the idea of Nick Vuci as an admin at least for now- I'll have to elaborate more on the problems in DMs on Discord seeing as I'm not totally comfortable clogging up this page with the details of that matter. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:06, 28 October 2025 (UTC)

Xenharmonic Wiki:Elections

2025-10-28T01:59:45Z

Aura:

Xenharmonic Wiki:Elections

2025-10-27T05:29:49Z

Aura:

Triad

2025-10-26T16:36:24Z

Aura: added some information and corrected the wording of a few sections.

{{Wikipedia|Triad (music)}}
A '''triad''' is a [[chord]] of three [[pitch class]]es.

== Quality ==
{{Category see also|Triads}}
Triads can be classified by [[quality]], based on the quality of the intervals above their root.

=== Tertian triads ===
Many triads in Western classical music are based on tertian harmony, i.e. stacks of thirds. These triads occur naturally in the [[diatonic scale]] by taking the third and the fifth above the root, with different notes of the scale leading to different qualities.

The most common triads in modern music are the ''[[major triad]]'' (root-M3-P5) and the ''[[minor triad]]'' (root-m3-P5). Also common are the ''[[augmented triad]]'' (root-M3-A5) and the ''[[diminished triad]]'' (root-m3-d5). Note that the augmented triad does not occur naturally in the diatonic scale, except if you count the {{w|harmonic minor scale}} and the {{w|harmonic major scale}} (which are both altered diatonic scales).

Suspended second and suspended fourth triads may be classified as tertian triads if they are used, as their name implies, as suspensions leading to other tertian triads. They may otherwise be classified as quartal or quintal triads, depending on their voicing, if they are used differently, such as a set of parallel suspended triads.

[https://www.youtube.com/watch?v=QusJLok_2oA The Perfect Triad (Claudi Meneghin)], illustrates the spectrum of "perfect triads" in which the root and the 5th are fixed (at 2:3) and the third of the triad gradually moves through a spectrum of [[Just intonation]] intervals from subminor [[7/6]] to supermajor [[9/7]].

It should be noted that triads framed by a perfect fifth- or, rather, as many of such as were known in the Medieval era- were referred to under the term "quinta fissa" or "split fifth" by Jacobus of Liege<ref>Speculum Musicae, c. 1325</ref>. Today, such a term could conceivably refer not only to Claudi Meneghin's "perfect triads", but also to suspended second and suspended fourth triads, and even other such triads that could be viable for voice-leading.

=== Xenharmonic triads ===
{{todo|expand|inline=1}}

In addition to the tertian and split fifth triads which are framed by the [[3/2]] perfect fifth, one could argue that there are other triads that could be referred to under the term "quarta fissa" or "split fourth". Such triads can be stacked multiple times in the span of a single octave due to two stacked instances of [[4/3]] being able to fit within the span of the octave. Because the perfect fourth sounds more alien than the perfect fifth as an interval for framing chords, chords formed this way only dial the strangeness up a notch when the divisions are of such sizes as to not cause crowding, and such triads are arguably xenharmonic.

== See also ==
* Chords by size:
** [[Monad]]
** [[Dyad]]
** [[Tetrad]]
** [[Pentad]]
** [[Hexad]]

== External links ==
* [http://tonalsoft.com/enc/t/triad.aspx Tonalsoft Encyclopedia | ''Triad'']

[[Category:Triad| ]]
[[Category:Terms]]

== References ==
<references />

{{Stub}}

Triad

2025-10-26T16:17:22Z

Aura:

{{Wikipedia|Triad (music)}}
A '''triad''' is a [[chord]] of three [[pitch class]]es.

== Quality ==
{{Category see also|Triads}}
Triads can be classified by [[quality]], based on the quality of the intervals above their root.

=== Tertian triads ===
Many triads in Western classical music are based on tertian harmony, i.e. stacks of thirds. These triads occur naturally in the [[diatonic scale]] by taking the third and the fifth above the root, with different notes of the scale leading to different qualities.

The most common triads in modern music are the ''[[major triad]]'' (root-M3-P5) and the ''[[minor triad]]'' (root-m3-P5). Also common are the ''[[augmented triad]]'' (root-M3-A5) and the ''[[diminished triad]]'' (root-m3-d5). Note that the augmented triad does not occur naturally in the diatonic scale, except if you count the {{w|harmonic minor scale}} and the {{w|harmonic major scale}} (which are both altered diatonic scales).

Suspended second and suspended fourth triads may be classified as tertian triads if they are used, as their name implies, as suspensions leading to other tertian triads. They may otherwise be classified as quartal or quintal triads, depending on their voicing, if they are used differently, such as a set of parallel suspended triads.

[https://www.youtube.com/watch?v=QusJLok_2oA The Perfect Triad (Claudi Meneghin)], illustrates the spectrum of "perfect triads" in which the root and the 5th are fixed (at 2:3) and the third of the triad gradually moves through a spectrum of [[Just intonation]] intervals from subminor [[7/6]] to supermajor [[9/7]]. It should be noted that such "perfect triads"- or, rather, as many of such as were known in the Medieval era- were referred to by the term "quinta fissa" or "split fifth" by Jacobus of Liege<ref>Speculum Musicae, c. 1325</ref>.

=== Xenharmonic triads ===
{{todo|expand|inline=1}}

== See also ==
* Chords by size:
** [[Monad]]
** [[Dyad]]
** [[Tetrad]]
** [[Pentad]]
** [[Hexad]]

== External links ==
* [http://tonalsoft.com/enc/t/triad.aspx Tonalsoft Encyclopedia | ''Triad'']

[[Category:Triad| ]]
[[Category:Terms]]

== References ==
<references />

{{Stub}}

User talk:Overthink

2025-10-25T17:37:02Z

Aura:

54:64:81

2025-10-25T17:08:21Z

Aura:

{{Infobox Chord|ColorName=wa or w}}
'''54:64:81''', is a [[minor triad]] found on the ii, iii, and vi of the [[3-limit]] [[5L 2s|diatonic scale]]. It is easily referred to as the '''Pythagorean minor triad''', and, unlike [[10:12:15]], 54:64:81 is [[ambitonal]].

In [[meantone]] this chord is [[tempered together]] with both 10:12:15 and [[27:32:40]], so it may be considered a 3-limit approximation of those chords, or a chord in its own right with a less-stable third. Accordingly, [[User:Aura|Aura]] takes advantage of these properties in his non-meantone [[5-limit]] music where it shows up as the triad built on the ii degree in Ionian mode in place of 27:32:40, thus allowing it to be more reliably used in the major scale's [[Wikipedia: Circle progression|circle progression]], while also cutting back on tonal ambiguity.

[[Category:Minor triads|##]]

64:81:96

2025-10-25T17:06:37Z

Aura:

{{Infobox Chord|ColorName=lawa or Lw}}
'''64:81:96''' is a [[major triad]] found on the I, IV, and V of the [[3-limit]] [[5L 2s|diatonic scale]]. It is easily referred to as the '''Pythagorean major triad''', and, unlike [[4:5:6]], 54:64:81 is [[ambitonal]].

In [[meantone]] this chord is tempered together with 4:5:6; it may be considered a 3-limit approximation of that chord, or used in its own right as a chord with a less stable third. Accordingly, [[User:Aura|Aura]] takes advantage of these properties in his non-meantone [[5-limit]] music where it shows up as the triad built on the IV degree in Ionian mode, thus preventing that scale degree from being tonicized by accident and lessening tonal ambiguity in plagal and semiplagal cadences, as well as other chord progressions that use both I and IV.

[[Category:Major triads|##]]

10:12:15

2025-10-25T16:50:02Z

Aura:

{{Infobox Chord|ColorName=gu or g}}

'''10:12:15''' is the classical [[minor triad]], and can also be referred to as the '''Ptolemaic minor triad'''. It is found on the iii ({{Frac|5|4}}) and vi ({{Frac|5|3}}) of Ptolemy's intense diatonic scale ([[Zarlino]]), which is perhaps the most common [[5-limit]] diatonic. Unlike [[27:32:40]], which appears on the ii of the same scale, 10:12:15 is [[utonal]].

However, there are other 5-limit diatonic scales which don't have the Ptolemaic minor triad occurring in all the same places. For instance, [[User:Aura|Aura]] is known to use a diatonic minor scale in which this chord only occurs on the i and iv scale degrees while using a Pythagorean minor triad (that is, [[54:64:81]]) on the v. Conversely, in the diatonic major scale that Aura uses, this chord only really appears on the iii. Compared to its Pythagorean counterpart, the Ptolemaic major triad sounds like it's more consonant. Because of these properties, the Ptolemaic major triad has earned its status as a bread-and-butter chord in 5-limit harmony.

There are a number of possible tetrads which can be reasonably built off of this triad, such as [[10:12:15:18]] in the 5-limit, as well as [[70:84:105:120]] in the 7-limit and [[110:132:165:192]] in the 11-limit.

[[Category:Minor triads|#@]]

27:32:40

2025-10-25T16:49:31Z

Aura: Added information

{{Infobox Chord|ColorName=wa yo-5 or w(y5)}}

'''27:32:40''' is a 5-limit [[minor triad]] found on the ii ({{Frac|9|8}}) of Ptolemy's intense diatonic scale ([[Zarlino]]), perhaps the most common [[5-limit]] diatonic. Unlike [[10:12:15]], which appears on the iii and vi of the same scale, 27:32:40 is [[otonal]].

As has been noted by multiple theorists of a more traditional Western Classical school of thought, this chord is not ideal when situated on the ii scale degree of a major scale, for a number of different possible reasons. However, because of the way 5-limit diatonic music works, the occurrence of this chord in a simple 5-limit diatonic scale is inevitable outside of [[meantone]]. Thus, [[User:Aura|Aura]] has decided to place this chord on the vi scale degree while using [[54:64:81]] on the ii scale degree in his diatonic major scales. This has the effect of allowing the vi-ii-V-I sequence in the major scale's [[Wikipedia: Circle progression|circle progression]] to actually function in such a way as to make each chord in the sequence seem progressively less tense, thus making the progression overall more coherent.

[[Category:Minor triads|##]]

10:12:15

2025-10-25T16:27:34Z

Aura:

4:5:6

2025-10-25T16:11:49Z

Aura:

{{Infobox Chord|ColorName=yo or y}}
'''4:5:6''' is the classical [[major triad]], and can also be referred to as the '''Ptolemaic major triad'''. It is found on the I ({{Frac|1|1}}), IV ({{Frac|4|3}}), and V ({{Frac|3|2}}) of Ptolemy's intense diatonic scale ([[Zarlino]]), which is perhaps the most common [[5-limit]] diatonic.

However, there are other 5-limit diatonic scales which don't have the Ptolemaic major triad occurring in all the same places. For instance, [[User:Aura|Aura]] is known to use a diatonic major scale in which this chord only occurs on the I and V scale degrees while using a Pythagorean major triad (that is, [[64:81:96]]) on the IV. Conversely, in the diatonic minor scale that Aura uses, this chord only really appears on the bVI. Compared to its Pythagorean counterpart, the Ptolemaic major triad sounds like it's more easily tonicized, a fact which Aura exploits in order to help stabilize Ionian mode in fixed pitch diatonic scales. Because of these properties, the Ptolemaic major triad has earned its status as a bread-and-butter chord in 5-limit harmony.

There are a number of possible tetrads which can be reasonably built off of this triad, such as [[36:45:54:64]] and [[20:25:30:36]] in the 5-limit, as well as [[4:5:6:7]] in the 7-limit and [[32:40:48:55]] in the 11-limit.

[[Category:Major triads|#]]

4:5:6

2025-10-25T16:05:24Z

Aura:

{{Infobox Chord|ColorName=yo or y}}
'''4:5:6''' is the classical [[major triad]], and can also be referred to as the '''Ptolemaic major triad'''. It is found on the I ({{Frac|1|1}}), IV ({{Frac|4|3}}), and V ({{Frac|3|2}}) of Ptolemy's intense diatonic scale ([[Zarlino]]), which is perhaps the most common [[5-limit]] diatonic.

However, there are other 5-limit diatonic scales which don't have the Ptolemaic major triad occurring in all the same places. For instance, [[User:Aura|Aura]] is known to use a diatonic major scale in which this chord only occurs on the I and V scale degrees while using a Pythagorean major triad (that is, [[64:81:96]]) on the IV. Compared to its Pythagorean counterpart, the Ptolemaic major triad sounds like it's more easily tonicized, a fact which Aura exploits in order to help stabilize Ionian mode in fixed pitch diatonic scales. Because of these properties, the Ptolemaic major triad has earned its status as a bread-and-butter chord in 5-limit harmony.

There are a number of possible tetrads which can be reasonably built off of this triad, such as [[36:45:54:64]] and [[20:25:30:36]] in the 5-limit, as well as [[4:5:6:7]] in the 7-limit and [[32:40:48:55]] in the 11-limit.

[[Category:Major triads|#]]

32:40:48:55

2025-10-25T08:50:02Z

Aura: Added link

{{Infobox Chord|ColorName=yo loyo-6}}
'''32:40:48:55''' is an [[11-limit]] tetrad that easily fits in what is otherwise a [[5-limit]] context, especially when the root of the chord is a Ptolemaic minor sixth ([[8/5]]) above the Tonic.

[[Category:Major tetrads|#]] 

{{stub}}

32:40:48:55

2025-10-25T08:47:54Z

Aura: Created artic stub. More to come.

{{Infobox Chord|ColorName=yo loyo-6}}
'''32:40:48:55''' is an [[11-limit]] tetrad that easily fits in what is otherwise a 5-limit context, especially when the root of the chord is a Ptolemaic minor sixth ([[8/5]]) above the Tonic.

[[Category:Major tetrads|#]] 

{{stub}}

64:81:96

2025-10-25T02:12:56Z

Aura: Expanded the article- will have to link articles on tetrads later.

{{Infobox Chord|ColorName=lawa or Lw}}
'''64:81:96''' is a [[major triad]] found on the I, IV, and V of the [[3-limit]] [[5L 2s|diatonic scale]]. It is easily referred to as the '''Pythagorean major triad'''.

In [[meantone]] this chord is tempered together with [[4:5:6]]; it may be considered a 3-limit approximation of that chord, or used in its own right as a chord with a less stable third. Accordingly, [[User:Aura|Aura]] takes advantage of these properties in his non-meantone [[5-limit]] music where it shows up as the triad built on the IV degree in Ionian mode, thus preventing that scale degree from being tonicized by accident and lessening tonal ambiguity in plagal and semiplagal cadences, as well as other chord progressions that use both I and IV.

[[Category:Major triads|##]]

4:5:6

2025-10-25T01:55:32Z

Aura: Fixed typo

{{Infobox Chord|ColorName=yo or y}}
'''4:5:6''' is the classical [[major triad]], and can also be referred to as the '''Ptolemaic major triad'''. It is found on the I ({{Frac|1|1}}), IV ({{Frac|4|3}}), and V ({{Frac|3|2}}) of Ptolemy's intense diatonic scale ([[Zarlino]]), which is perhaps the most common [[5-limit]] diatonic.

However, there are other 5-limit diatonic scales which don't have the Ptolemaic major triad occurring in all the same places. For instance, [[User:Aura|Aura]] is known to use a diatonic scale in which this chord only occurs on the I and V scale degrees while using a Pythagorean major triad (that is, [[64:81:96]]) on the IV. Compared to its Pythagorean counterpart, the Ptolemaic major triad sounds like it's more easily tonicized, a fact which Aura exploits in order to help stabilize Ionian mode in fixed pitch diatonic scales. Because of these properties, the Ptolemaic major triad has earned its status as a bread-and-butter chord in 5-limit harmony.

There are a number of possible tetrads which can be reasonably built off of this triad, such as [[36:45:54:64]] and [[20:25:30:36]] in the 5-limit, as well as [[4:5:6:7]] in the 7-limit and [[32:40:48:55]] in the 11-limit.

[[Category:Major triads|#]] 

{{stub}}

4:5:6

2025-10-25T01:54:57Z

Aura: Added information about potential usages and the tetrads that can be built off from it

{{Infobox Chord|ColorName=yo or y}}
'''4:5:6''' is the classical [[major triad]], and can also be referred to as the '''Ptolemaic major triad'''. It is found on the I ({{Frac|1|1}}), IV ({{Frac|4|3}}), and V ({{Frac|3|2}}) of Ptolemy's intense diatonic scale ([[Zarlino]]), which is perhaps the most common [[5-limit]] diatonic.

However, there are other 5-limit diatonic scales which don't have the Ptolemaic major triad occurring in all the same places. For instance, [[User:Aura|Aura]] is known to us a diatonic scale in which this chord only occurs on the I and V scale degrees while using a Pythagorean major triad (that is, [[64:81:96]]) on the IV. Compared to its Pythagorean counterpart, the Ptolemaic major triad sounds like it's more easily tonicized, a fact which Aura exploits in order to help stabilize Ionian mode in fixed pitch diatonic scales. Because of these properties, the Ptolemaic major triad has earned its status as a bread-and-butter chord in 5-limit harmony.

There are a number of possible tetrads which can be reasonably built off of this triad, such as [[36:45:54:64]] and [[20:25:30:36]] in the 5-limit, as well as [[4:5:6:7]] in the 7-limit and [[32:40:48:55]] in the 11-limit.

[[Category:Major triads|#]] 

{{stub}}