2.3.7 subgroup: Difference between revisions

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The '''2.3.7 subgroup'''<ref>Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, and "limit" used alone usually implies prime limit.</ref> ('''za''' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, and 7 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, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.
The '''2.3.7 subgroup'''<ref group="note">Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, and "limit" used alone usually implies prime limit.</ref> sometimes called '''septal''' or, in [[color notation]], '''za''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, and 7 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, and 7. This is an infinite set, even when restricted to a single octave. Some examples within the [[octave]] include [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.


The 2.3.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 5. Its simplest expansion is the [[2.3.7.11 subgroup]], which adds prime 11.
The 2.3.7 subgroup is a retraction of the [[7-limit]], obtained by removing [[prime interval|prime]] [[5/1|5]]. Its simplest expansion is the [[2.3.7.11 subgroup]], which adds prime [[11/1|11]].


A notable subset of the 2.3.7 subgroup is the 1.3.7 [[tonality diamond]], comprised of all intervals in which 1, 3 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.3.7 tonality diamond within the octave is [[1/1]], [[8/7]], [[7/6]], [[4/3]], [[3/2]], [[12/7]], [[7/4]], and [[2/1]].
A notable subset of the 2.3.7 subgroup is the {1, 3, 7} [[tonality diamond]], comprising all intervals in which 1, 3 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in this tonality diamond within the octave is [[1/1]], [[8/7]], [[7/6]], [[4/3]], [[3/2]], [[12/7]], [[7/4]], and [[2/1]].


Another such subset is the 1.3.7.9 tonality diamond, which adds the following intervals to the previous list: [[9/8]], [[9/7]], [[14/9]], and [[16/9]].
Another such subset is the {1, 3, 7, 9} tonality diamond, which adds the following intervals to the previous list: [[9/8]], [[9/7]], [[14/9]], and [[16/9]].


When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].
When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].


== Scales ==
== Chords and harmony ==
* Zo minor pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1
There are a number of ways to approach harmony in this subgroup, most of which are discussed in [[Superpyth #Chords and harmony]]. The basic forms of chords include the {{w|tertian harmony|tertian}} triad [[6:7:9]], the tetrad [[6:7:8:9]], and their utonal inverses. The fourth-spanning triads [[6:7:8]] and [[21:24:28]] can also be used, as well as their wide voicing 4:7:12 and 7:12:21. Extensions of these chords include but are not limited to the [[9-odd-limit]] [[anomalous saturated suspension|saturated suspensions]], [[12:14:18:21]] and [[14:18:21:24]].
* Ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1
 
* Zo [[wikipedia:In_scale|in]]: 1/1 9/8 7/6 3/2 14/9 2/1
Like in [[5-limit]] [[JI]], one quickly runs into [[wolf interval]]s without care, but the 2.3.7 wolf being [[21/16]] or [[32/21]] may be considered less discordant, and useful in its own ways. For example, 1–9/8–21/16–3/2 and 1–8/7–4/3–3/2 may be considered [[21-odd-limit]] concords. They are conflated in superpyth as the sus2-4 chord, but subtlely contrast each other in JI.
* Zo minor: 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1
 
* Za harmonic minor: 1/1 9/8 7/6 4/3 3/2 14/9 27/14 2/1 (zo minor with a ru 7th)
== Properties ==
* Za [[diasem]]/Tas[9] ([[Chiral|left-handed]]): 1/1 9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1
The simpler ratios fall into 3 categories:
* Ratios without a 7 are pythagorean and sound much like 12edo intervals;
* Ratios with a 7 in the numerator (7-over or '''zo''' in color notation) sound [[supermajor and subminor|subminor]];
* Ratios with a 7 in the denominator (7-under or '''ru''' in color notation) sound [[supermajor and subminor|supermajor]].
 
This subgroup is notably well-represented by [[5edo]] for its size, and therefore many of its simple intervals tend to cluster around the notes of 5edo: [[9/8]]~[[8/7]]~[[7/6]] representing a pentatonic "second", [[9/7]]~[[21/16]]~[[4/3]] representing a pentatonic "third", and so on. Therefore, one way to approach the 2.3.7 subgroup is to think of a pentatonic framework for composition as natural to it, rather than the diatonic framework associated with the [[5-limit]], and a few of the scales below reflect that nature.
 
=== Scales ===
==== Minor ====
* zo pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1
* zo {{w|in scale|in}}: 1/1 9/8 7/6 3/2 14/9 2/1 (the in scale is a minor scale with no fourth or seventh)
* zo: 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1
* za harmonic minor: 1/1 9/8 7/6 4/3 3/2 14/9 27/14 2/1 (zo scale with a ru seventh)
 
==== Major ====
* ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1
* ru: 1/1 9/8 9/7 4/3 3/2 12/7 27/14 2/1
 
==== Misc ====
* [[Diasem]]/Tas[9] ([[chirality|left-handed]]): 1/1 9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1


== Regular temperaments ==
== Regular temperaments ==
=== Rank-1 temperaments (edos) ===
=== Rank-1 temperaments (edos) ===
A list of edos with progressively better tunings for the 2.3.7 subgroup: {{EDOs| 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277 }} and so on.
A list of edos with progressively better tunings for the 2.3.7 subgroup (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs| '''5''', 12, 14, 17, 22, 31, '''36''', 77, 94, 130, '''135''', 171, 265, 306, 400, '''571''', 706, '''1277''', … }}


Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.  
Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{Main|Tour of regular temperaments#Clans defined by a 2.3.7 (za) comma}}
{{Main| Tour of regular temperaments #Clans defined by a 2.3.7 (za) comma }}
 
In the below tables, the generator of the temperament is highlighted in bold. Intervals in the tables reflect the {1, 3, 7, 9, 21} tonality diamond. It is structurally notable that the intervals come in four triplets, each centered around one note of 5edo, consisting of a "minor", "middle", and "major" interval of each set.
 
==== Semaphore ====
{{Main| Semaphore and godzilla }}
 
Semaphore tempers out the comma [[49/48]] ({{S|7}}) in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].
 
The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 696.230{{c}}; a chart of mistunings of simple intervals is below.
 
{| class="wikitable center-1 center-2 center-3 center-4"
|+ style="font-size: 105%;" | Semaphore (49/48)
|-
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*
|-
! Optimal tuning !! Deviation
|-
| 9/8 || 203.910 || 192.460 || -11.450
|-
| '''8/7''' || 231.174 || '''251.885''' || +20.711
|-
| 7/6 || 266.871 || '''251.885''' || -14.986
|-
| 9/7 || 435.084 || 444.345 || +9.261
|-
| 21/16 || 470.781 || 444.345 || -26.436
|-
| 4/3 || 498.045 || 503.770 || +5.725
|-
| 3/2 || 701.955 || 696.230 || -5.725
|-
| 32/21 || 729.219 || 755.655 || +26.436
|-
| 14/9 || 764.916 || 755.655 || -9.261
|-
| 12/7 || 933.129 || 948.115 || +14.986
|-
| 7/4 || 968.826 || 948.115 || -20.711
|-
| 16/9 || 996.090 || 1007.540 || +11.450
|}
<nowiki />* In 2.3.7-targeted DKW tuning
 
==== Archy ====
{{Main| Superpyth }}
 
Archy tempers out the comma [[64/63]] ({{S|8}}) in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.
 
The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712.585{{c}} though most other optimizations tune this a few cents flatter; a chart of mistunings of simple intervals is below.
 
{| class="wikitable center-1 center-2 center-3 center-4"
|+ style="font-size: 105%;" | Archy (64/63)
|-
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*
|-
! Optimal tuning !! Deviation
|-
| 9/8 || 203.910 || 225.171 || +21.261
|-
| 8/7 || 231.174 || 225.171 || -6.003
|-
| 7/6 || 266.871 || 262.244 || -4.627
|-
| 9/7 || 435.084 || 450.341 || +15.257
|-
| 21/16 || 470.781 || 487.415 || +16.634
|-
| 4/3 || 498.045 || 487.415 || -10.630
|-
| '''3/2''' || 701.955 || '''712.585''' || +10.630
|-
| 32/21 || 729.219 || '''712.585''' || -16.634
|-
| 14/9 || 764.916 || 749.659 || -15.257
|-
| 12/7 || 933.129 || 937.756 || +4.627
|-
| 7/4 || 968.826 || 974.829 || +6.003
|-
| 16/9 || 996.090 || 974.829 || -21.261
|}
<nowiki />* In 2.3.7-targeted DKW tuning
 
==== Slendric ====
{{Main| Slendric }}
 
Slendric tempers out the comma [[1029/1024]] ([[S-expression|S7/S8]]) in the 2.3.7 subgroup, which splits the perfect fifth into three intervals of [[8/7]]. It is one of the most accurate temperaments of its simplicity. While semaphore and archy equate each middle interval of each triplet with either the major or the minor, gamelic makes it a true "neutral" intermediate between them.
 
The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 699.126{{c}}, and therefore ~8/7 to 233.042{{c}}; a chart of mistunings of simple intervals is below.
 
{| class="wikitable center-1 center-2 center-3 center-4"
|+ style="font-size: 105%;" | Gamelic (1029/1024)
|-
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*
|-
! Optimal tuning !! Deviation
|-
| 9/8 || 203.910 || 198.253 || -5.657
|-
| '''8/7''' || 231.174 || '''233.042''' || +1.868
|-
| 7/6 || 266.871 || 267.831 || +0.960
|-
| 9/7 || 435.084 || 431.295 || -3.789
|-
| 21/16 || 470.781 || 466.084 || -4.697
|-
| 4/3 || 498.045 || 500.874 || +2.829
|-
| 3/2 || 701.955 || 699.126 || -2.829
|-
| 32/21 || 729.219 || 733.916 || +4.697
|-
| 14/9 || 764.916 || 768.705 || +3.789
|-
| 12/7 || 933.129 || 932.169 || -0.960
|-
| 7/4 || 968.826 || 966.958 || -1.868
|-
| 16/9 || 996.090 || 1001.747 || +5.657
|}
<nowiki>*</nowiki> In 2.3.7-targeted DKW tuning
 
==== Exotemperaments ====
[[Trienstonian]] tempers out 28/27 in the 2.3.7 subgroup, mapping 7/4 to +3 fifths, and equating 9/8 with 7/6 and 9/7 with 4/3. This temperament equates the outer intervals in each cluster of three intervals without setting the middle to them. Due to its inaccuracy, it can reasonably be considered an exotemperament, which in a pentatonic system can be considered an analog to [[mavila]].
 
The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 723.494{{c}}, and therefore ~8/7 to 229.517{{c}}; a chart of mistunings of simple intervals is below.
 
{| class="wikitable center-1 center-2 center-3 center-4"
|+ style="font-size: 105%;" | Trienstonian (28/27)
|-
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*
|-
! Optimal tuning !! Deviation
|-
| 9/8 || 203.910 || 246.989 || +43.079
|-
| 8/7 || 231.174 || 229.517 || -1.868
|-
| 7/6 || 266.871 || 246.989 || -19.882
|-
| 9/7 || 435.084 || 476.506 || +41.422
|-
| 21/16 || 470.781 || 493.977 || +23.196
|-
| 4/3 || 498.045 || 476.506 || -21.539
|-
| '''3/2''' || 701.955 || '''723.494''' || +21.539
|-
| 32/21 || 729.219 || 706.023 || -23.196
|-
| 14/9 || 764.916 || 723.494 || -41.422
|-
| 12/7 || 933.129 || 953.011 || +19.882
|-
| 7/4 || 968.826 || 970.483 || +1.868
|-
| 16/9 || 996.090 || 953.011 || -43.079
|}
<nowiki>*</nowiki> In 2.3.7-targeted DKW tuning


== Music ==
== Music ==
Line 30: Line 209:
* From Ancient Worlds (for harmonic piano), 1992
* From Ancient Worlds (for harmonic piano), 1992
* Revelation: Music in Pure Intonation, 2007
* Revelation: Music in Pure Intonation, 2007
; [[La Monte Young]]
* The Well-Tuned Piano, 1974


== Notes ==
== Notes ==
<references />
<references group="note"/>


[[Category:Subgroup]]
[[Category:Just intonation subgroups|#]]
[[Category:7-limit]]
[[Category:Rank-3 temperaments|#]]
[[Category:Rank 3]]
[[Category:7-limit|#]]
[[Category:Lists of scales]]
[[Category:Lists of scales|#]]

Latest revision as of 12:02, 17 March 2026

The 2.3.7 subgroup[note 1] sometimes called septal or, in color notation, za is a just intonation subgroup consisting of rational intervals where 2, 3, and 7 are the only allowable prime factors, so that every such interval may be written as a ratio of integers which are products of 2, 3, and 7. This is an infinite set, even when restricted to a single octave. Some examples within the octave include 3/2, 7/4, 7/6, 9/7, 9/8, 21/16, and so on.

The 2.3.7 subgroup is a retraction of the 7-limit, obtained by removing prime 5. Its simplest expansion is the 2.3.7.11 subgroup, which adds prime 11.

A notable subset of the 2.3.7 subgroup is the {1, 3, 7} tonality diamond, comprising all intervals in which 1, 3 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in this tonality diamond within the octave is 1/1, 8/7, 7/6, 4/3, 3/2, 12/7, 7/4, and 2/1.

Another such subset is the {1, 3, 7, 9} tonality diamond, which adds the following intervals to the previous list: 9/8, 9/7, 14/9, and 16/9.

When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in a 2-dimensional lattice diagram, each prime represented by a different dimension, such that each point on the lattice represents a different interval class.

Chords and harmony

There are a number of ways to approach harmony in this subgroup, most of which are discussed in Superpyth #Chords and harmony. The basic forms of chords include the tertian triad 6:7:9, the tetrad 6:7:8:9, and their utonal inverses. The fourth-spanning triads 6:7:8 and 21:24:28 can also be used, as well as their wide voicing 4:7:12 and 7:12:21. Extensions of these chords include but are not limited to the 9-odd-limit saturated suspensions, 12:14:18:21 and 14:18:21:24.

Like in 5-limit JI, one quickly runs into wolf intervals without care, but the 2.3.7 wolf being 21/16 or 32/21 may be considered less discordant, and useful in its own ways. For example, 1–9/8–21/16–3/2 and 1–8/7–4/3–3/2 may be considered 21-odd-limit concords. They are conflated in superpyth as the sus2-4 chord, but subtlely contrast each other in JI.

Properties

The simpler ratios fall into 3 categories:

  • Ratios without a 7 are pythagorean and sound much like 12edo intervals;
  • Ratios with a 7 in the numerator (7-over or zo in color notation) sound subminor;
  • Ratios with a 7 in the denominator (7-under or ru in color notation) sound supermajor.

This subgroup is notably well-represented by 5edo for its size, and therefore many of its simple intervals tend to cluster around the notes of 5edo: 9/8~8/7~7/6 representing a pentatonic "second", 9/7~21/16~4/3 representing a pentatonic "third", and so on. Therefore, one way to approach the 2.3.7 subgroup is to think of a pentatonic framework for composition as natural to it, rather than the diatonic framework associated with the 5-limit, and a few of the scales below reflect that nature.

Scales

Minor

  • zo pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1
  • zo in: 1/1 9/8 7/6 3/2 14/9 2/1 (the in scale is a minor scale with no fourth or seventh)
  • zo: 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1
  • za harmonic minor: 1/1 9/8 7/6 4/3 3/2 14/9 27/14 2/1 (zo scale with a ru seventh)

Major

  • ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1
  • ru: 1/1 9/8 9/7 4/3 3/2 12/7 27/14 2/1

Misc

Regular temperaments

Rank-1 temperaments (edos)

A list of edos with progressively better tunings for the 2.3.7 subgroup (decreasing TE error, bold ones do particularly well in this subgroup): 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277, …

Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 and so on.

Rank-2 temperaments

Main article: Tour of regular temperaments § Clans defined by a 2.3.7 (za) comma

In the below tables, the generator of the temperament is highlighted in bold. Intervals in the tables reflect the {1, 3, 7, 9, 21} tonality diamond. It is structurally notable that the intervals come in four triplets, each centered around one note of 5edo, consisting of a "minor", "middle", and "major" interval of each set.

Semaphore

Main article: Semaphore and godzilla

Semaphore tempers out the comma 49/48 (S7) in the 2.3.7 subgroup, which equates 8/7 with 7/6, creating a single neutral semifourth which serves as the generator. Similarly to dicot, semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's 25/24.

The DKW (2.3.7) optimum tuning states ~3/2 is tuned to 696.230 ¢; a chart of mistunings of simple intervals is below.

Semaphore (49/48)
Interval Just tuning Tunings*
Optimal tuning Deviation
9/8 203.910 192.460 -11.450
8/7 231.174 251.885 +20.711
7/6 266.871 251.885 -14.986
9/7 435.084 444.345 +9.261
21/16 470.781 444.345 -26.436
4/3 498.045 503.770 +5.725
3/2 701.955 696.230 -5.725
32/21 729.219 755.655 +26.436
14/9 764.916 755.655 -9.261
12/7 933.129 948.115 +14.986
7/4 968.826 948.115 -20.711
16/9 996.090 1007.540 +11.450

* In 2.3.7-targeted DKW tuning

Archy

Main article: Superpyth

Archy tempers out the comma 64/63 (S8) in the 2.3.7 subgroup, which equates 9/8 with 8/7, and 4/3 with 21/16. It serves as a septimal analogue of meantone, favoring fifths sharp of just rather than flat.

The DKW (2.3.7) optimum tuning states ~3/2 is tuned to 712.585 ¢ though most other optimizations tune this a few cents flatter; a chart of mistunings of simple intervals is below.

Archy (64/63)
Interval Just tuning Tunings*
Optimal tuning Deviation
9/8 203.910 225.171 +21.261
8/7 231.174 225.171 -6.003
7/6 266.871 262.244 -4.627
9/7 435.084 450.341 +15.257
21/16 470.781 487.415 +16.634
4/3 498.045 487.415 -10.630
3/2 701.955 712.585 +10.630
32/21 729.219 712.585 -16.634
14/9 764.916 749.659 -15.257
12/7 933.129 937.756 +4.627
7/4 968.826 974.829 +6.003
16/9 996.090 974.829 -21.261

* In 2.3.7-targeted DKW tuning

Slendric

Main article: Slendric

Slendric tempers out the comma 1029/1024 (S7/S8) in the 2.3.7 subgroup, which splits the perfect fifth into three intervals of 8/7. It is one of the most accurate temperaments of its simplicity. While semaphore and archy equate each middle interval of each triplet with either the major or the minor, gamelic makes it a true "neutral" intermediate between them.

The DKW (2.3.7) optimum tuning states ~3/2 is tuned to 699.126 ¢, and therefore ~8/7 to 233.042 ¢; a chart of mistunings of simple intervals is below.

Gamelic (1029/1024)
Interval Just tuning Tunings*
Optimal tuning Deviation
9/8 203.910 198.253 -5.657
8/7 231.174 233.042 +1.868
7/6 266.871 267.831 +0.960
9/7 435.084 431.295 -3.789
21/16 470.781 466.084 -4.697
4/3 498.045 500.874 +2.829
3/2 701.955 699.126 -2.829
32/21 729.219 733.916 +4.697
14/9 764.916 768.705 +3.789
12/7 933.129 932.169 -0.960
7/4 968.826 966.958 -1.868
16/9 996.090 1001.747 +5.657

* In 2.3.7-targeted DKW tuning

Exotemperaments

Trienstonian tempers out 28/27 in the 2.3.7 subgroup, mapping 7/4 to +3 fifths, and equating 9/8 with 7/6 and 9/7 with 4/3. This temperament equates the outer intervals in each cluster of three intervals without setting the middle to them. Due to its inaccuracy, it can reasonably be considered an exotemperament, which in a pentatonic system can be considered an analog to mavila.

The DKW (2.3.7) optimum tuning states ~3/2 is tuned to 723.494 ¢, and therefore ~8/7 to 229.517 ¢; a chart of mistunings of simple intervals is below.

Trienstonian (28/27)
Interval Just tuning Tunings*
Optimal tuning Deviation
9/8 203.910 246.989 +43.079
8/7 231.174 229.517 -1.868
7/6 266.871 246.989 -19.882
9/7 435.084 476.506 +41.422
21/16 470.781 493.977 +23.196
4/3 498.045 476.506 -21.539
3/2 701.955 723.494 +21.539
32/21 729.219 706.023 -23.196
14/9 764.916 723.494 -41.422
12/7 933.129 953.011 +19.882
7/4 968.826 970.483 +1.868
16/9 996.090 953.011 -43.079

* In 2.3.7-targeted DKW tuning

Music

Michael Harrison
  • From Ancient Worlds (for harmonic piano), 1992
  • Revelation: Music in Pure Intonation, 2007
La Monte Young
  • The Well-Tuned Piano, 1974

Notes

  1. ↑ Sometimes incorrectly named 2.3.7-limit or 2.3.7-prime limit; a prime limit is a subgroup spanned by all primes up to a given prime, and "limit" used alone usually implies prime limit.
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