2.3.7 subgroup: Difference between revisions
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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'''<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 | 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 | 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]]. | ||
== 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 {{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]]. | |||
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. | |||
== Properties == | == Properties == | ||
The simpler ratios fall into 3 categories: | 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. | 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 === | === Scales === | ||
==== Minor ==== | ==== Minor ==== | ||
* zo pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1 | * zo pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1 | ||
* zo | * 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 | * 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 | * 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 ==== | ==== Major ==== | ||
* ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1 | * 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 | * ru: 1/1 9/8 9/7 4/3 3/2 12/7 27/14 2/1 | ||
==== Misc ==== | ==== 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 }} | 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 === | ||
{{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 | 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 ==== | ==== 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. | 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" | {| class="wikitable center-1 center-2 center-3 center-4" | ||
| Line 84: | Line 88: | ||
==== Archy ==== | ==== Archy ==== | ||
{{Main| Superpyth }} | |||
The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712. | 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" | {| class="wikitable center-1 center-2 center-3 center-4" | ||
| Line 122: | Line 128: | ||
==== Slendric ==== | ==== Slendric ==== | ||
{{Main| Slendric }} | |||
The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 699. | 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" | {| class="wikitable center-1 center-2 center-3 center-4" | ||
| Line 159: | Line 167: | ||
<nowiki>*</nowiki> In 2.3.7-targeted DKW tuning | <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. | |||
The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 723. | |||
{| class="wikitable center-1 center-2 center-3 center-4" | {| class="wikitable center-1 center-2 center-3 center-4" | ||
| Line 203: | 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]] | ; [[La Monte Young]] | ||
* The Well-Tuned Piano, 1974 | * The Well-Tuned Piano, 1974 | ||
| Line 209: | Line 216: | ||
<references group="note"/> | <references group="note"/> | ||
[[Category: | [[Category:Just intonation subgroups|#]] | ||
[[Category: | [[Category:Rank-3 temperaments|#]] | ||
[[Category: | [[Category:7-limit|#]] | ||
[[Category:Lists of scales]] | [[Category:Lists of scales|#]] | ||