2.3.7 subgroup
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
- Diasem/Tas[9] (left-handed): 1/1 9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1
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
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 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.
| 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
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.
| 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
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.
| 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.
| 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
- From Ancient Worlds (for harmonic piano), 1992
- Revelation: Music in Pure Intonation, 2007
- The Well-Tuned Piano, 1974
Notes
- ↑ 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.