2.3.7 subgroup
The 2.3.7 subgroup[1] (za in color notation) 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 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 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, 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.
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.
Properties
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
- Zo minor pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1
- Ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1
- Zo in: 1/1 9/8 7/6 3/2 14/9 2/1
- 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)
- Za 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: 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277 and so on.
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.
Commas and 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 they come in four triplets, each centered around one note of 5edo, consisting of a "minor", "middle", and "major" interval of each set.
Semaphore
Semaphore temperament 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 696.230c; 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 temperament 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 712.585c (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
Gamelic
Gamelic temperament, better known as 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 699.126c, and therefore ~8/7 to 233.042c; 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
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.