2.5.7 subgroup
The 2.5.7 subgroup, or the no-threes 7-limit (yaza nowa in color notation) is a just intonation subgroup consisting of rational intervals where 2, 5, 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, 5, 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 5/4, 7/4, 8/7, 7/5, 28/25, 35/32, and so on.
The 2.5.7 subgroup is a retraction of the 7-limit, obtained by removing prime 3. Its simplest expansion is the 2.5.7.11 subgroup, which adds prime 11.
A notable subset of the 2.5.7 subgroup is the 1.5.7 tonality diamond, comprised of all intervals in which 1, 5 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 1.5.7 tonality diamond within the octave is 1/1, 8/7, 5/4, 7/5, 10/7, 8/5, 7/4, and 2/1.
Another such subset is the 1.5.7.25.35 tonality diamond, which adds the following intervals to the previous list: 25/16, 25/14, 35/32, 64/35, 28/25, and 32/25.
When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 5 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 modestly well-represented by 6edo for its size, enough so that many of its simple intervals tend to cluster around the notes of 6edo. Therefore, one way to approach the 2.5.7 subgroup is to think of a hexatonic framework for composition as natural to it, rather than the diatonic framework associated with the 5-limit.
Scales
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Regular temperaments
Rank-1 temperaments (edos)
A list of edos with progressively better* tunings for the 2.5.7 subgroup: 6, 15, 16, 21, 25, 27, 31, 68, 103, 134, 140, 171, 239, 379, 410, 550, 618, 789, 5902, 6691, 7480, 8269, 9058, 9847, and so on.
Another list of edos is those with progressively smaller relative error for the 2.5.7 subgroup: 1, 2, 4, 6, 31, 379, 789, 103169, and so on.
* in absolute DKW distance
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Commas and rank-2 temperaments
In the below tables, the generator of the temperament is highlighted in bold. Intervals in the tables reflect the 1.5.7.25.35 tonality diamond. It is structurally notable that they come in five clusters, each centered around one note of 6edo - 35/32 ~ 28/25 ~ 8/7 close to a wholetone, 5/4 and 32/25 close to a ditone, 7/5 and 10/7 close to a tritone, and so on - with intervals within each cluster separated by the commas 50/49 and 128/125.
Jubilic
Jubilic temperament tempers out the comma 50/49 = S5/S7 in the 2.5.7 subgroup, which makes 7/5 ~ 10/7 a single half-octave tritone which serves as the period. Similarly to dicot, jubilic can be regarded as an exotemperament that elides fundamental distinctions within the subgroup, though the comma involved is half the size of dicot's 25/24.
The DKW (2.5.7) optimum tuning states ~5/4 is tuned to 385.002c and therefore that ~8/7 is tuned to 214.998c; a chart of mistunings of simple intervals is below.
| Interval | Just tuning | Tunings* | |
|---|---|---|---|
| Optimal tuning | Deviation | ||
| 35/32 | 155.140 | 170.005 | +14.865 |
| 28/25 | 196.198 | 214.998 | +18.799 |
| 8/7 | 231.174 | 214.998 | -16.176 |
| 5/4 | 386.314 | 385.002 | -1.311 |
| 32/25 | 427.373 | 429.995 | +2.623 |
| 7/5 | 582.512 | 600.000 | +17.488 |
| 10/7 | 617.488 | 600.000 | -17.488 |
| 25/16 | 772.627 | 770.005 | -2.623 |
| 8/5 | 813.686 | 814.998 | +1.311 |
| 7/4 | 968.826 | 985.002 | +16.176 |
| 25/14 | 1003.802 | 985.002 | -18.799 |
| 64/35 | 1044.860 | 1029.995 | -14.865 |
Augmented
Augmentsept temperament tempers out the comma 128/125 = S4/S5. Strictly speaking, augmented is a 2.3.5 temperament, with which the 2.5.7 temperament shares only the same comma, hence this version should be known under a different name. However, both temperaments make the interval 5/4 an exact third of the octave, which serves as the period. The generator in this case is then naturally ~8/7.
The DKW (2.5.7) optimum tuning states ~8/7 is tuned to 218.297c; a chart of mistunings of simple intervals is below.
| Interval | Just tuning | Tunings* | |
|---|---|---|---|
| Optimal tuning | Deviation | ||
| 35/32 | 155.140 | 181.703 | -26.563 |
| 28/25 | 196.198 | 181.703 | -14.495 |
| 8/7 | 231.174 | 218.297 | -12.877 |
| 5/4 | 386.314 | 400.000 | +13.686 |
| 32/25 | 427.373 | 400.000 | -27.373 |
| 7/5 | 582.512 | 581.703 | -0.809 |
| 10/7 | 617.488 | 618.297 | +0.809 |
| 25/16 | 772.627 | 800.000 | +27.373 |
| 8/5 | 813.686 | 800.000 | -13.686 |
| 7/4 | 968.826 | 981.703 | +12.877 |
| 25/14 | 1003.802 | 1018.297 | +14.495 |
| 64/35 | 1044.860 | 1018.297 | -26.563 |
Bapbo
Bapbo temperament tempers out the comma 256/245 = S6 * S8 = S4/S7, equating 35/32 with 8/7, and making 5/4 the square of 8/7, which hence serves as the generator. This comma is over 70 cents, and pathologically equates the outer edges of the clusters around 1\6 and 5\6 to each other without also equating the middle interval with them (if this is done, the tuning reduces to 6edo). Therefore, bapbo can safely be considered an exotemperament - perhaps an analogue to mavila.
The DKW (2.5.7) optimum tuning states ~5/4 is tuned to 410.773c and therefore ~8/7 is tuned to 205.386c; a chart of mistunings of simple intervals is below.
| Interval | Just tuning | Tunings* | |
|---|---|---|---|
| Optimal tuning | Deviation | ||
| 35/32 | 155.140 | 205.386 | +50.247 |
| 28/25 | 196.198 | 173.068 | -23.130 |
| 8/7 | 231.174 | 205.386 | -25.788 |
| 5/4 | 386.314 | 410.773 | +24.459 |
| 32/25 | 427.373 | 378.454 | -48.918 |
| 7/5 | 582.512 | 583.841 | +1.329 |
| 10/7 | 617.488 | 616.159 | -1.329 |
| 25/16 | 772.627 | 821.546 | +48.918 |
| 8/5 | 813.686 | 789.227 | -24.459 |
| 7/4 | 968.826 | 994.614 | +25.788 |
| 25/14 | 1003.802 | 1026.932 | +23.130 |
| 64/35 | 1044.860 | 994.614 | -50.247 |
Didacus
Didacus temperament, the 2.5.7 restriction of hemimean and septimal meantone, tempers out the comma 3136/3125 in the 2.5.7 subgroup, which splits the major third (5/4) into two intervals of 28/25. It is one of the most accurate temperaments of its simplicity. While augmented and jubilic merge the middle interval of the triplets surrounding 1\6 and 5\6 with one of the two nearby intervals, didacus makes it an exact mean between the two.
The DKW (2.5.7) optimum tuning states ~5/4 is tuned to 388.122c, and therefore ~28/25 to 194.061c; a chart of mistunings of simple intervals is below.
| Interval | Just tuning | Tunings* | |
|---|---|---|---|
| Optimal tuning | Deviation | ||
| 35/32 | 155.140 | 158.427 | +3.288 |
| 28/25 | 196.198 | 194.061 | -2.137 |
| 8/7 | 231.174 | 229.695 | -1.479 |
| 5/4 | 386.314 | 388.122 | +1.808 |
| 32/25 | 427.373 | 423.756 | -3.617 |
| 7/5 | 582.512 | 582.183 | -0.329 |
| 10/7 | 617.488 | 617.817 | +0.329 |
| 25/16 | 772.627 | 776.244 | +3.617 |
| 8/5 | 813.686 | 811.878 | -1.808 |
| 7/4 | 968.826 | 970.305 | +1.479 |
| 25/14 | 1003.802 | 1005.939 | +2.137 |
| 64/35 | 1044.860 | 1041.573 | -3.288 |
Music
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