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 (which is the 7-odd-limit (1.3.5.7) with intervals of 3 removed) 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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Todo: expand |
Regular 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.
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
Birds
As 31edo is very strong in the 2.5.7 subgroup so that it is a weakly consistent circle of 5/4's and 7/4's (and thus 8/5's and 8/7's) and a strongly consistent circle of 35/32's (and thus 64/35's), it makes sense for those interested in high-complexity fractional-octave temperaments to consider 31st-octave temperaments (temperaments with a 1\31 period) that preserve this representation of 2.5.7, which can be seen as combining the simplificatory logics of didacus, rainy and quince/mercy, which is the 2.5.7-subgroup restriction of miracle. See 31st-octave_temperaments#Birds for details on the canonical extension of it to the full 19-limit that utilises 31edo's good approximation of the interval 11/9 and of the 13:17:19 chord.
Commas & rank-2 temperaments
Didacus (3136/3125)
Didacus temperament, which is the 2.5.7 restriction of orion, 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, and as (7/5)/(5/4) = 28/25 this implies 7/5 is found at (28/25)3 and thus 7/4 is found at (28/25)5. It has the unique property among 2.5.7 temperaments being simultaneously extremely abundant in simple 2.5.7 harmony and surprisingly accurate. 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 |
* In 2.5.7-targeted DKW tuning
Rainy (2100875/2097152)
Rainy is related to a number of high-limit rank 3 temperaments such as valentine, dwynwyn and tertiaseptal in rank 2, and eros and sophia in rank 3, though is good as a (rank 2) pure 2.5.7 temperament also. Its generator is ~256/245 sharpened by approximately 1 ¢ which acts as the square root of ~35/32, the cube root of ~8/7 and the fifth root of ~5/4. Four generators reaches an ambiguous interval whose 2.5.7 interpretation is rather complex, being 2048/1715~1225/1024, which is the starting point for extensions. The most simple extension, valentine, interprets this interval as a flat ~6/5 by tempering (6/5)/(1225/1024) = 6144/6125, and the generator as a sharp ~25/24, so that 3/2 is found at 9 generators, that is, at (~8/7)3, so also tempering 1029/1024 = S7/S8 = (6/5)/(2048/1715). Meanwhile, a much more accurate and complex mapping is to find 4/3 at 22 generators octave reduced, which is the strategy taken by tertiaseptal, notable as the very high-limit 140 & 311 temperament. These two mappings of 3 merge in 31edo (which serves as a trivial tuning of tertiaseptal, as another tuning of tertiaseptal is 311edo - 140edo = 171edo, and 171 - 140 = 31).
2.5.7[6 & 60] = 2.5.7-subgroup restriction of Waage (244140625/240945152)
This temperament sharpens ~28/25 by 3.8 ¢ to make it equal to 1\6 so that 6edo is made a strongly consistent circle of 28/25's, so it is one of the 6th-octave temperaments. It relates the close relation of the 2.5.7 subgroup to hexatonic structure in an intriguing way by contrasting it with equalized hexatonic structure, chosen to represent ~28/25.
Cloudy (16807/16384)
Cloudy interprets 1\5 = 240 ¢ as a 9 ¢ sharp ~8/7 so that 5edo is made to be a strongly consistent circle of 8/7's, by tempering 2/(8/7)5.
Augmented (128/125)
It is debated whether augmented counts as a normal temperament or an exotemperament, though it is clear it can produce the JI-like effects of DR in good tunings of it like 27edo. 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 |
* In 2.5.7-targeted DKW tuning
Jubilic (50/49)
It is debated whether jubilic counts as a normal temperament or an exotemperament with respect to 7/5 and 10/7 specifically (which it equates), though it is clear it can produce the JI-like effects of DR in good tunings of it that focus on optimizing the convincingness of the harmonic seventh chord ~4:5:6:7 and more generally on other intervals of the 7-odd-limit than 7/5 and 10/7. As 7/5 needs to be sharpened (and 10/7 flattened), an optimized tuning flattens 5/4 and sharpens 7/4 so that it makes sense to temper 225/224 so that (5/4)2 ~ 14/9, implying tempering 64/63, which is especially natural as it makes use of the fact that 7/4 is sharp in an optimized tuning to equate it with 16/9 to achieve a half-octave-period analogue of archy (2.3.7 superpyth). This leads to pajara.
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, showcasing that the DKW optimum tuning can give strange results for high-damage temperaments, so check out jubilic and pajara for more reasonable tunings; a chart of mistunings of simple intervals in the DKW optimum tuning is given below to examine how DKW attempts to balance error between higher-complexity intervals of 2.5.7 not usually considered:
| 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 |
* In 2.5.7-targeted DKW tuning
Dieses & rank-2 exotemperaments
Some important dieses in the 2.5.7 subgroup are:
- 2/(5/4)3, the enharmonic diesis (or just "diesis"), leading to augmented if tempered.
- 2/(8/7)5, the cloudy comma, leading to cloudy if tempered, which can be likened in structural role to the enharmonic diesis but for prime 7 instead of prime 5. Equating it with the enharmonic diesis to form a general-purpose diesis results in rainy temperament, as their difference is the rainy comma.
- (10/7)/(7/5), the septimal tritonic diesis, leading to jubilic if tempered.
- (8/7)2/(5/4), which can be likened in structural role to the septimal tritonic diesis, leading to Bapbo if tempered, considered below:
Bapbo (256/245)
The bapbo exotemperament tempers out 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 |
* In 2.5.7-targeted DKW tuning
Music
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Todo: expand |