3.5.7 subgroup

The 3.5.7 subgroup is a non-octave just intonation subgroup consisting of rational intervals where 3, 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 3, 5, and 7. This is an infinite set. Some examples of intervals in this subgroup are 5/3, 7/3, 7/5, 25/21, 35/27, and so on.

The 3.5.7 subgroup is a retraction of the 7-limit, obtained by removing prime 2. Its simplest expansion is the 3.5.7.11 subgroup, which adds prime 11.

The approaches to composition in this subgroup can be categorized as follows:

1. By not having an interval of equivalence, by composing in the 7-limit but without octavation;
2. By treating the 3rd harmonic as the interval of equivalence.

Since octavation is usually trivial to implement, this article focuses on the second approach.

Introduction to tritave equivalence

See EDT #Introduction to tritave equivalence.

Temperaments

Rank-2 temperaments

Todo: cleanup, improve readability Rewrite for clarity

If factors of two are eliminated, the search for consonant intervals begins with the odd harmonic series, 1:3:5:7:9:.... We can take the second tritave of the series, 3:5:7:9, and find within it the two isoharmonic triads 3:5:7 and 5:7:9; the analogy here is with the third octave of the full harmonic series, 4:5:6:7:8, and the isoharmonic triad 4:5:6, the foundation of triadic harmony in 5-limit theory. Hence, 3:5:7 or 5:7:9 can be viewed as the fundamental consonant triad of no-twos music, and if we then apply the 5-limit analogy one more time, these triads are bounded by the intervals 7/3 or 9/5 respectively, either of them filling the role of the "fifth" in diatonicism.

The standard Bohlen–Pierce theory takes 3:5:7 to be the fundamental triad, and therefore naturally goes together with scales generated by 7/3, or equivalently 9/7 (the latter being convention), against the tritave. 7/3 generates pentatonic (4L 1s⟨3/1⟩) and enneatonic (4L 5s⟨3/1⟩) MOS scales, and therefore the enneatonic, known as the "Lambda" scale, can be seen as the analog of the diatonic scale. As generators of the Lambda scale run from 7\9 to 3\4, 13edt is the smallest equal temperament supporting it, and can be seen as an equivalent of 12edo. However, 13edt's accuracy in the 3.5.7 subgroup is much better than 12edo's in the 5-limit, more comparable to that of 31edo. Therefore, higher multiples of 13edt remain excellent 3.5.7 subgroup tunings as well, and can be used to introduce higher harmonics (39edt is especially notable in this regard, with a good representation of both the 11th and 13th harmonics).

The linear temperament generated by 7/3 that is satisfied in 13edt's 3.5.7 subgroup representation is Bohlen–Pierce–Stearns, which tempers out the comma 245/243 and thereby equates the interval 5/3 to two generators down (81/49 considering tritave-reduction)—therefore flattening 7/3 by a fraction of this comma. It is also the 4 & 9 temperament in the 3.5.7 subgroup, and for these reasons serves a function very analogous to that of meantone in the 5-limit.

If we instead take 9/5, or more simply 5/3, as a generator, the temperament supported by 13edt is Arcturus, which equates 7/3, two tritaves up (i.e. 21/1) to six steps of 5/3. Naively, 5/3 as generator would be the most natural application of the Pythagorean principle of using the next higher prime harmonic (5) as a generator against the tritave. However, a larger MOS scale is needed to get full use out of the 7th harmonic, and due to the proximity of 5/3 to half the tritave, most simple MOS scales of Arcturus are quite hard. It is advisable to use (2L 9s⟨3/1⟩) or (2L 11s⟨3/1⟩) scales—and therefore, higher EDTs such as 28edt or 43edt.

Moving through the scales of 13edt, we find the temperament Sirius, defined so that two generator-steps represent 7/5 and three represent 5/3; therefore the comma 3125/3087 is tempered out and the generator represents 25/21. The smallest complete proper MOS of Sirius is the hard 6L 1s⟨3/1⟩, though some say that it is disadvantageous to have a non-octave scale with a single step of distinct size from the others, because it creates a strong sense of a second equal division of a y (in this case, 25/9) less than 3/1 and therefore competes with the relatively fragile tritave equivalence. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it. Otherwise, the 13-note MOS of Sirius, 6L 7s⟨3/1⟩, is usable although this is very soft and close to 13edt in good tunings.

At higher accuracies, the rank two 3.5.7 temperament tempering out 16875/16807 called Canopus begins to predominate. This has a mapping [⟨1 3 3], ⟨0 -5 -4]] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 ¢; the temperament's property is that four 7/5s reach 27/7. This can be extended, notably, to include the fractional harmonic 11/4, which is quite close to three 7/5s. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even.

The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the 5L 3s⟨3/1⟩ unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an EDO. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen–Pierce temperament is its index-2 subtemperament.

Due to its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas EDOs supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the Superpythagorean or Parapythagorean diatonic scale.

Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDT supporting the BP nonatonic scale—13EDT, the traditional tempered BP scale—is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26, 39, and 52 as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored.

For example, CompactStar suggested the alternative fundamental chord 11:13:15 to avoid the highly-dissonant 7/5 tritone present in the simpler 3:5:7 chord, with the best temperament for this being Electra temperament. 15EDT very well approximates the 5th and 13th harmonics, and 12EDT, the 13th and 17th. 39EDT makes for a fine 3.5.7.11.13 system, tempering out 245/243, 275/273, 847/845 and 1331/1343, and so supporting among other things the [⟨13 19 23 0 2], ⟨0 0 0 1 1]] temperament supported by the whole suite of 13nEDTs: 13, 26, 39, 52, 65, 78 etc.

One should bear in mind that, in the world of tritave equivalence, ratios of 3 in the denominator are fungible instead of those of 2. For example, while the octave-reduced fifth harmonic is 5:4, the tritave-reduced fifth harmonic would be 5:3 instead, which would be a "major sixth" by conventional pitch class terminology.