EDT: Difference between revisions

== Rank two temperaments ==

If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [{{val|1 1 2}}, {{val|0 2 -1}}] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses [[MOSScales|MOS]] of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit.

At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus]] begins to predominate. This has a mapping [{{val|1 3 3}}, {{val|0 -5 -4}}] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even.

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 26EDT, 39EDT and 52EDT 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, 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 [{{val|13 19 23 0 2}}, {{val|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, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.''' For example making the fifth harmonic 5:3 a "major sixth" by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology.

There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2 cents sharp octave which is relevant to inharmonicity.