EDT: Difference between revisions
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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. | 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. | ||
[[User:CompactStar|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. The best temperament for this is [[Electra]] temperament, using ~[[15/11]] as a generator | For example, [[User:CompactStar|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. The best temperament for this is [[Electra]] temperament, using ~[[15/11]] as a generator. Electra possesses MOS of the forms [[4L 3s (tritave-equivalent)|4L 3s<3/1>]] (heptatonic) and [[7L 4s (tritave-equivalent)|4L 7s<3/1>]] (hendecatonic), and larger MOS of size 18 and 25. | ||
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. s | |||
'''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. | '''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. | ||