EDT: Difference between revisions
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If factors of two are eliminated, the | 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-equivalent)|4L 1s]]) and enneatonic ([[4L 5s (3/1-equivalent)|4L 5s]]) [[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 [[9edt|7\9]] to [[4edt|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 [[11/9|11th]] and [[13/9|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 tuning|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-equivalent)|2L 9s]]) or ([[2L 11s (3/1-equivalent)|2L 11s]]) scales - and therefore, higher EDTs such as [[28edt]] or [[41edt]]. | |||
The named but not necessarily no twos rank two temperament which 13EDT "supports" is [[Sirius]], which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13EDT supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it. | The named but not necessarily no twos rank two temperament which 13EDT "supports" is [[Sirius]], which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13EDT supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it. | ||
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. | |||
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 (tritave-equivalent)|5L 3s]] 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. | 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 (tritave-equivalent)|5L 3s]] 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. | ||