19edo: Difference between revisions

The most salient characteristic of 19et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for [[meantone]] temperament. It is also suitable for [[magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its ''twelfths''. For all of these there are more optimal tunings: the fifth of 19et is flatter than the usual for meantone, and [[31edo|31]] and [[43edo|43 equal temperament]] are more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[22edo|22]] and [[41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19edo is the same as magic. 19edo's 7-step supermajor third can be used for [[sensi]], whose generator is a very sharp major third, two of which make an approximate 5/3 minor sixth, though [[27edo]] and [[46edo]] are better sensi tunings for the 13-limit approximations of sensi.

However, for all of these 19et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build, and many 19et instruments have been built. 19et is in fact the second equal temperament, after 12et which is able to deal with [[5-limit]] music in a tolerable manner, and is the fifth (after 12) [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]]. It is less successful with [[7-limit]] (but still better than 12edo), as it eliminates the distinction between a septimal minor third ([[7/6]]), and a septimal whole tone ([[8/7]]). 19edo also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and negri are of particular note for being very simple 7-limit temperaments, with their [[mos scale]]s in 19edo offering a great abundance of septimal tetrads. The [[Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.

Being a zeta integral tuning, the 13-limit is represented relatively well, though only the 2.3.5.7.13 subgroup is represented [[consistent]]ly. Practically 19edo can be used ''adaptively'' on instruments which allow you to bend notes up: by different amounts, the 3rd, 5th, 7th and 13th harmonics are all tuned flat. The same cannot be said of 12edo, in which the 5th and 7th are - not only farther than they are in 19, but fairly sharp already. 19edo's [[negri]], [[sensi]] and [[semaphore]] scales have many 13-limit chords. (You can think of the sensi[8] [[3L 5s]] mos scale as 19edo's answer to the diminished scale. Both are made of two diminished seventh chords, but sensi[8] gives you additional ratios of 7 and 13.)

Another option would be to use stretched octaves; the [[The Riemann Zeta Function and Tuning|zeta function]]-optimal tuning has an octave of roughly 1203 cents. Stringed instruments, in particular the piano, are frequently tuned with stretched octaves anyway due to the inharmonicity inherent in strings, which makes 19edo a promising option for them. Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-just. For instance, if we are using [[93ed30]] (a variant of 19edo in which 30:1 is just), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well.