19edo: Difference between revisions

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However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build, and many 19edo 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) [[zeta integral edo]]. It is less successful with [[7-limit]] (but still better than 12et), 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.)

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. In 12edo, this is not possible, since the 5 and 7 are not only much farther from just 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 #Optimal octave stretch|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.

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 However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build, and many 19edo 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) [[zeta integral edo]]. It is less successful with [[7-limit]] (but still better than 12et), 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.)
+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. In 12edo, this is not possible, since the 5 and 7 are not only much farther from just 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 #Optimal octave stretch|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.