19edo: Difference between revisions

Line 20:

=== As an approximation of other temperaments ===

The most salient characteristic of 19-et 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 family|meantone]] temperament. It is also a suitable for [[Regular Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its ''twelfths.'' For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.

The most salient characteristic of 19-et 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 family|meantone]] temperament. It is also a suitable for [[Regular Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its ''twelfths.'' For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic. Sensi has a very sharp major third as a generator and 19edo's 7-step supermajor third can be used, though [[27edo]] and [[46edo]] are better sensi tunings.

However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with [[Harmonic Limit|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 12-et), as it eliminates the distinction between a septimal minor third ([[7/6]]), and a septimal whole tone ([[8/7]]). 19-EDO 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 scales in 19-EDO 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 19-edo 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]] and [[sensi]] scales have many 13-limit chords~~, though [[27edo]] and [[46edo]] are better tunings for sensi~~. (You can think of sensi[8] 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 19-edo 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]] and [[sensi]] scales have many 13-limit chords. (You can think of sensi[8] 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 a stretched octave; 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're 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.

@@ Line 20: / Line 20: @@
 === As an approximation of other temperaments ===
-The most salient characteristic of 19-et 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 family|meantone]] temperament. It is also a suitable for [[Regular Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its ''twelfths.'' For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.
+The most salient characteristic of 19-et 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 family|meantone]] temperament. It is also a suitable for [[Regular Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its ''twelfths.'' For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic. Sensi has a very sharp major third as a generator and 19edo's 7-step supermajor third can be used, though [[27edo]] and [[46edo]] are better sensi tunings.
 However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with [[Harmonic Limit|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 12-et), as it eliminates the distinction between a septimal minor third ([[7/6]]), and a septimal whole tone ([[8/7]]). 19-EDO 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 scales in 19-EDO 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 19-edo 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]] and [[sensi]] scales have many 13-limit chords, though [[27edo]] and [[46edo]] are better tunings for sensi. (You can think of sensi[8] 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 19-edo 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]] and [[sensi]] scales have many 13-limit chords. (You can think of sensi[8] 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 a stretched octave; 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're 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.