19edo: Difference between revisions

Line 20:

19edo's most salient characteristic is that, having an almost just minor third and perfect fifths and major thirds about seven cents flat, it serves as a good tuning for [[meantone]]. 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 19edo is flatter than the usual for meantone, and [[31edo]] is more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[41edo]] more closely matches it. It does make for a good tuning for muggles, but in 19edo it 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 major sixth, though [[46edo]] is a better sensi tuning.

However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19edo is in fact the second edo, after [[12edo]] which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy (unless you count [[15edo]]), and is the fifth [[zeta integral edo]], after 12edo. It is less successful in the [[7-limit]] (but still better than 12edo), as it conflates the septimal subminor third ([[7/6]]) with the 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.

However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19edo is in fact the second edo, after [[12edo]] which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy (unless you count [[15edo]]), and is the fifth [[zeta integral edo]], after 12edo. It is less successful in the [[7-limit]] (but still better than 12edo), as it conflates the septimal subminor third ([[7/6]]) with the 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 no-11's 13-limit is represented relatively well and consistently. 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. This is in contrast to 12edo, where this is not possible since the 5th and 7th harmonics are not only much farther from just than they are in 19edo, 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.)

=== Adaptive tuning and octave stretch ===

Being a zeta integral tuning, the no-11's 13-limit is represented relatively well and consistently. 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. This is in contrast to 12edo, where this is not possible since the 5th and 7th harmonics are not only much farther from just than they are in 19edo, but fairly sharp already. 19edo's negri, sensi and godzilla 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 [[octave stretching]]; the closest [[the Riemann zeta function and tuning #Optimal octave stretch|local zeta peak]] to 19 occurs at 18.9481, which makes the octaves 1203.29{{c}}, and a step size of between 63.2–63.4{{c}} would be preferable in theory. Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos with split sharps. 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 example, if we are using [[49ed6]] or [[30ed3]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57{{c}}, respectively), 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. The most extreme of these options would be [[11edf]], which has octaves stretched by 12.47{{c}}.

=== As a means of extending harmony ===

Because 19edo's 5-limit chords are more blended and consonant than those of 12edo, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.

Because 19edo's 5-limit chords are more blended and consonant than those of 12edo, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non-diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.

19edo's diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3{{c}} off [[23/16]].

In addition, [[Joseph Yasser]] talks about the idea of a 12 tone supra diatonic scale where the 7 tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.

In addition, [[Joseph Yasser]] talks about the idea of a 12-tone supra-diatonic scale where the 7-tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra-diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.

19edo also closely approximates most of the intervals of [[Bozuji tuning]] (a 21st century tuning based on Gioseffo Zarlino's approach to just intonation). with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.

19edo also closely approximates most of the intervals of [[Bozuji tuning]], a 21st century tuning based on Gioseffo Zarlino's approach to just intonation. with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.

Due to the narrow whole tones and wide diatonic semitones, 19edo's diatonic scale tends to sound somewhat dull compared to 12edo, but the pentatonic scale is said by many to sound much more expressive owing to the significantly larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.

@@ Line 20: / Line 20: @@
 edo's most salient characteristic is that, having an almost just minor third and perfect fifths and major thirds about seven cents flat, it serves as a good tuning for [[meantone]]. 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 19edo is flatter than the usual for meantone, and [[31edo]] is more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[41edo]] more closely matches it. It does make for a good tuning for muggles, but in 19edo it 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 major sixth, though [[46edo]] is a better sensi tuning.
-However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19edo is in fact the second edo, after [[12edo]] which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy (unless you count [[15edo]]), and is the fifth [[zeta integral edo]], after 12edo. It is less successful in the [[7-limit]] (but still better than 12edo), as it conflates the septimal subminor third ([[7/6]]) with the 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.
+However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19edo is in fact the second edo, after [[12edo]] which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy (unless you count [[15edo]]), and is the fifth [[zeta integral edo]], after 12edo. It is less successful in the [[7-limit]] (but still better than 12edo), as it conflates the septimal subminor third ([[7/6]]) with the 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 no-11's 13-limit is represented relatively well and consistently. 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. This is in contrast to 12edo, where this is not possible since the 5th and 7th harmonics are not only much farther from just than they are in 19edo, but fairly sharp already. 19edo's [[negri]], [[sensi]] and [[semaphore]] scales have many 13-limit chords. (You can think of the sensi[8] [[3L&nbsp;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.)
+=== Adaptive tuning and octave stretch ===
+Being a zeta integral tuning, the no-11's 13-limit is represented relatively well and consistently. 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. This is in contrast to 12edo, where this is not possible since the 5th and 7th harmonics are not only much farther from just than they are in 19edo, but fairly sharp already. 19edo's negri, sensi and godzilla scales have many 13-limit chords. (You can think of the Sensi[8] [[3L&nbsp;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 [[octave stretching]]; the closest [[the Riemann zeta function and tuning #Optimal octave stretch|local zeta peak]] to 19 occurs at 18.9481, which makes the octaves 1203.29{{c}}, and a step size of between 63.2–63.4{{c}} would be preferable in theory. Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos with split sharps. 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 example, if we are using [[49ed6]] or [[30ed3]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57{{c}}, respectively), 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. The most extreme of these options would be [[11edf]], which has octaves stretched by 12.47{{c}}.
 === As a means of extending harmony ===
-Because 19edo's 5-limit chords are more blended and consonant than those of 12edo, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.
+Because 19edo's 5-limit chords are more blended and consonant than those of 12edo, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non-diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.
 edo's diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3{{c}} off [[23/16]].
-In addition, [[Joseph Yasser]] talks about the idea of a 12 tone supra diatonic scale where the 7 tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.
+In addition, [[Joseph Yasser]] talks about the idea of a 12-tone supra-diatonic scale where the 7-tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra-diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.
-edo also closely approximates most of the intervals of [[Bozuji tuning]] (a 21st century tuning based on Gioseffo Zarlino's approach to just intonation). with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.
+edo also closely approximates most of the intervals of [[Bozuji tuning]], a 21st century tuning based on Gioseffo Zarlino's approach to just intonation. with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.
 Due to the narrow whole tones and wide diatonic semitones, 19edo's diatonic scale tends to sound somewhat dull compared to 12edo, but the pentatonic scale is said by many to sound much more expressive owing to the significantly larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.