19edo: Difference between revisions

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Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning, which he defined as dividing the just major second into three approximately equal parts. Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.

In 1577 music theorist Francisco de Salinas proposed [[1/3-comma meantone]], in which the fifth is of size 694.786 cents; the fifth of 19edo is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which ~~fails to close by~~ less than a cent, so that his suggestion is effectively 19edo.

In 1577 music theorist Francisco de Salinas proposed [[1/3-comma meantone]], in which the fifth is of size 694.786 cents; the fifth of 19edo is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which comes within less than one cent of closing exactly, so that his suggestion is effectively 19edo.

In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([http://www.tonalsoft.com/sonic-arts/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).

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

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 ~~a more accurate approximation is~~ [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[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.

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~~. (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 ~~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.

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 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 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.

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.

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

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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.

~~The~~ narrow whole tones and wide diatonic semitones of 19edo ~~give the~~ diatonic scale a somewhat ~~duller quality~~, but ~~has the opposite effect on~~ the pentatonic scale~~, which~~ becomes much more expressive owing to the 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.

Due to the narrow whole tones and wide diatonic semitones, many consider 19edo's diatonic scale to sound somewhat dull compared to 12edo, but the pentatonic scale becomes 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.

=== Prime harmonics ===

Line 571:

| 137.14

|

| 19-comma

|-

| 5

Line 712:

| Zoquadyo

| Ragisma

|-

| 11

| [[45/44]]

| {{monzo| -2 2 1 0 -1 }}

| 38.91

| Luyo

| Undecimal 1/5-tone

|-

| 11

| [[56/55]]

| {{monzo| 3 0 -1 1 -1 }}

| 31.19

| Luzogu

| Undecimal tritonic comma

|-

| 11

Line 747:

Line 761:

| Lururuyo

| Swetisma

|-

| 13

| [[39/38]]

| {{monzo| -1 1 0 0 0 1 0 -1 }}

| 44.97

| Nutho

| Undevicesimal 2/9-tone

|-

| 13

@@ Line 12: / Line 12: @@
 Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning, which he defined as dividing the just major second into three approximately equal parts.  Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.
-In 1577 music theorist Francisco de Salinas proposed [[1/3-comma meantone]], in which the fifth is of size 694.786 cents; the fifth of 19edo is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which fails to close by less than a cent, so that his suggestion is effectively 19edo.
+In 1577 music theorist Francisco de Salinas proposed [[1/3-comma meantone]], in which the fifth is of size 694.786 cents; the fifth of 19edo is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which comes within less than one cent of closing exactly, so that his suggestion is effectively 19edo.
 In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([http://www.tonalsoft.com/sonic-arts/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).
 === As an approximation of other temperaments ===
-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 a more accurate approximation is [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[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.
+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. (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 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.
+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 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 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.
+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.
 === As a means of extending harmony ===
@@ Line 32: / Line 32: @@
 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.
-The narrow whole tones and wide diatonic semitones of 19edo give the diatonic scale a somewhat duller quality, but has the opposite effect on the pentatonic scale, which becomes much more expressive owing to the 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.
+Due to the narrow whole tones and wide diatonic semitones, many consider 19edo's diatonic scale to sound somewhat dull compared to 12edo, but the pentatonic scale becomes 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.
 === Prime harmonics ===
@@ Line 571: / Line 571: @@
 | 137.14
 |
-|
+| 19-comma
 |-
 | 5
@@ Line 712: / Line 712: @@
 | Zoquadyo
 | Ragisma
+|-
+| 11
+| [[45/44]]
+| {{monzo| -2 2 1 0 -1 }}
+| 38.91
+| Luyo
+| Undecimal 1/5-tone
+|-
+| 11
+| [[56/55]]
+| {{monzo| 3 0 -1 1 -1 }}
+| 31.19
+| Luzogu
+| Undecimal tritonic comma
 |-
 | 11
@@ Line 747: / Line 761: @@
 | Lururuyo
 | Swetisma
+|-
+| 13
+| [[39/38]]
+| {{monzo| -1 1 0 0 0 1 0 -1 }}
+| 44.97
+| Nutho
+| Undevicesimal 2/9-tone
 |-
 | 13