19edo: Difference between revisions

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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. Finally, 19edo can be used as a tuning for sensi, though [[46edo]] provides 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]], which has a ~~18c~~ sharp fifth). 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]], muggles, and [[triton]]/[[liese]]. 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]], which has a 18-cent-sharp fifth). 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]], muggles, and [[triton]]/[[liese]]. 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.

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

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=== Adaptive tuning ~~and octave stretch~~ ===

=== Adaptive tuning ===

The no-11's 13-limit is represented relatively well and consistently. 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.) Its diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3{{c}} off [[23/16]].

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.

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}}~~.

Another option would be to use [[octave stretching]], which has similar benefits to adaptive use, but it also works for fixed-pitch Instruments. For more on that see the section: [[19edo#Octave stretch]].

=== Subsets and supersets ===

19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]].

19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]]. As such, it does not contain any nontrivial subset edos, though it contains [[19ed4]].

[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. See [[undevigintone]]. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best. See [[meanmag]].

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=== Interval mappings ===

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 65~~

~~| steps = 18.9480867166984~~

~~| step size = 63.3309324546460~~

~~| tempered height = 5.980169~~

~~| pure height = 5.214351~~

~~| integral = 1.313799~~

~~| gap = 16.699651~~

~~| octave = 1203.28771663827~~

~~| consistent = 10~~

~~| distinct = 7~~

}}

== Regular temperament properties ==

Line 1,026:

Line 1,012:

| [[Liese]] / [[pycnic]]<br>[[Triton]]

|}

== Octave stretch or compression ==

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 [[30edt]] (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-odd-limit [[tonality diamond]]. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well.

What follows is a comparison of stretched-octave 19edo tunings.

; 19edo

* Step size: 63.158{{c}}, octave size: 1200.000{{c}}

Pure-octaves 19edo approximates all harmonics up to 16 within 21.5{{c}}.

{{Harmonics in equal|19|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19edo}}

{{Harmonics in equal|19|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19edo (continued)}}

; [[WE|19et, 5-limit WE tuning]]

* Step size: 63.293{{c}}, octave size: 1202.569{{c}}

Stretching the octave of 19edo by about 2.6{{c}} results in [[JND|just noticeably]] improved primes 3, 5, 7 and 13, but a just noticeably worse prime 11. This approximates all harmonics up to 16 but 11 within 14.3{{c}}. Both 5-limit TE and WE tuning do this.

{{Harmonics in cet|intervals=integer|63.293100|columns=11|collapsed=true|title=Approximation of harmonics in 19et, 5-limit WE tuning}}

{{Harmonics in cet|intervals=integer|63.293100|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19et, 5-limit WE tuning (continued)}}

; [[49ed6]]

* Step size: 63.305{{c}}, octave size: 1202.799{{c}}

Stretching the octave of 19edo by about 2.8{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 but 11 within 13.7{{c}}. The tuning 49ed6 does this.

{{Harmonics in equal|49|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 49ed6}}

{{Harmonics in equal|49|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 49ed6 (continued)}}

; [[ZPI|65zpi]]

* Step size: 63.331{{c}}, octave size: 1203.288{{c}}

Stretching the octave of 19edo by around 3.5{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 but 11 within 13.2{{c}}. The tuning 65zpi does this.

{{Harmonics in cet|63.330932|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65zpi}}

{{Harmonics in cet|63.330932|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65zpi (continued)}}

; [[WE|19et, 2.3.5.7.13-subgroup WE tuning]]

* Step size: 63.374{{c}}, octave size: 1204.109{{c}}

Stretching the octave of 19edo by around 4.1{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 but 11 within 16.4{{c}}. Both 2.3.5.7.13-subgroup TE and WE tuning do this.

{{Harmonics in cet|63.374142|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19et, 2.3.5.7.13-subgroup WE tuning}}

{{Harmonics in cet|63.374142|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19et, 2.3.5.7.13-subgroup WE tuning (continued)}}

; [[30edt]]

* Step size: 63.399{{c}}, octave size: 1204.572{{c}}

Stretching the octave of 19edo by around 4.5{{c}} has similar results to 65zpi, but it overshoots the optimum, meaning the improvements are less and the drawbacks are greater compared to 65zpi. The damage to the octave has also started to become [[JND|noticeable]] when it is stretched this far. This approximates all harmonics up to 16 but 11 within 18.3{{c}}. The tuning 30edt does this.

{{Harmonics in equal|30|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 30edt}}

{{Harmonics in equal|30|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 30edt (continued)}}

One can stretch the octave even further – 12.5 cents – to get the tuning [[11edf]], but its approximations of most harmonics are worse than pure-octaves 19. So it is hard to see a use case for 11edf.

== Scales ==

@@ Line 24: / Line 24: @@
 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. Finally, 19edo can be used as a tuning for sensi, though [[46edo]] provides 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]], which has a 18c sharp fifth). 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]], muggles, and [[triton]]/[[liese]]. 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]], which has a 18-cent-sharp fifth). 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]], muggles, and [[triton]]/[[liese]]. 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.
 === As a means of extending harmony ===
@@ Line 38: / Line 38: @@
 {{Harmonics in equal|19|columns=12}}
-=== Adaptive tuning and octave stretch ===
+=== Adaptive tuning ===
 The no-11's 13-limit is represented relatively well and consistently. 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.) Its diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3{{c}} off [[23/16]].
 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.
-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}}.
+Another option would be to use [[octave stretching]], which has similar benefits to adaptive use, but it also works for fixed-pitch Instruments. For more on that see the section: [[19edo#Octave stretch]].
 === Subsets and supersets ===
-edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]].
+edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]]. As such, it does not contain any nontrivial subset edos, though it contains [[19ed4]].
 [[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. See [[undevigintone]]. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best. See [[meanmag]].
@@ Line 560: / Line 560: @@
 === Interval mappings ===
 {{Q-odd-limit intervals|19}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 65
-| steps = 18.9480867166984
-| step size = 63.3309324546460
-| tempered height = 5.980169
-| pure height = 5.214351
-| integral = 1.313799
-| gap = 16.699651
-| octave = 1203.28771663827
-| consistent = 10
-| distinct = 7
-}}
 == Regular temperament properties ==
@@ Line 1,026: / Line 1,012: @@
 | [[Liese]] / [[pycnic]]<br>[[Triton]]
 |}
+== Octave stretch or compression ==
+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 [[30edt]] (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-odd-limit [[tonality diamond]]. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well.
+What follows is a comparison of stretched-octave 19edo tunings.
+; 19edo
+* Step size: 63.158{{c}}, octave size: 1200.000{{c}}
+Pure-octaves 19edo approximates all harmonics up to 16 within 21.5{{c}}.
+{{Harmonics in equal|19|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19edo}}
+{{Harmonics in equal|19|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19edo (continued)}}
+; [[WE|19et, 5-limit WE tuning]]
+* Step size: 63.293{{c}}, octave size: 1202.569{{c}}
+Stretching the octave of 19edo by about 2.6{{c}} results in [[JND|just noticeably]] improved primes 3, 5, 7 and 13, but a just noticeably worse prime 11. This approximates all harmonics up to 16 but 11 within 14.3{{c}}. Both 5-limit TE and WE tuning do this.
+{{Harmonics in cet|intervals=integer|63.293100|columns=11|collapsed=true|title=Approximation of harmonics in 19et, 5-limit WE tuning}}
+{{Harmonics in cet|intervals=integer|63.293100|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19et, 5-limit WE tuning (continued)}}
+; [[49ed6]]
+* Step size: 63.305{{c}}, octave size: 1202.799{{c}}
+Stretching the octave of 19edo by about 2.8{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 but 11 within 13.7{{c}}. The tuning 49ed6 does this.
+{{Harmonics in equal|49|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 49ed6}}
+{{Harmonics in equal|49|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 49ed6 (continued)}}
+; [[ZPI|65zpi]]
+* Step size: 63.331{{c}}, octave size: 1203.288{{c}}
+Stretching the octave of 19edo by around 3.5{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 but 11 within 13.2{{c}}. The tuning 65zpi does this.
+{{Harmonics in cet|63.330932|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65zpi}}
+{{Harmonics in cet|63.330932|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65zpi (continued)}}
+; [[WE|19et, 2.3.5.7.13-subgroup WE tuning]]
+* Step size: 63.374{{c}}, octave size: 1204.109{{c}}
+Stretching the octave of 19edo by around 4.1{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 but 11 within 16.4{{c}}. Both 2.3.5.7.13-subgroup TE and WE tuning do this.
+{{Harmonics in cet|63.374142|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19et, 2.3.5.7.13-subgroup WE tuning}}
+{{Harmonics in cet|63.374142|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19et, 2.3.5.7.13-subgroup WE tuning (continued)}}
+; [[30edt]]
+* Step size: 63.399{{c}}, octave size: 1204.572{{c}}
+Stretching the octave of 19edo by around 4.5{{c}} has similar results to 65zpi, but it overshoots the optimum, meaning the improvements are less and the drawbacks are greater compared to 65zpi. The damage to the octave has also started to become [[JND|noticeable]] when it is stretched this far. This approximates all harmonics up to 16 but 11 within 18.3{{c}}. The tuning 30edt does this.
+{{Harmonics in equal|30|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 30edt}}
+{{Harmonics in equal|30|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 30edt (continued)}}
+One can stretch the octave even further – 12.5 cents – to get the tuning [[11edf]], but its approximations of most harmonics are worse than pure-octaves 19. So it is hard to see a use case for 11edf.
 == Scales ==