19edo: Difference between revisions

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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 ===

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== Octave stretch ==

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.

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

; 19edo

* Step size: 63.158{{c}}, octave size: 1200.0{{c}}

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

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

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

; [[WE|19et, 2.3.5.11 WE tuning]]

* Step size: 63.192{{c}}, octave size: 1200.6{{c}}

Stretching the octave of 19edo by around half a cent 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 within 19.6{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.

{{Harmonics in cet|63.192|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning}}

{{Harmonics in cet|63.192|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning (continued)}}

; [[WE|19et, 13-limit WE tuning]]

* Step size: 63.291{{c}}, octave size: 1202.5{{c}}

Stretching the octave of 19edo by around 2.5{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 within 25.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}}

{{Harmonics in cet|63.291|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning (continued)}}

; [[49ed6]]

* Step size: NNN{{c}}, octave size: 1202.8{{c}}

Stretching the octave of 19edo by just under 3{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 within 26.8{{c}}. The tuning 49ed6 does this.

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

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

; [[zpi|65zpi]]

* Step size: 63.331{{c}}, octave size: 1203.3{{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 within 28.5{{c}}. The tuning 65zpi does this.

{{Harmonics in cet|63.331|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi}}

{{Harmonics in cet|63.331|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi (continued)}}

; [[30edt]]

* Step size: NNN{{c}}, octave size: 1204.6{{c}}

Stretching the octave of 19edo by around 4.5{{c}} has similar results to 65zpi, but it "overshoot", 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 within 30.4{{c}}. The tuning 30edt does this.

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

{{Harmonics in equal|30|3|1|columns=12|start=12|collapsed=true|intervals=integer|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 43: / Line 43: @@
 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 ===
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 | [[Liese]] / [[pycnic]]<br>[[Triton]]
 |}
+== Octave stretch ==
+== Octave stretch ==
+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.
+What follows is a comparison of stretched-octave 19edo tunings.
+; 19edo
+* Step size: 63.158{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 19edo approximates all harmonics up to 16 within 21.5{{c}}.
+{{Harmonics in equal|19|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo}}
+{{Harmonics in equal|19|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo (continued)}}
+; [[WE|19et, 2.3.5.11 WE tuning]]
+* Step size: 63.192{{c}}, octave size: 1200.6{{c}}
+Stretching the octave of 19edo by around half a cent 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 within 19.6{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.
+{{Harmonics in cet|63.192|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning}}
+{{Harmonics in cet|63.192|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning (continued)}}
+; [[WE|19et, 13-limit WE tuning]]
+* Step size: 63.291{{c}}, octave size: 1202.5{{c}}
+Stretching the octave of 19edo by around 2.5{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 within 25.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}}
+{{Harmonics in cet|63.291|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning (continued)}}
+; [[49ed6]]
+* Step size: NNN{{c}}, octave size: 1202.8{{c}}
+Stretching the octave of 19edo by just under 3{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 within 26.8{{c}}. The tuning 49ed6 does this.
+{{Harmonics in equal|49|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6}}
+{{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6 (continued)}}
+; [[zpi|65zpi]]
+* Step size: 63.331{{c}}, octave size: 1203.3{{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 within 28.5{{c}}. The tuning 65zpi does this.
+{{Harmonics in cet|63.331|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi}}
+{{Harmonics in cet|63.331|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi (continued)}}
+; [[30edt]]
+* Step size: NNN{{c}}, octave size: 1204.6{{c}}
+Stretching the octave of 19edo by around 4.5{{c}} has similar results to 65zpi, but it "overshoot", 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 within 30.4{{c}}. The tuning 30edt does this.
+{{Harmonics in equal|30|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt}}
+{{Harmonics in equal|30|3|1|columns=12|start=12|collapsed=true|intervals=integer|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 ==