19edo: Difference between revisions

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== Octave stretch or compression ==

Pianos are frequently tuned with stretched octaves anyway due to the slight [[~~timbre|~~inharmonicity]] inherent in their strings, which makes 19edo a promising option for pianos with split sharps.

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

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.

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; [[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 within 30.4{{c}}. The tuning 30edt does this.

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)}}

@@ Line 1,014: / Line 1,014: @@
 == Octave stretch or compression ==
-Pianos are frequently tuned with stretched octaves anyway due to the slight [[timbre|inharmonicity]] inherent in their strings, which makes 19edo a promising option for pianos with split sharps.
+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-limit [[tonality diamond]]. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well.
+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.
@@ Line 1,052: / Line 1,052: @@
 ; [[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 within 30.4{{c}}. The tuning 30edt does this.
+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)}}