19-limit: Difference between revisions

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Intervals of [[harmonic class|HC19]] are relatively clear in terms of their categories. [[19/16]] is most commonly considered a minor third, as 1-19/16-3/2 is an important {{w|tertian}} chord. The [[Functional Just System]] and [[Helmholtz-Ellis notation]] agree. However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if [[17/16]] is considered an augmented unison and/or if [[23/16]] is considered an augmented fourth. Perhaps most interestingly, [[Sagittal notation]] provides an accidental to enharmonically spell intervals of HC19 this way.

== Edo ~~approximations~~ ==

== Edo approximation ==

A list of [[edo]]s with progressively better tunings for 19-limit intervals: {{EDOs| 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h }} and so on.

Here is a list of [[edo]]s with progressively better tunings for 19-limit intervals (decreasing [[TE error]]): {{EDOs| 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h }} and so on.

~~Another~~ list of edos which provides relatively good tunings for 19-limit intervals ([[TE ~~relative error|~~relative error]] < 5%): {{EDOs| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.

Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "152fg" means taking the second closest approximation of harmonics 13 and 17.

== Intervals ==

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 Intervals of [[harmonic class|HC19]] are relatively clear in terms of their categories. [[19/16]] is most commonly considered a minor third, as 1-19/16-3/2 is an important {{w|tertian}} chord. The [[Functional Just System]] and [[Helmholtz-Ellis notation]] agree. However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if [[17/16]] is considered an augmented unison and/or if [[23/16]] is considered an augmented fourth. Perhaps most interestingly, [[Sagittal notation]] provides an accidental to enharmonically spell intervals of HC19 this way.
-== Edo approximations ==
+== Edo approximation ==
-A list of [[edo]]s with progressively better tunings for 19-limit intervals: {{EDOs| 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h }} and so on.
+Here is a list of [[edo]]s with progressively better tunings for 19-limit intervals (decreasing [[TE error]]): {{EDOs| 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h }} and so on.
-Another list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error|relative error]] < 5%): {{EDOs| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.
+Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.
 : '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "152fg" means taking the second closest approximation of harmonics 13 and 17.
 == Intervals ==