19edo: Difference between revisions
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In music, '''19 equal temperament''', called 19-TET, 19-[[EDO]], or 19-ET, is the scale derived by dividing the [[octave]] into 19 [[Equal|equally]] large steps. Each step represents a frequency ratio of the 19th root of 2, or about 63.2 [[cent|cents]]. It is the 8th [[prime EDO]], following [[17edo]] and preceding [[23edo]]. | In music, '''19 equal temperament''', called 19-TET, 19-[[EDO]], or 19-ET, is the scale derived by dividing the [[octave]] into 19 [[Equal|equally]] large steps. Each step represents a frequency ratio of the 19th root of 2, or about 63.2 [[cent|cents]]. It is the 8th [[prime EDO]], following [[17edo]] and preceding [[23edo]]. | ||
== Theory | == Theory == | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! colspan="2" | <!-- empty cell --> | ! colspan="2" | <!-- empty cell --> | ||
| Line 46: | Line 46: | ||
| +5 | | +5 | ||
|- | |- | ||
! colspan="2" | [[ | ! colspan="2" | [[Nearest edomapping]] | ||
| 19 | | 19 | ||
| 11 | | 11 | ||
| Line 55: | Line 55: | ||
| 10 | | 10 | ||
|- | |- | ||
! colspan="2" | [[ | ! colspan="2" | [[Fifthspan]] | ||
| 0 | | 0 | ||
| +1 | | +1 | ||
| Line 328: | Line 328: | ||
Key signatures are the same, but with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of Bbb would have double-flats on B and E, and flats on C, D, F, G, and A. Thinking of rewriting this key as A# might seem better, but then the key signature would contain double-sharps on C, F, and G, and sharps on A, B, D, and E, which is actually worse. | Key signatures are the same, but with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of Bbb would have double-flats on B and E, and flats on C, D, F, G, and A. Thinking of rewriting this key as A# might seem better, but then the key signature would contain double-sharps on C, F, and G, and sharps on A, B, D, and E, which is actually worse. | ||
== Chord | == Chord names == | ||
All 19edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Here are the zo, gu, yo and ru triads: | All 19edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Here are the zo, gu, yo and ru triads: | ||
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The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo tends to conflate zo and ru ratios. | The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo tends to conflate zo and ru ratios. | ||
For a more complete list, see [[19edo Chord Names]] and [[Ups and Downs Notation# | For a more complete list, see [[19edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]]. | ||
== | == JI approximation == | ||
=== 15-odd-limit interval mappings === | |||
The following table shows how [[15-odd-limit intervals]] are represented in 19edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | The following table shows how [[15-odd-limit intervals]] are represented in 19edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | ||
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|} | |} | ||
=== Selected 17-limit intervals === | |||
[[File:19ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:19ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
=== | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! colspan="2" | | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | |||
! | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! | ! colspan="2" | Tuning error | ||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | |||
| {{monzo| -30 19 }} | |||
| [{{val| 19 30 }}] | |||
| +2.28 | | +2.28 | ||
| 2.28 | |||
| 3.61 | |||
|- | |||
| 2.3.5 | |||
| 81/80, 3125/3072 | |||
| [{{val| 19 30 44 }}] | |||
| +2.58 | | +2.58 | ||
| 1.91 | |||
| 3.02 | |||
|- | |||
| 2.3.5.7 | |||
| 49/48, 81/80, 126/125 | |||
| [{{val| 19 30 44 53 }}] | |||
| +3.85 | | +3.85 | ||
| 2.76 | |||
| 4.35 | |||
|- | |||
| 2.3.5.7.13 | |||
| 49/48, 65/64, 81/80, 91/90 | |||
| [{{val| 19 30 44 53 70 }}] | |||
| +4.14 | | +4.14 | ||
| 2.53 | | 2.53 | ||
| 3.99 | | 3.99 | ||
|} | |} | ||
== Commas == | 19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit – ''both'' 19 and 19e val achieve this in the case of 13-limit, 19eg val in the 17-limit, and 19egh val in the 19-limit. The next ETs better in those subgroups are 34, 31, 27e, 22, and 26, respectively. | ||
19et is prominent in the 2.3.5.7.13 subgroup, and the next ET that does better in this is 53. | |||
=== Commas === | |||
19 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 19 30 44 53 66 70 }}.) | 19 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 19 30 44 53 66 70 }}.) | ||
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| 26.84 | | 26.84 | ||
| Thoyo | | Thoyo | ||
| | | Wilsorma | ||
|- | |- | ||
| 13 | | 13 | ||
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<references/> | <references/> | ||
== Linear temperaments == | === Linear temperaments === | ||
* [[List of 19et rank two temperaments by badness]] | * [[List of 19et rank two temperaments by badness]] | ||
* [[List of 19et rank two temperaments by complexity]] | * [[List of 19et rank two temperaments by complexity]] | ||