17edo: Difference between revisions
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'''17 equal divisions of the octave''' ('''17edo'''), or '''17(-tone) equal temperament''' ('''17tet''', '''17et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived from dividing the octave in 17 [[equal]] steps, each around 70.6 [[cent]]s in size | '''17 equal divisions of the octave''' ('''17edo'''), or '''17(-tone) equal temperament''' ('''17tet''', '''17et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived from dividing the octave in 17 [[equal]] steps, each around 70.6 [[cent]]s in size. | ||
== Theory == | == Theory == | ||
17edo can plausibly be treated as a 2.3.25.7.11.13.23 subgroup temperament, for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because the 3, 7, 11, and 13 are all sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]]. | |||
As a no-fives system, it is best used with timbres in which harmonic multiples of 5 are attenuated or absent. Also, the standard major chord (4:5:6) cannot be used since it includes the fifth harmonic. | As a no-fives system, it is best used with timbres in which harmonic multiples of 5 are attenuated or absent. Also, the standard major chord (4:5:6) cannot be used since it includes the fifth harmonic. | ||
Instead, the tonic chords of | Instead, the tonic chords of 17edo could be considered to be the tetrad 6:7:8:9 and its utonal inversion (representing 14:16:18:21 as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the [[wikipedia: Mu chord|mu chord]] of Steely Dan fame). These are realized in 17edo as 0-4-7-10 and 0-3-6-10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0-3-7-10 (which is a sus4 with added second, or sus2 with added fourth). These three chords comprise the three ways to divide the 17edo perfect fifth into two whole tones and one subminor third. Chromatic alterations of them also exist, for example, the 0-3-7-10 chord may be altered to 0-2-7-10 (which approximates 12:13:16:18) or 0-3-8-10 (which approximates 8:9:11:12). The 0-3-8-10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0-3-6-10. | ||
17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|17|intervals=odd}} | |||
== Intervals == | == Intervals == | ||
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For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. | For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. | ||
== | == Notation == | ||
=== Sagittal === | === Sagittal === | ||
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 17edo in the Revo flavor of Sagittal: | |||
From the appendix to [[The Sagittal Songbook]] by [[ | |||
[[File:17edo Sagittal.png|800px]] | [[File:17edo Sagittal.png|800px]] | ||
== JI approximation == | == JI approximation == | ||
=== 15-odd-limit mappings === | === 15-odd-limit interval mappings === | ||
The following table shows how [[15-odd-limit intervals]] are represented in 17edo (ordered by absolute error). Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | The following table shows how [[15-odd-limit intervals]] are represented in 17edo (ordered by absolute error). Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | ||
{| class="wikitable sortable center- | {| class="wikitable sortable center-all mw-collapsible mw-collapsed" | ||
|+ | |+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent) | ||
|- | |- | ||
! class="unsortable" | Interval, complement | ! class="unsortable" | Interval, complement | ||
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| ''34.612'' | | ''34.612'' | ||
|} | |} | ||
{{15-odd-limit|17}} | |||
{ | |||
|} | |||
=== Selected 13-limit intervals === | === Selected 13-limit intervals === | ||
[[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
== Tuning | == Tuning by ear == | ||
17edo is very close to a circle of seventeen [[25/24]] chromatic semitones: (25/24)^17 is only 1.43131 cents sharp of an octave. This means that if you can tune seventeen 25/24's accurately (by say, tuning 5/4 up, 3/2 down and 5/4 up, taking care to minimize the error at each step), you have a shot at approximating 17edo within melodic just noticeable difference. | 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones: (25/24)^17 is only 1.43131 cents sharp of an octave. This means that if you can tune seventeen 25/24's accurately (by say, tuning 5/4 up, 3/2 down and 5/4 up, taking care to minimize the error at each step), you have a shot at approximating 17edo within melodic just noticeable difference. | ||
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|} | |} | ||
17et is lower in relative error than any previous equal temperaments in the no-5 11- and 13-limit. The next ETs better in these subgroups are 41 and 207, respectively. | 17et is lower in relative error than any previous equal temperaments in the no-5 11- and 13-limit. The next ETs doing better in these subgroups are 41 and 207, respectively. | ||
=== Commas === | === Commas === | ||
17et [[tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 17 27 39 48 59 63 }}, cent values rounded to 5 digits.) | |||
{| class="commatable wikitable center-all left-3 right-4 left-6" | {| class="commatable wikitable center-all left-3 right-4 left-6" | ||
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=== MOS scales === | === MOS scales === | ||
{{ | {{Main| MOS scales of 17edo }} | ||
* diatonic ([[leapfrog]]/[[archy]]) 5L2s 3331331 (10\17, 1\1) | * diatonic ([[leapfrog]]/[[archy]]) 5L2s 3331331 (10\17, 1\1) | ||
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== Music == | == Music == | ||
{{See also|:Category:17edo tracks}} | {{See also|:Category:17edo tracks}} | ||
=== Scores === | === Scores === | ||
* [http://home.snafu.de/djwolf/PreludeIn17tet.pdf Prelude] (PDF) by [[Daniel Wolf]] | * [http://home.snafu.de/djwolf/PreludeIn17tet.pdf Prelude] (PDF) by [[Daniel Wolf]] | ||
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[[Category:Pythagorean]] | [[Category:Pythagorean]] | ||
[[Category:Teentuning]] | [[Category:Teentuning]] | ||