Xenharmonic Wiki

17edo: Difference between revisions

← Older edit
VisualWikitext
→Commas: Add Major minthma
Yourmusic Productions (talk | contribs)
autopatrolled, emailconfirmed
1,896 edits
→Instruments: add new entry, consolidate existing ones.
 
(9 intermediate revisions by 3 users not shown)
Line 12: Line 12:
17edo is the next smallest edo to have a [[5L 2s|diatonic]] [[3/2|perfect fifth]] after [[12edo]], and is quite popular for that reason. The perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of 12edo's, lending itself to a diatonic scale with more constrasting large and small steps, so it can be seen as a tuning that emphasizes the [[hard]]ness of [[Pythagorean tuning]] rather than mellowing it out as in [[meantone]]. It completely misses [[harmonic]] [[5/1|5]], with [[5/4]] and [[6/5]] both being about halfway between its steps,  but it approximates harmonics [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] acceptably, with a sharp tuning for all of them. It can thus be treated as a temperament of the 2.3.25.7.11.13.23 [[subgroup]] or any of its subsets, where it is quite accurate for its size.  
17edo is the next smallest edo to have a [[5L 2s|diatonic]] [[3/2|perfect fifth]] after [[12edo]], and is quite popular for that reason. The perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of 12edo's, lending itself to a diatonic scale with more constrasting large and small steps, so it can be seen as a tuning that emphasizes the [[hard]]ness of [[Pythagorean tuning]] rather than mellowing it out as in [[meantone]]. It completely misses [[harmonic]] [[5/1|5]], with [[5/4]] and [[6/5]] both being about halfway between its steps,  but it approximates harmonics [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] acceptably, with a sharp tuning for all of them. It can thus be treated as a temperament of the 2.3.25.7.11.13.23 [[subgroup]] or any of its subsets, where it is quite accurate for its size.  


A notable [[comma]] it [[tempering out|tempers out]] is [[64/63]], which equates the harmonic seventh [[7/4]] with the pythagorean minor seventh [[16/9]], while its patent val does not temper out [[81/80]]. This makes 17edo by default a [[superpyth]]agorean system rather than a [[meantone]] one, being very close to 1/7-comma superpyth. Other commas it tempers out include [[78/77]], [[99/98]], [[144/143]], [[169/168]], [[243/242]], and many more, each of which has its own effect on the structure of 17edo. If one wants to approximate JI with prime 5, then 17edo would not be the best option, and it would be better to use other systems like [[19edo]], [[22edo]], [[27edo]], or [[31edo]] instead. That said, the 17[[wart|c]] [[val]] does temper out 81/80 (while improving consistency as shown below in [[#Approximation to JI]]), while still tempering out 64/63, thus placing it on the meantone spectrum with the [[dominant (temperament)|dominant]] [[extension]].
A notable [[comma]] it [[tempering out|tempers out]] is [[64/63]], which equates the harmonic seventh [[7/4]] with the pythagorean minor seventh [[16/9]], while its patent val does not temper out [[81/80]]. This makes 17edo by default a [[superpyth]]agorean system rather than a [[meantone]] one, being very close to 1/7-comma superpyth. Other commas it tempers out can be found in the [[#Commas]] section, each of which has its own effect on the structure of 17edo. If one wants to approximate JI with prime 5, then 17edo would not be the best option, and it would be better to use other systems like [[19edo]], [[22edo]], [[27edo]], or [[31edo]] instead. That said, the 17c [[val]] (written using [[wart notation]]) does temper out 81/80 (while improving consistency as shown below in [[#Approximation to JI]]), while still tempering out 64/63, thus placing it on the meantone spectrum with the [[dominant (temperament)|dominant]] [[extension]].


=== As a means of extending harmony ===
=== As a means of extending harmony ===
Line 555: Line 555:


=== Commas ===
=== Commas ===
17et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[patent val]] {{val| 17 27 39 48 59 63 69 }}, cent values ​​rounded to 5 digits.)
17et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[patent val]] {{val| 17 27 39 48 59 63 69 72 77}}, cent values ​​rounded to 1/100 of a cent.)


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints.</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
Line 567: Line 567:
|-
|-
| 3
| 3
| [[134217728/129140163|(18 digits)]]
| <abbr title="134217728/129140163">(18 digits)</abbr>
| {{Monzo| 27 -17 }}
| {{Monzo| 27 -17 }}
| 66.765
| 66.76
| Sasawa
| Sasawa
| [[Gothic comma]]
| [[Gothic comma]]
Line 576: Line 576:
| [[25/24]]
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| {{Monzo| -3 -1 2 }}
| 70.762
| 70.76
| Yoyo
| Yoyo
| Dicot comma
| Dicot comma
Line 583: Line 583:
| [[32805/32768]]
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| {{Monzo| -15 8 1 }}
| 1.9537
| 1.95
| Layo
| Layo
| Schisma
| Schisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
|-
| 7
| 7
| [[525/512]]
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| {{Monzo| -9 1 2 1 }}
| 43.408
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicennma
| Avicennma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.264
| Ru
| Septimal comma
|-
|-
| 7
| 7
| [[245/243]]
| [[245/243]]
| {{Monzo| 0 -5 1 2 }}
| {{Monzo| 0 -5 1 2 }}
| 14.191
| 14.19
| Zozoyo
| Zozoyo
| Sensamagic comma
| Sensamagic comma
Line 611: Line 611:
| [[1728/1715]]
| [[1728/1715]]
| {{Monzo| 6 3 -1 -3 }}
| {{Monzo| 6 3 -1 -3 }}
| 13.074
| 13.07
| Triru-agu
| Triru-agu
| Orwellisma
| Orwellisma
|-
| 7
| [[17496/16807]]
| {{Monzo| 3 7 0 -5 }}
| 69.56
| Quinru
| Bleu comma
|-
| 7
| [[19683/19208]]
| {{Monzo| -3 9 0 -4 }}
| 42.29
| Laquadru
| Skwares comma
|-
|-
| 7
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| <abbr title="420175/419904">(12 digits)</abbr>
| {{Monzo| -6 -8 2 5 }}
| {{Monzo| -6 -8 2 5 }}
| 1.1170
| 1.12
| Quinzo-ayoyo
| Quinzo-ayoyo
| [[Wizma]]
| [[Wizma]]
|-
| 11
| [[45/44]]
| {{Monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Cake comma
|-
|-
| 11
| 11
| [[99/98]]
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| {{Monzo| -1 2 0 -2 1 }}
| 17.576
| 17.58
| Loruru
| Loruru
| Mothwellsma
| Mothwellsma
Line 632: Line 653:
| [[896/891]]
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| {{Monzo| 7 -4 0 1 -1 }}
| 9.6880
| 9.69
| Saluzo
| Saluzo
| Pentacircle comma
| Pentacircle comma
Line 639: Line 660:
| [[243/242]]
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| {{Monzo| -1 5 0 0 -2 }}
| 7.1391
| 7.14
| Lulu
| Lulu
| Rastma
| Rastma, neutral thirds comma
|-
|-
| 11
| 11
| [[385/384]]
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| {{Monzo| -7 -1 1 1 1 }}
| 4.5026
| 4.50
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|-
|-
| 13
| 13
| [[1352/1331]]
| [[40/39]]
| {{Monzo| 3 0 0 0 -3 2 }}
| {{Monzo| 3 -1 1 0 0 -1 }}
| 27.101
| 43.83
| Bithotrilu
| Thuyo
| Lovecraft comma
| Unintendo comma
|-
| 13
| [[65/64]]
| {{Monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[78/77]]
| {{Monzo| 1 1 0 -1 -1 1 }}
| 22.34
| Tholuru
| Negustma
|-
| 13
| [[144/143]]
| {{Monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[169/168]]
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| Thothoru
| Buzurgisma, dhanvantarisma
|-
|-
|13
| 13
|[[352/351]]
| [[352/351]]
|[5 -3 0 0 1 -1⟩
| [5 -3 0 0 1 -1⟩
|4.925
| 4.93
|Thulo
| Thulo
|Major minthma
| Major minthma
|-
|-
| 13
| 13
| [[364/363]]
| [[364/363]]
| {{Monzo| 2 -1 0 1 -2 1 }}
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.763
| 4.76
| Tholuluzo
| Tholuluzo
| Minor minthma
| Minor minthma
|-
|-
| 17
| 13
| [[136/135]]
| [[512/507]]
| {{Monzo| 3 -3 -1 0 0 0 1 }}
| {{Monzo| 9 -1 0 0 0 -2 }}
| 12.776
| 16.99
| Sogu 2nd
| Thuthu
| Diatisma
| Tridecimal neutral thirds comma
|-
| 13
| [[1352/1331]]
| {{Monzo| 3 0 0 0 -3 2 }}
| 27.10
| Bithotrilu
| Lovecraft comma
|-
| 13
| [[2197/2187]]
| {{Monzo| 0 -7 0 0 0 3 }}
| 7.90
| Satritho
| Threedie
|-
| 23
| [[162/161]]
| {{Monzo| 1 4 0 -1 0 0 0 0 -1 }}
| 10.72
| Twethuru
| Minor kirnbergerisma
|-
| 23
| [[208/207]]
| {{Monzo| 4 -2 0 0 0 1 0 0 -1 }}
| 8.34
| Twethutho
| Vicetone comma
|-
| 23
| [[253/252]]
| {{Monzo| -2 -2 0 -1 1 0 0 0 1 }}
| 6.86
| Twetholoru
| Middle neutravicema
|-
| 23
| [[529/528]]
| {{Monzo| -4 -1 0 0 -1 0 0 0 2 }}
| 3.28
| Bitwetho-alu
| Preziosisma
|-
|-
|23
| 23
|[[208/207]]
| [[736/729]]
|[4 -2 0 0 0 1 0 0 -1⟩
| {{Monzo| 5 -6 0 0 0 0 0 0 1 }}
|8.343
| 16.54
|Twethutho 2nd
| Satwetho
|Vicetone comma
| 23-limit Tenney/Cage comma (HEJI)
|}
|}
<references group="note" />
<references group="note" />
Note that despite their relatively large size, the 17-comma, the avicennma and the chromatic semitone are all tempered out by the 13-limit patent val, as stated.
Note that due to the inaccurate prime 5, the rather large commas [[25/24]], [[525/512]], [[45/44]], and [[40/39]] are all tempered out by 17edo's patent val.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 769: Line 860:


* diatonic ([[leapfrog]]/[[archy]]) [[5L&nbsp;2s]] 3 3 3 1 3 3 1 (10\17, 1\1)
* diatonic ([[leapfrog]]/[[archy]]) [[5L&nbsp;2s]] 3 3 3 1 3 3 1 (10\17, 1\1)
* [[neutrominant]] [[3L&nbsp;4s]] 3 2 3 2 3 2 2 (5\17, 1\1)
* [[neutrominant]] [[3L&nbsp;4s]] 3 2 3 2 3 2 2 (5\17, 1\1) (''dedicated article: [[17edo neutral scale]]'')
* [[neutrominant]] [[7L&nbsp;3s]] 2 2 2 1 2 2 1 2 2 1 (5\17, 1\1)
* [[neutrominant]] [[7L&nbsp;3s]] 2 2 2 1 2 2 1 2 2 1 (5\17, 1\1)
* [[squares]] [[3L&nbsp;5s]] 1 1 4 1 4 1 4 (6\17, 1\1)
* [[squares]] [[3L&nbsp;5s]] 1 1 4 1 4 1 4 (6\17, 1\1)
Line 791: Line 882:


[[File:17edo soprano ukulele with 3D printed fretboard.jpg|frameless|640x640px]]
[[File:17edo soprano ukulele with 3D printed fretboard.jpg|frameless|640x640px]]
Keyboards have also been built for 17edo &mdash; see [[Stephen Weigel]]'s [https://www.youtube.com/watch?v=2B14mttkavA ''Take This Stone (17-TET microtonal cover)''] (2025).


=== Keyboards ===
=== Keyboards ===
[[Lumatone mapping for 17edo|Lumatone mappings for 17edo]] are available.
[[Lumatone mapping for 17edo|Lumatone mappings for 17edo]] are available.
The Striso Board can be tuned in many ways, but as it has 17 notes per octave and is organised in a circle of fifths based layout, it works particularly well with 17edo, letting you play far wider stretches of notes than a standard keyboard.
[[File:Strisoboard_piano2a_s.jpg|frameless]]
It is possible to rebuild some standard MIDI keyboards to have 17 note per octave by combining parts from multiple keyboards, as with the finished product shown in the following videos by [[Stephen Weigel]] and [[Chris Vaisvil]]:
* [https://www.youtube.com/watch?v=2B14mttkavA ''Take This Stone (17-TET microtonal cover)''] (2025)
* [https://www.youtube.com/watch?v=nboggmtayk0 ''DIY microtonal piano - 17 notes per octave''] (2026)


== Music ==
== Music ==
Retrieved from "https://en.xen.wiki/w/17edo"