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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. 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 ===
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Since the intervals of the 2.3.7-subgroup cluster around [[5edo]], a [[Pentatonic Functional Just System|pentatonic system of interval classification]] may be preferred over the [[heptatonic]] one, with [[7/6]] becoming a major interval and [[8/7]]~[[9/8]] becoming a minor one.
Since the intervals of the 2.3.7-subgroup cluster around [[5edo]], a [[Pentatonic Functional Just System|pentatonic system of interval classification]] may be preferred over the [[heptatonic]] one, with [[7/6]] becoming a major interval and [[8/7]]~[[9/8]] becoming a minor one.


Of course, scales generated by the perfect fifth are not the only scales 17edo contains. Another type of scale is [[neutral third scales]], which are generated by half a fifth (5\17), and take the mos patterns [[4L 3s]] (mosh) and [[7L 3s]] (dicoid). Other notable scales include that of [[bleu]] (generated by 2\17), and [[skwares]] (generated by 6\17). Non-mos scales also exist; a more complete list can be found in the [[#Scales]] section.
Of course, scales generated by the perfect fifth are not the only scales 17edo contains. Another type of scale is [[neutral third scales]], which are generated by half a fifth (5\17), and take the mos patterns [[4L 3s]] (mosh) and [[7L 3s]] (dicoid). Other notable scales include that of [[bleu]] and [[glacier]] (generated by 2\17), and [[skwares]] (generated by 6\17). Non-mos scales also exist; a more complete list can be found in the [[#Scales]] section.


Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.
Because the 5th harmonic is not well approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.


=== Odd harmonics ===
=== Odd harmonics ===
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=== 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
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|-
|-
| 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
| [[17-comma]]
| [[Gothic comma]]
|-
|-
| 5
| 5
| [[25/24]]
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| {{Monzo| -3 -1 2 }}
| 70.762
| 70.76
| Yoyo
| Yoyo
| Dicot comma
| Dicot comma
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| [[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
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| [[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
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| [[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
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| [[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
| [[40/39]]
| {{Monzo| 3 -1 1 0 0 -1 }}
| 43.83
| Thuyo
| 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
| [[352/351]]
| [5 -3 0 0 1 -1⟩
| 4.93
| Thulo
| Major minthma
|-
| 13
| [[364/363]]
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| Tholuluzo
| Minor minthma
|-
| 13
| [[512/507]]
| {{Monzo| 9 -1 0 0 0 -2 }}
| 16.99
| Thuthu
| Tridecimal neutral thirds comma
|-
|-
| 13
| 13
| [[1352/1331]]
| [[1352/1331]]
| {{Monzo| 3 0 0 0 -3 2 }}
| {{Monzo| 3 0 0 0 -3 2 }}
| 27.101
| 27.10
| Bithotrilu
| Bithotrilu
| Lovecraft comma
| Lovecraft comma
|-
|-
| 13
| 13
| [[364/363]]
| [[2197/2187]]
| {{Monzo| 2 -1 0 1 -2 1 }}
| {{Monzo| 0 -7 0 0 0 3 }}
| 4.763
| 7.90
| Tholuluzo
| Satritho
| Minor minthma
| 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
|-
|-
| 17
| 23
| [[136/135]]
| [[736/729]]
| {{Monzo| 3 -3 -1 0 0 0 1 }}
| {{Monzo| 5 -6 0 0 0 0 0 0 1 }}
| 12.776
| 16.54
| Sogu 2nd
| Satwetho
| Diatisma
| 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 ===
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| 8/7~9/8
| 8/7~9/8
| [[Machine]]
| [[Machine]]
|-
| 1
| 3\17
| 211.76
| 26/23
| [[Shoal|Shoal (trivial tuning)]]
|-
|-
| 1
| 1
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| 352.94
| 352.94
| 11/9
| 11/9
| [[Suhajira]] / [[neutrominant]] (17c) / [[beatles]] (17c) / [[dicotic]] (17) <br>[[Hemif]] / [[mohamaq]] (17c) / [[salsa]] (17)
| [[Suhajira]] / [[neutrominant]] (17c) / [[beatles]] (17c) / [[dichotic]] (17) <br>[[Hemif]] / [[mohamaq]] (17c) / [[salsa]] (17)
|-
|-
| 1
| 1
Line 749: 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)
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* [[User:FloraC/Flora's 17-note well temperament|Flora's 17-note well temperament]]
* [[User:FloraC/Flora's 17-note well temperament|Flora's 17-note well temperament]]


== Introductory materials ==
{{Todo|expand scales list}}
* [[SeventeenTheory]], an introduction to 17edo theory, through the eyes of the [[SeventeenTonePianoProject]].
 
* [http://anaphoria.com/Secor17puzzle.pdf The 17-tone Puzzle] by George Secor, another introduction into 17edo theory.
== Instruments ==
* [[17edo tetrachords]]
=== Fretted String Instruments ===
* [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú] {{forbidden}}
* [http://chrisvaisvil.com/?p=436 17 note per octave conversion from a "standard" Stratocaster copy] - conversion by Brad Smith
 
[[File:17P1050829r.JPG|alt=17P1050829r.JPG|17P1050829r.JPG]]
 
* 17edo soprano Harmony ukulele with a 3D printed fretboard - conversion by [[User:Tristanbay|Tristan Bay]]
 
[[File:17edo soprano ukulele with 3D printed fretboard.jpg|frameless|640x640px]]
 
=== Keyboards ===
[[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 ==
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-->
-->


== Instruments ==
== Introductory Materials ==
* [http://chrisvaisvil.com/?p=436 17 note per octave conversion from a "standard" Stratocaster copy] - conversion by Brad Smith
* [[SeventeenTheory]], an introduction to 17edo theory, through the eyes of the [[SeventeenTonePianoProject]].
 
* [http://anaphoria.com/Secor17puzzle.pdf The 17-tone Puzzle] by George Secor, another introduction into 17edo theory.  
[[File:17P1050829r.JPG|alt=17P1050829r.JPG|17P1050829r.JPG]]
* [[17edo tetrachords]]
 
* [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú] {{forbidden}}
* 17edo soprano Harmony ukulele with a 3D printed fretboard - conversion by [[User:Tristanbay|Tristan Bay]]
 
[[File:17edo soprano ukulele with 3D printed fretboard.jpg|frameless|640x640px]]
 
== See also ==
* [[Lumatone mapping for 17edo]]


[[Category:Teentuning]]
[[Category:Teentuning]]
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