17edo: Difference between revisions

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 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]].

The diatonic [[major triad]], which is 0–6–10 steps, is quite [[dissonant]] compared to [[4:5:6]], as the major third is over 37 cents sharp from the traditional [[5/4]], and is instead closer to [[9/7]] or [[14/11]]. Instead, a different construction based on the [[2.3.7 subgroup]] follows naturally from its [[support]] of [[superpyth]], and may be preferred. Such chords include the tetrads [[6:7:8:9]] and its utonal inverse, realized in 17edo as 0–4–7–10 and 0–3–6–10, respectively, in addition to the sus2-4 chord, realized as 0–3–7–10. Possible chromatic alterations include but are not limited to an approximation of 12:13:16:18, 0–2–7–10, and an approximation of 8:9:11:12, 0–3–8–10. It is important to note that the chromatic semitone in 17edo is 2 steps, rather than 1 step as in [[12edo]] or [[19edo]]. Similarly, the fourth-spanning triad [[6:7:8]] and its inverse can be used, with their wide voicing realized in 17edo as 0–14–27 and 0–13–27, respectively. Extensions of these chords include 0–12–14–27, representing 8:13:14:24, and 0–13–15–27, representing 7:12:13:21.

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]] 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.

 

17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. It does not contain any nontrivial subset edos, though it contains [[17ed4]] and [[17ed8]]. 17ed8, built by taking every third step of 17edo, is a system where all odd harmonics up to the 21st are mapped exactly as in 17edo, except for the 11th. Beyond that, the 27th, 31st, 35th, and 39th harmonics are likewise mapped identically.

 

[[34edo]], which doubles 17edo, provides a great correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.

{{See also| 17edo solfege }}

 

! colspan="2" | [[Circle-of-fifths notation]]<ref group="note">Half-sharps and half-flats (denoted "t" and "d", respectively) can be used to alter the note by a single step, since sharps and flats each span two edosteps. Using half-sharps and half-flats may be preferable for compatibility with the ups-and-downs notation in 34edo, in which an up or down respectively constitute a quarter-sharp or quarter-flat. </ref>

| [[1/1]]

<references group="note" />

{{Ups and downs sharpness}}

{{Sagittal chart|Evo-SZ}}

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.)

! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints.</ref>

| <abbr title="134217728/129140163">(18 digits)</abbr>

| 66.76

| [[Gothic comma]]

| 70.76

| 1.95

| 43.41

| 14.19

| 13.07

| 1.12

| 17.58

| 9.69

| 7.14

| Rastma, neutral thirds comma

| 4.50

| 27.10

| [[2197/2187]]

| {{Monzo| 0 -7 0 0 0 3 }}

| {{Monzo| 1 4 0 -1 0 0 0 0 -1 }}

| {{Monzo| 4 -2 0 0 0 1 0 0 -1 }}

| 8.34

| {{Monzo| -4 -1 0 0 -1 0 0 0 2 }}

| 3.28

| Bitwetho-alu

| Preziosisma

| 23

| [[736/729]]

| {{Monzo| 5 -6 0 0 0 0 0 0 1 }}

| 16.54

| Satwetho

| 23-limit Tenney/Cage comma (HEJI)

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.

| [[Suhajira]] / [[neutrominant]] (17c) / [[beatles]] (17c) / [[dichotic]] (17) <br>[[Hemif]] / [[mohamaq]] (17c) / [[salsa]] (17)

* [[neutrominant]] [[3L&nbsp;4s]] 3 2 3 2 3 2 2 (5\17, 1\1) (''dedicated article: [[17edo neutral scale]]'')

 

=== Fretted String Instruments ===

* [http://chrisvaisvil.com/?p=436 17 note per octave conversion from a "standard" Stratocaster copy] - conversion by Brad Smith

 

 

 

 

[[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.

 

 

It is possible to rebuild some standard MIDI keyboards to have 17 notes 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)

== Introductory Materials ==

* [http://anaphoria.com/Secor17puzzle.pdf The 17-tone Puzzle] by George Secor, another introduction into 17edo theory.  

* [[17edo tetrachords]]

* [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú] {{forbidden}}