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<span style="display: block; text-align: right;"><span style="color: #1300ff;">[[:de:17edo|Deutsch]] - [[17平均律|日本語]]</span>
{{interwiki
| de = 17-EDO
| en = 17edo
| es = 17 EDO
| ja = 17平均律
}}
{{Infobox ET}}
{{ED intro}}
{{Wikipedia|17 equal temperament}}


=<span style="color: #1300ff;">17 tone equal temperament</span>=
== Theory ==
__FORCETOC__
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.


'''17-EDO''' divides the octave in 17 equal steps, each 70.588 [[cent|cents]] in size. It is the seventh [[prime_numbers|prime]] edo, following [[13edo|13edo]] and coming before [[19edo|19edo]].
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]].


==Theory==
=== As a means of extending harmony ===
An introduction to 17-EDO theory, through the eyes of the [[SeventeenTonePianoProject|SeventeenTonePianoProject]]: [[SeventeenTheory|SeventeenTheory]].
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.


Another introduction into 17-EDO theory: [http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf The 17-tone Puzzle] by George Secor.
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.


[[17edo_Solfege|17edo Solfege]]. [[17edo_tetrachords|17edo tetrachords]]. [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú]
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.


17-EDO 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|27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6|44ed6]].
Because the 5th harmonic is not well approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.


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.
=== Odd harmonics ===
{{Harmonics in equal|17|intervals=odd|columns=11}}
{{Harmonics in equal|17|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 17edo (continued)}}


Instead, the tonic chords of 17-EDO could be considered to be the tetrad 6:7:8:9 and its utonal inversion, the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the famous [https://en.wikipedia.org/wiki/Mu_chord mu major chord] of Steely Dan fame). These are realized in 17-EDO 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 17-EDO 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.
=== Subsets and supersets ===
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.


=Intervals=
[[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.


{| class="wikitable"
== Intervals ==
{{See also| 17edo solfege }}
 
{| class="wikitable center-all right-2 left-3"
|-
|-
! | Degree
! #
! | Cents
! Cents
! | Names ofIntervals
! Approximate ratios<ref group="note">{{sg|limit=2.3.25.7.11.13.85.23&nbsp;subgroup}}</ref>
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]]
! 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>
! | Approximate Ratios*
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and ^d2)
! | Temperament(s)generated
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U {{=}} 1)}}
|-
|-
| | 0
| 0
| | 0
| 0.0
| | Unison
| [[1/1]]
| style="text-align:center;" | unison
| Unison
| style="text-align:center;" | P1
| D
| style="text-align:center;" | C
| unison
| style="text-align:center;" | 1/1
| P1
| |
| D
| unison
| P1
| D
|-
|-
| | 1
| 1
| | 70.59
| 70.6
| | Super Unison/Minor Second
| [[24/23]], [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| style="text-align:center;" | minor 2nd
| Minor 2nd<br>(Semiaugmented 1sn)
| style="text-align:center;" | m2
| Eb<br>(Dt)
| style="text-align:center;" | Db
| up unison, <br>minor 2nd
| style="text-align:center;" | [[25/24|25/24]], [[26/25|26/25]], [[33/32|33/32]]
| ^1, m2
| |  
| Eb
| uber unison, <br>minor 2nd
| U1, m2
| UD, Eb
|-
|-
| | 2
| 2
| | 141.18
| 141.2
| | Augmented Unison/Neutral Second
| [[12/11]], [[13/12]], [[14/13]], [[25/23]]
| style="text-align:center;" | mid 2nd
| Augmented 1sn<br>(Neutral 2nd)
| style="text-align:center;" | ~2
| D#<br>(Ed)
| style="text-align:center;" | Dv
| augmented 1sn, <br>mid 2nd
| style="text-align:center;" | [[13/12|13/12]], [[12/11|12/11]], [[14/13|14/13]]
| A1, ~2
| | [[Bleu|Bleu]]
| vE
| neutral 2nd
| N2
| UEb, uE
|-
|-
| | 3
| 3
| | 211.765
| 211.8
| | Major Second/Sub Third
| [[8/7]], [[9/8]], [[17/15]], [[25/22]], [[26/23]]
| style="text-align:center;" | major 2nd
| Major 2nd
| style="text-align:center;" | M2
| E
| style="text-align:center;" | D
| major 2nd
| style="text-align:center;" | [[9/8|9/8]], [[25/22|25/22]], [[8/7|8/7]], [[28/25|28/25]]
| M2
| | [[Machine|Machine]]
| E
| major 2nd
| M2
| E
|-
|-
| | 4
| 4
| | 282.35
| 282.4
| | Minor Third/Super Second
| [[7/6]], [[13/11]], [[20/17]]
| style="text-align:center;" | minor 3rd
| Minor 3rd
| style="text-align:center;" | m3
| F
| style="text-align:center;" | Eb
| minor 3rd
| style="text-align:center;" | [[13/11|13/11]], [[7/6|7/6]]
| m3
| | [[Huxley|Huxley]]
| F
| minor 3rd
| m3
| F
|-
|-
| | 5
| 5
| | 352.94
| 352.9
| | Augmented Second/Neutral Third/
| [[11/9]], [[27/22]], [[16/13]], [[39/32]]
Diminished Fourth
| Diminished 4th<br>(Neutral 3rd)
| style="text-align:center;" | mid 3rd
| Gb<br>(Ft)
| style="text-align:center;" | ~3
| mid 3rd
| style="text-align:center;" | Ev
| ~3
| style="text-align:center;" | [[11/9|11/9]], [[16/13|16/13]]
| ^F
| | [[maqamic|Maqamic]]/[[Hemif|hemif]]
| neutral 3rd
| N3
| UF, uF#
|-
|-
| | 6
| 6
| | 423.53
| 423.5
| | Major Third/Sub Fourth
| [[9/7]], [[14/11]], [[23/18]], [[32/25]], [[51/40]]
| style="text-align:center;" | major 3rd
| Major 3rd<br>(Semidiminished 4th)
| style="text-align:center;" | M3
| F#<br>(Gd)
| style="text-align:center;" | E
| major 3rd
| style="text-align:center;" | [[32/25|32/25]], [[33/26|33/26]], [[9/7|9/7]], [[14/11|14/11]]
| M3
| | [[Skwares|Skwares]]
| F#
| major 3rd
| M3
| F#
|-
|-
| | 7
| 7
| | 494.11
| 494.1
| | Perfect Fourth
| [[4/3]], [[21/16]], [[85/64]]
| style="text-align:center;" | perfect 4th
| Perfect 4th
| style="text-align:center;" | P4
| G
| style="text-align:center;" | F
| perfect 4th
| style="text-align:center;" | [[4/3|4/3]]
| P4
| | [[Supra|Supra]]
| G
| perfect 4th
| P4
| G
|-
|-
| | 8
| 8
| | 564.71
| 564.7
| | Super FourthDiminished Fifth
| [[11/8]], [[18/13]], [[25/18]], [[32/23]]
| style="text-align:center;" | up 4th,
| Diminished 5th<br>(Semiaugmented 4th)
diminished 5th
| Ab<br>(Gt)
| style="text-align:center;" | ^4, d5
| mid 4th, <br>diminished 5th
| style="text-align:center;" | F^, Gb
| ~4, d5
| style="text-align:center;" | [[11/8|11/8]], 18/13
| ^G, Ab
| | [[Progress|Progress]]
| uber 4th/<br>neutral 4th
| U4/N4
| UG
|-
|-
| | 9
| 9
| | 635.29
| 635.3
| | Augmented Fourth/Sub Fifth
| [[13/9]], [[16/11]], [[23/16]], [[36/25]]
| style="text-align:center;" | augmented 4th,
| Augmented 4th<br>(Semidiminished 5th)
down 5th
| G#<br>(Ad)
| style="text-align:center;" | A4, v5
| augmented 4th, <br>mid 5th
| style="text-align:center;" | F#, Gv
| A4, ~5
| style="text-align:center;" | [[16/11|16/11]], 13/9, [[23/16|23/16]]
| G#, vA
| | Progress
| unter 5th/<br>neutral 5th
| u5/N5
| uA
|-
|-
| | 10
| 10
| | 705.88
| 705.9
| | Perfect Fifth
| [[3/2]], [[32/21]], [[128/85]]
| style="text-align:center;" | perfect 5th
| Perfect 5th
| style="text-align:center;" | P5
| A
| style="text-align:center;" | G
| perfect 5th
| style="text-align:center;" | [[3/2|3/2]]
| P5
| | Supra
| A
| perfect 5th
| P5
| A
|-
|-
| | 11
| 11
| | 776.47
| 776.5
| | Super Fifth/Minor Sixth
| [[11/7]], [[14/9]], [[25/16]], [[36/23]], [[80/51]]
| style="text-align:center;" | minor 6th
| Minor 6th<br>(Semiaugmented 5th)
| style="text-align:center;" | m6
| Bb<br>(At)
| style="text-align:center;" | Ab
| minor 6th
| style="text-align:center;" | 25/16, 14/9, [[11/7|11/7]]
| m6
| | Skwares
| Bb
| minor 6th
| m6
| Bb
|-
|-
| | 12
| 12
| | 847.06
| 847.1
| | Augmented Fifth/Neutral Sixth/
| [[13/8]], [[18/11]], [[44/27]], [[64/39]]
Diminished Seventh
| Augmented 5th<br>(Neutral 6th)
| style="text-align:center;" | mid 6th
| A#<br>(Bd)
| style="text-align:center;" | ~6
| mid 6th
| style="text-align:center;" | Av
| ~6
| style="text-align:center;" | [[13/8|13/8]], 18/11
| vB
| | Maqamic/hemif
| neutral 6th
| N6
| UBb, uB
|-
|-
| | 13
| 13
| | 917.65
| 917.6
| | Major Sixth/Sub Seventh
| [[12/7]], [[17/10]], [[22/13]]
| style="text-align:center;" | major 6th
| Major 6th
| style="text-align:center;" | M6
| B
| style="text-align:center;" | A
| major 6th
| style="text-align:center;" | [[17/10|17/10]], [[22/13|22/13]],[[12/7|12/7]]
| M6
| | Huxley
| B
| major 6th
| M6
| B
|-
|-
| | 14
| 14
| | 988.235
| 988.2
| | Minor Seventh/Super Sixth
| [[7/4]], [[16/9]], [[23/13]], [[30/17]], [[44/25]]
| style="text-align:center;" | minor 7th
| Minor 7th
| style="text-align:center;" | m7
| C
| style="text-align:center;" | Bb
| minor 7th
| style="text-align:center;" | [[16/9|16/9]], 7/4, 25/14
| m7
| | Machine
| C
| minor 7th
| m7
| C
|-
|-
| | 15
| 15
| | 1058.82
| 1058.8
| | Augmented Sixth/Neutral Seventh/
| [[11/6]], [[13/7]], [[24/13]], [[46/25]]
Diminished Octave
| Diminished 8ve<br>(Neutral 7th)
| style="text-align:center;" | mid 7th
| Db<br>(Ct)
| style="text-align:center;" | ~7
| mid 7th
| style="text-align:center;" | Bv
| ~7
| style="text-align:center;" | [[11/6|11/6]], 24/13, [[13/7|13/7]]
| ^C
| | Bleu
| neutral 7th
| N7
| UC, uC#
|-
|-
| | 16
| 16
| | 1129.41
| 1129.4
| | Major Seventh/Sub Octave
| [[23/12]], [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| style="text-align:center;" | major 7th
| Major 7th<br>(Semidiminished 8ve)
| style="text-align:center;" | M7
| C#<br>(Dd)
| style="text-align:center;" | B
| major 7th,<br>down 8ve
| style="text-align:center;" | 25/13, 48/25, 64/33
| M7, v8
| |
| C#
| major 7th,<br>unter octave
| M7, u8
| C#, uD
|-
|-
| | 17
| 17
| | 1200
| 1200.0
| | Perfect Octave
| [[2/1]]
| style="text-align:center;" | octave
| Octave
| style="text-align:center;" | P8
| D
| style="text-align:center;" | C
| octave
| style="text-align:center;" | [[2/1|2/1]]
| P8
| |
| D
| octave
| P8
| D
|}
|}
<nowiki>*</nowiki>Ratios based on treating 17edo as a 2.3.7.11.13.25 subgroup
<references group="note" />


Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
=== Interval quality and chord names in color notation ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | quality
! Quality
! | color
! Color
! | monzo format
! Monzo format
! | examples
! Examples
|-
|-
| style="text-align:center;" | minor
| rowspan="2" | minor
| style="text-align:center;" | zo
| zo
| style="text-align:center;" | {a, b, 0, 1}
| (a, b, 0, 1)
| style="text-align:center;" | 7/6, 7/4
| 7/6, 7/4
|-
|-
| style="text-align:center;" | "
| fourthward wa
| style="text-align:center;" | fourthward wa
| (a, b), b < -1
| style="text-align:center;" | {a, b}, b &lt; -1
| 32/27, 16/9
| style="text-align:center;" | 32/27, 16/9
|-
|-
| style="text-align:center;" | mid
| rowspan="2" | mid
| style="text-align:center;" | lova
| ilo
| style="text-align:center;" | {a, b, 0, 0, 1}
| (a, b, 0, 0, 1)
| style="text-align:center;" | 11/9, 11/6
| 11/9, 11/6
|-
|-
| style="text-align:center;" | "
| lu
| style="text-align:center;" | lu
| (a, b, 0, 0, -1)
| style="text-align:center;" | {a, b, 0, 0, -1}
| 12/11, 18/11
| style="text-align:center;" | 12/11, 18/11
|-
|-
| style="text-align:center;" | major
| rowspan="2" | major
| style="text-align:center;" | fifthward wa
| fifthward wa
| style="text-align:center;" | {a, b}, b &gt; 1
| (a, b), b > 1
| style="text-align:center;" | 9/8, 27/16
| 9/8, 27/16
|-
|-
| style="text-align:center;" | "
| ru
| style="text-align:center;" | ru
| (a, b, 0, -1)
| style="text-align:center;" | {a, b, 0, -1}
| 9/7, 12/7
| style="text-align:center;" | 9/7, 12/7
|}
|}


=Chord Names=
All 17edo chords can be named using ups and downs. Here are the zo, ilo and ru triads:


All 17edo chords can be named using ups and downs. Here are the blue, jade and red triads:
{| class="wikitable center-all"
 
{| class="wikitable"
|-
|-
! | color of the 3rd
! [[Color notation|Color of the 3rd]]
! | JI chord
! JI chord
! | notes as edosteps
! Notes as edosteps
! | notes of C chord
! Notes of C chord
! | written name
! Written name
! | spoken name
! Spoken name
|-
|-
| style="text-align:center;" | zo
| zo
| style="text-align:center;" | 6:7:9
| 6:7:9
| style="text-align:center;" | 0-4-10
| 0-4-10
| style="text-align:center;" | C Eb G
| C Eb G
| style="text-align:center;" | Cm
| Cm
| style="text-align:center;" | C minor
| C minor
|-
|-
| style="text-align:center;" | lova
| ilo
| style="text-align:center;" | 18:22:27
| 18:22:27
| style="text-align:center;" | 0-5-10
| 0-5-10
| style="text-align:center;" | C Ev G
| C vE G
| style="text-align:center;" | C~
| C~
| style="text-align:center;" | C mid
| C mid
|-
|-
| style="text-align:center;" | ru
| ru
| style="text-align:center;" | 14:18:27
| 14:18:21
| style="text-align:center;" | 0-6-10
| 0-6-10
| style="text-align:center;" | C E G
| C E G
| style="text-align:center;" | C
| C
| style="text-align:center;" | C major or C
| C major or C
|}
|}


0-4-9 = C Eb Gv = Cm(v5) = C minor down-five
Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).


0-5-9 = C Ev G = C~(v5) = C mid down-five
0-4-9 = C Eb vG = Cm(v5) = C minor down-five


0-6-11 = C E G^ = C(^5) = C up-five
0-5-9 = C vE vG = C~(v5) = C mid down-five
 
0-6-11 = C E ^G = C(^5) = C up-five


0-4-10-14 = C Eb G Bb = Cm7 = C minor seven
0-4-10-14 = C Eb G Bb = Cm7 = C minor seven


0-5-10-14 = C Ev G Bb = C7(~3) = C seven mid-three
0-5-10-14 = C vE G Bb = C~,7 = C mid add seven
 
0-6-10-15 = C E G vB = C,~7 = C add mid-seven
 
0-5-10-15 = C vE G vB = C~7 = C mid-seven
 
For a more complete list, see [[Ups and downs notation #Chords and chord progressions]].
 
== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp. The gamut runs D, ^D/Eb, D#/vE, E, F etc.
{{Ups and downs sharpness}}
 
=== Quarter tone notation ===
Since a sharp raises by 2 steps, 17edo can be notated using quarter-tone accidentals.
{{Sharpness-sharp2}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[24edo #Sagittal notation|24]], [[31edo #Sagittal notation|31]], and [[38edo #Sagittal notation|38]], and is a subset of the notation for [[34edo #Sagittal notation|34edo]].
 
==== Evo and Revo flavors ====
{{Sagittal chart|}}


0-6-10-15 = C E G Bv = C,~7 = C mid seven
==== Alternative Evo flavor ====
{{Sagittal chart|Alternative_Evo}}


0-5-10-15 = C Ev G Bv = C.~7 = C dot mid seven
==== Evo-SZ flavor ====
{{Sagittal chart|Evo-SZ}}


For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to the Stein-Zimmerman notation.


=Selected just intervals by error=
==== Sagittal songbook diagram ====  
The following table shows how [[Just-24|some prominent just intervals]] are represented in 17edo (ordered by absolute error).
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:


===Best direct mapping, even if inconsistent===
[[File:17edo Sagittal.png|800px]]


{| class="wikitable"
=== 3L 4s (mosh) notation ===
The notation of Neutral[7]. The generator is the perfect 3rd. Notes are denoted as {{nowrap|sLsLsLs {{=}} DEFGABCD}}, and raising and lowering by a chroma {{nowrap|(L − s)}}, 1 edostep in this instance, is denoted by ♯ and ♭.
 
{| class="wikitable center-all right-2 left-4 left-5 mw-collapsible mw-collapsed"
|-
|-
| | '''Interval, complement'''
! #
| | '''Error (abs., in [[cent|cents]])'''
! Cents
! Note
! Name
! Associated ratios
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| 0
| style="text-align:center;" | 1.324
| 0.0
| D
| Perfect 1sn
| 1/1
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| 1
| style="text-align:center;" | 2.604
| 70.6
| D#
| Augmented 1sn
| 33/32
|-
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| 2
| style="text-align:center;" | 3.927
| 141.2
| Eb
| Minor 2nd
| 12/11
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| 3
| style="text-align:center;" | 5.533
| 211.8
| E
| Major 2nd
| 9/8
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| 4
| style="text-align:center;" | 6.021
| 282.4
| Fb
| Diminished 3rd
| 32/27
|-
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| 5
| style="text-align:center;" | 6.531
| 352.9
| F
| Perfect 3rd
| 11/9, 27/22
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| 6
| style="text-align:center;" | 6.857
| 423.5
| F#
| Augmented 3rd
| 81/64
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| 7
| style="text-align:center;" | 7.855
| 494.1
| G
| Minor 4th
| 4/3
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| 8
| style="text-align:center;" | 9.461
| 564.7
| G#
| Major 4th
| 11/8
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| 9
| style="text-align:center;" | 11.555
| 635.3
| Ab
| Minor 5th
| 16/11
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| 10
| style="text-align:center;" | 12.878
| 705.9
| A
| Major 5th
| 3/2
|-
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| 11
| style="text-align:center;" | 13.388
| 776.5
| Bb
| Diminished 6th
| 128/81
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| 12
| style="text-align:center;" | 15.482
| 847.1
| B
| Perfect 6th
| 18/11, 44/27
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| 13
| style="text-align:center;" | 17.806
| 917.6
| B#
| Augmented 6th
| 27/16
|-
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| 14
| style="text-align:center;" | 19.409
| 988.2
| Cb
| Minor 7th
| 16/9
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| 15
| style="text-align:center;" | 21.734
| 1058.8
| C
| Major 7th
| 11/6
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| 16
| style="text-align:center;" | 23.828
| 1129.4
| Db
| Diminished 8ve
| 64/33
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| 17
| style="text-align:center;" | 27.755
| 1200.0
| D
| Perfect 8ve
| 2/1
|}
 
== Approximation to JI ==
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|17}}
{{Q-odd-limit intervals|17.04|apx=val|header=none|tag=none|title=15-odd-limit intervals by 17c val mapping}}
 
=== Selected 13-limit intervals ===
[[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]]
 
== Tuning by ear ==
17edo is very close to a circle of seventeen [[25/24]] chromatic semitones: (25/24)<sup>17</sup> 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.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
! [[TE error|Absolute]] (¢)
| style="text-align:center;" | 29.361
! [[TE simple badness|Relative]] (%)
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| 2.3
| style="text-align:center;" | 29.445
| {{Monzo| 27 -17 }}
| {{Mapping| 17 27 }}
| −1.24
| 1.24
| 1.76
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| 2.3.7
| style="text-align:center;" | 30.685
| 64/63, 17496/16807
| {{Mapping| 17 27 48 }}
| −3.13
| 2.85
| 4.05
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| 2.3.7.11
| style="text-align:center;" | 33.288
| 64/63, 99/98, 243/242
| {{Mapping| 17 27 48 59 }}
| −3.31
| 2.49
| 3.54
|-
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| 2.3.7.11.13
| style="text-align:center;" | 33.373
| 64/63, 78/77, 99/98, 144/143
|-
| {{Mapping| 17 27 48 59 63 }}
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| −3.00
| style="text-align:center;" | 34.612
| 2.31
| 3.28
|}
|}
* 17et is lower in relative error than any previous equal temperaments in the no-5 11- and 13-limit. The next equal temperaments doing better in these subgroups are [[41edo|41]] and [[207edo|207]], respectively.


===Patent val mapping===
=== Uniform maps ===
{{Uniform map|edo=17}}


{| class="wikitable"
=== Commas ===
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"
|-
|-
| | '''Interval, complement'''
! [[Harmonic limit|Prime<br>limit]]
| | '''Error (abs., in [[cent|cents]])'''
! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints.</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| 3
| style="text-align:center;" | 1.324
| <abbr title="134217728/129140163">(18 digits)</abbr>
| {{Monzo| 27 -17 }}
| 66.76
| Sasawa
| [[Gothic comma]]
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| 5
| style="text-align:center;" | 2.604
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.76
| Yoyo
| Dicot comma
|-
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| 5
| style="text-align:center;" | 3.927
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| 7
| style="text-align:center;" | 5.533
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| 7
| style="text-align:center;" | 6.021
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| 7
| style="text-align:center;" | 6.531
| [[245/243]]
| {{Monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| 7
| style="text-align:center;" | 6.857
| [[1728/1715]]
| {{Monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| 7
| style="text-align:center;" | 7.855
| [[17496/16807]]
| {{Monzo| 3 7 0 -5 }}
| 69.56
| Quinru
| Bleu comma
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| 7
| style="text-align:center;" | 9.461
| [[19683/19208]]
| {{Monzo| -3 9 0 -4 }}
| 42.29
| Laquadru
| Skwares comma
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| 7
| style="text-align:center;" | 11.555
| <abbr title="420175/419904">(12 digits)</abbr>
| {{Monzo| -6 -8 2 5 }}
| 1.12
| Quinzo-ayoyo
| [[Wizma]]
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| 11
| style="text-align:center;" | 12.878
| [[45/44]]
| {{Monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Cake comma
|-
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| 11
| style="text-align:center;" | 13.388
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| 11
| style="text-align:center;" | 15.482
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| 11
| style="text-align:center;" | 19.409
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma, neutral thirds comma
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| 11
| style="text-align:center;" | 29.445
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| 13
| style="text-align:center;" | 33.373
| [[40/39]]
| {{Monzo| 3 -1 1 0 0 -1 }}
| 43.83
| Thuyo
| Unintendo comma
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| 13
| style="text-align:center;" | 35.976
| [[65/64]]
| {{Monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| 13
| style="text-align:center;" | 37.300
| [[78/77]]
| {{Monzo| 1 1 0 -1 -1 1 }}
| 22.34
| Tholuru
| Negustma
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| 13
| style="text-align:center;" | 39.904
| [[144/143]]
| {{Monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| 13
| style="text-align:center;" | 41.227
| [[169/168]]
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| Thothoru
| Buzurgisma, dhanvantarisma
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| 13
| style="text-align:center;" | 42.833
| [[352/351]]
| [5 -3 0 0 1 -1⟩
| 4.93
| Thulo
| Major minthma
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| 13
| style="text-align:center;" | 46.760
| [[364/363]]
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| Tholuluzo
| Minor minthma
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| 13
| style="text-align:center;" | 48.855
| [[512/507]]
| {{Monzo| 9 -1 0 0 0 -2 }}
| 16.99
| Thuthu
| Tridecimal neutral thirds comma
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| 13
| style="text-align:center;" | 52.782
| [[1352/1331]]
|}
| {{Monzo| 3 0 0 0 -3 2 }}
 
| 27.10
[[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]]
| Bithotrilu
 
| Lovecraft comma
[[:File:17ed2-001.svg|17ed2-001.svg]]
 
=Commas=
17 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes [[val|val]] &lt; 17 27 39 48 59 63 |, cent values ​​rounded to 5 digits.)
 
{| class="wikitable"
|-
|-
! | Comma
| 13
! | Monzo
| [[2197/2187]]
! | Value (Cents)
| {{Monzo| 0 -7 0 0 0 3 }}
! | Name 1
| 7.90
! | Name 2
| Satritho
! | Name 3
| Threedie
|-
|-
| style="text-align:center;" | <span style="display: block; text-align: center;"></span>
| 23
| |<nowiki> | 27 -17 </nowiki>&gt;
| [[162/161]]
| style="text-align:right;" | 66.765
| {{Monzo| 1 4 0 -1 0 0 0 0 -1 }}
| | <span style="display: block; text-align: center;">17-Comma</span>
| 10.72
| style="text-align:center;" |
| Twethuru
| |
| Minor kirnbergerisma
|-
|-
| style="text-align:center;" | 25/24
| 23
| |<nowiki> |-3 -1 2</nowiki>&gt;
| [[208/207]]
| style="text-align:right;" | 70.762
| {{Monzo| 4 -2 0 0 0 1 0 0 -1 }}
| | Chromatic semitone
| 8.34
| style="text-align:center;" | Dicot comma
| Twethutho
| |
| Vicetone comma
|-
|-
| style="text-align:center;" | 32805/32768
| 23
| |<nowiki> | -15 8 1 </nowiki>&gt;
| [[253/252]]
| style="text-align:right;" | 1.9537
| {{Monzo| -2 -2 0 -1 1 0 0 0 1 }}
| style="text-align:center;" | Schisma
| 6.86
| style="text-align:center;" |
| Twetholoru
| style="text-align:center;" |
| Middle neutravicema
|-
|-
| style="text-align:center;" | 64/63
| 23
| |<nowiki> | 6 -2 0 -1 </nowiki>&gt;
| [[529/528]]
| style="text-align:right;" | 27.264
| {{Monzo| -4 -1 0 0 -1 0 0 0 2 }}
| style="text-align:center;" | Septimal Comma
| 3.28
| style="text-align:center;" | Archytas' Comma
| Bitwetho-alu
| style="text-align:center;" | Leipziger Komma
| Preziosisma
|-
|-
| style="text-align:center;" | 245/243
| 23
| |<nowiki> | 0 -5 1 2 </nowiki>&gt;
| [[736/729]]
| style="text-align:right;" | 14.191
| {{Monzo| 5 -6 0 0 0 0 0 0 1 }}
| style="text-align:center;" | Sensamagic
| 16.54
| style="text-align:center;" |  
| Satwetho
| style="text-align:center;" |  
| 23-limit Tenney/Cage comma (HEJI)
|}
<references group="note" />
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 ===
* [[List of 17edo rank two temperaments by badness]]
* [[List of edo-distinct 17c rank two temperaments]]
* [[List of edo-distinct 17et rank two temperaments]]
* [[List of edo-distinct 17et no-fives rank two temperaments]]
 
{| class="wikitable center-all right-3 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator
! Cents
! Associated<br>ratio
! Temperament
|-
| 1
| 2\17
| 141.18
| 13/12
| [[Bleu]] / [[progression]] (17c) / [[glacier]]
|-
|-
| style="text-align:center;" | 1728/1715
| 1
| |<nowiki> | 6 3 -1 -3 </nowiki>&gt;
| 3\17
| style="text-align:right;" | 13.074
| 211.76
| style="text-align:center;" | Orwellisma
| 8/7~9/8
| style="text-align:center;" | Orwell Comma
| [[Machine]]
| style="text-align:center;" |
|-
|-
| style="text-align:center;" |
| 1
| |<nowiki> | -6 -8 2 5 </nowiki>&gt;
| 3\17
| style="text-align:right;" | 1.1170
| 211.76
| style="text-align:center;" | Wizma
| 26/23
| style="text-align:center;" |
| [[Shoal|Shoal (trivial tuning)]]
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 99/98
| 1
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| 4\17
| style="text-align:right;" | 17.576
| 282.35
| style="text-align:center;" | Mothwellsma
| 13/11
| style="text-align:center;" |
| [[Huxley]] / [[lovecraft]] / [[subklei]] (17c)
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 896/891
| 1
| |<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
| 5\17
| style="text-align:right;" | 9.6880
| 352.94
| style="text-align:center;" | Pentacircle
| 11/9
| style="text-align:center;" |
| [[Suhajira]] / [[neutrominant]] (17c) / [[beatles]] (17c) / [[dichotic]] (17) <br>[[Hemif]] / [[mohamaq]] (17c) / [[salsa]] (17)
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 243/242
| 1
| |<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
| 6\17
| style="text-align:right;" | 7.1391
| 423.53
| style="text-align:center;" | Rastma
| 9/7
| style="text-align:center;" |
| [[Skwares]] / [[squares]] (17c) / [[sentinel]] (17) / [[sidi]] (17)
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 385/384
| 1
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| 7\17
| style="text-align:right;" | 4.5026
| 494.12
| style="text-align:center;" | Keenanisma
| 4/3
| style="text-align:center;" |
| [[Archy]] / [[supra]] / [[quasisuper]] (17c) / [[dominant (temperament)|dominant]] (17c) / [[superpyth]] (17) / [[schism]] (17)<br>[[Fiventeen]]
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | <span style="display: block; text-align: center;">525/512</span>
| 1
| | <span style="display: block; text-align: left;"><nowiki>| -9 1 2 1 </nowiki>&gt;</span>
| 8\17
| style="text-align:right;" | 43.408
| 564.71
| style="text-align:center;" | <span style="display: block; text-align: center;">Avicennma</span>
| 7/5
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| [[Lee]] / [[liese]] (17c) / [[pycnic]] (17)<br>[[Progress]] (17c)
| style="text-align:center;" |
|}
|}


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.
== Octave stretch or compression ==
17edo's approximations of harmonics 3, 7, 11, and 13 are all tempered sharp, so 17edo adapts well to slightly [[stretched and compressed tuning|compressing the octave]], if that is acceptable. [[44ed6]], [[27edt]] and [[zpi|56zpi]] are good demonstrations of this, where the octaves are flattened by about 1.5, 2.5 cents and 3 cents respectively.
 
== Scales ==
* [[Antipental blues]]: 4 3 1 2 4 3
* [https://web.archive.org/web/20140215081520/http://microtonalismo.com/proyecto-xvii Blues Peruvian]: 4 3 1 1 1 4 3
* [[Hydra]]: 3 3 1 1 2 3 2 1 1
* [[Maqam|Husayni]] Ascending: 2 2 3 3 2 2 3
* [[Otonal 17]]: 3 2 3 2 2 2 3
* [[Scorp]]: 3 2 3 1 3 2 3
* [[Screamapillar]]: 3 3 2 2 3 3 1
* sLmLs: 2 5 3 5 2
 
=== MOS scales ===
{{Main| MOS scales of 17edo }}
 
* 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) (''dedicated article: [[17edo neutral scale]]'')
* [[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;8s]] 1 3 1 1 3 1 1 3 (6\17, 1\1)
* lovecraft [[4L&nbsp;5s]] 3 1 3 1 3 1 3 1 1 (4\17, 1\1)
 
=== Well temperaments ===
* [[Secor wt17|George Secor's well temperament]]
* [[User:CritDeathX/Sam's 17-note Well Temperament|Nicolai's 17-note well temperament]]
* [[User:FloraC/Flora's 17-note well temperament|Flora's 17-note well temperament]]
 
{{Todo|expand scales list}}


=Scales=
== Instruments ==
<ul><li>[[Otonal_17|Otonal 17]]</li><li>[https://web.archive.org/web/20140215081520/http://microtonalismo.com/proyecto-xvii Blues Peruvian 17edo]</li><li>[[17edo_neutral_scale|17edo neutral scale]]</li><li>[[Scorp|Scorp]]</li><li>[[Screamapillar|Screamapillar]]</li><li>[[Hydra|Hydra]]</li><li>[[MOS_scales_of_17edo|MOS scales of 17edo]] (horograms)</li></ul>
=== Fretted String Instruments ===
* [http://chrisvaisvil.com/?p=436 17 note per octave conversion from a "standard" Stratocaster copy] - conversion by Brad Smith


[[List_of_17edo_rank_two_temperaments_by_badness|List of 17edo rank two temperaments by badness]]
[[File:17P1050829r.JPG|alt=17P1050829r.JPG|17P1050829r.JPG]]


[[List_of_edo-distinct_17c_rank_two_temperaments|List of edo-distinct 17c rank two temperaments]]
* 17edo soprano Harmony ukulele with a 3D printed fretboard - conversion by [[User:Tristanbay|Tristan Bay]]


=Music=
[[File:17edo soprano ukulele with 3D printed fretboard.jpg|frameless|640x640px]]


==Compositions==
=== Keyboards ===
<ul><li>[https://soundcloud.com/overtoneshock/demanding-two-faces-17-edo Demanding Two Faces] (xen-pop) by [https://soundcloud.com/overtoneshock Stephen Weigel]</li><li>[https://soundcloud.com/overtoneshock/where-were-you-during-the-apocalypse-17-edo Where were you at the Apocalypse?] (xen-pop) by [https://soundcloud.com/overtoneshock Stephen Weigel]</li><li>[http://home.snafu.de/djwolf/PreludeIn17tet.pdf Prelude] by [[Daniel_Wolf|Daniel Wolf]]</li><li>[http://www.georghajdu.de/heptadecatonic_drops.html Heptadecatonic Drops] by Georg Hajdu</li><li>[http://www.georghajdu.de/63.html?&L=1%20 Klangmoraste] by Georg Hajdu</li><li>[http://www.microtonalismo.com/proyecto-xvii Charles Loli 17edo] music for guitar heptadecatonic (2001) and armony inductive microtonaly (1993)</li><li>[http://microtonalismo.com/ microtonalismo] Heptadecatonic Peruvian</li><li>[[Multiverse|Multiverse]] by [[JacobBarton|Jacob Barton]]</li></ul>
[[Lumatone mapping for 17edo|Lumatone mappings for 17edo]] are available.


==Sound files==
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.
<ul><li>[http://chrisvaisvil.com/shanidar-cave-17-edo/ Shanidar Cave] is a piece in 17 edo that features an electric 17 edo guitar and what is essentially an electric tanpura which ends up making this a sort of fusion of middle eastern and Indian music in a sense. by[[Chris_Vaisvil| Chris Vaisvil]]</li><li><span style=""><span style=""><span style="">''[https://soundcloud.com/aaron-krister-johnson/puhlops-and-lauguas-big-adventure Puhlops and Laugua's Big Adventure]'' </span></span> </span> by [http://www.untwelve.org/board |Aaron Krister Johnson]</li><li><span style=""><span style=""><span style="">''[https://soundcloud.com/aaron-krister-johnson/adagio-for-margo Adagio for Margo]'' </span></span> </span> by [http://www.untwelve.org/board Aaron Krister Johnson]</li><li><span style=""><span style=""><span style="">''[http://home.comcast.net/~teamouse/Transformation.mp3 Transformation]'' </span></span> </span> by [http://home.comcast.net/%7Eteamouse/ Herman Miller]</li><li><span style=""><span style=""><span style="">''[http://www.emcollective.org/chris/items/17tet/17waltz.mp3 Waltz]'' </span></span> </span> by Christopher Bailey</li><li><span style=""><span style=""><span style="">''[http://www.emcollective.org/chris/items/17Chorus/17tppp4_lostfound.mp3 Lostfound]'' </span></span> </span> by Christopher Bailey</li><li>excerpt from <span style=""><span style=""><span style="">''[http://music.columbia.edu/~chris/sounds/balladei_exc.mp3 Balladei]'' </span></span> </span> (in 17, 29 and 12) by [http://music.columbia.edu/%7Echris/ Christopher Bailey] , CD available [http://www.amazon.com/Christopher-Bailey-Shiau-Uen-Jacob-Rhodebeck/dp/B000TDZSAU/ref=sr_1_19/105-0183009-3913216?ie=UTF8&s=music&qid=1184598550&sr=1-19 here] .</li><li><span style=""><span style=""><span style="">''[http://www.h-pi.com/mp3/17ET.mp3 Two-Part Invention in 17ET]'' </span></span></span> by Aaron Andrew Hunt</li><li>''[http://micro.soonlabel.com/gene_ward_smith/Others/McGowan/MSND-Ovtr.mp3 Overture to A Midsummer Night's Dream]'' by [http://azuma-asobi.com/ Rick McGowan]</li><li>''[http://micro.soonlabel.com/gene_ward_smith/Others/McGowan/FairyLullaby-1.mp3 Fairy Lullaby from A Midsummer Night's Dream]'' by [http://azuma-asobi.com/ Rick McGowan]</li><li><span style="">''[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2017-A%20Calamitous%20Simultaneity.mp3 A Calamitous Simultaneity]'' </span> by Igliashon Jones (17edo and 22edo)</li><li><span style="">''[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/First%20Impressions.mp3 First Impressions]'' </span> by Igliashon Jones</li><li>[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/I%20Insist.mp3 I Insist] by Igliashon Jones</li><li>''[http://archive.org/download/EtudeNo1For2PianosIn17EqualTemperament/EtudeForTwoPianosIn17edo.mp3 Etude no1 for 2 Pianos in 17 Equal Temperament]'' and ''[http://archive.org/download/EtudeNo2For2PianosIn17EqualTemperament/EtudeNo2ForTwoPianosIn17edo.mp3 Etude no2 for 2 Pianos in 17 Equal Temperament]'' Jon Lyle Smith</li><li>A number of compositions from [http://www.archive.org/details/seventeenTPP_01 seventeen-tone piano project phase I] , [[SeventeenTPPPhaseTwo|seventeen-tone piano project phase II]], [[SeventeenTPPPhaseThree|seventeen-tone piano project phase III]].</li><li>[http://www.archive.org/search.php?query=subject%3A%2217-edo%22 17edo] - 17edo-tagged compositions on www.archive.org</li><li>[http://soundclick.com/share?songid=8839072 sing a blue] by Andrew Heathwaite (composed 2008, recorded 2010). This and the other pieces below by Andrew for cümbüş, steel tubes &amp; voice.</li><li>[http://soundclick.com/share?songid=8839073 stringfinger it everybean] by Andrew Heathwaite (composed 2008, recorded 2010).</li><li>[http://soundclick.com/share?songid=8839069 cat feet belly] by Andrew Heathwaite (composed 2008, recorded 2010).</li><li><span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=a9N_T6LNDaE 17 Tone Jam]'' </span></span></span> by Marmalade Man</li><li>[http://www.youtube.com/results?search_query=17edo&search=tag youtube videos tagged with 17edo]</li><li>[http://chrisvaisvil.com/?p=784 On the Shores of the Dead Sea blog] <span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=D39MVFhb0Ho video of On the Shores of the Dead Sea]'' </span></span> </span> by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[http://chrisvaisvil.com/?p=466 Only in Disneyland 17 blog] edo guitar solo <span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/daily20110205-17et-disneyland.mp3 play Only in Disneyland]'' </span></span> </span> by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[http://chrisvaisvil.com/?p=460 17 Reasons I Hate the Blues blog] <span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/daily20110125b-17-reasons-I-hate-the-blues.mp3 play 17 Reasons I Hate the Blues]'' </span></span> </span> by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[http://chrisvaisvil.com/?p=422 Klingon Opera Overture blog] <span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/daily20110109-albino-17et.mp3 play Klingon Opera Overture]'' </span></span> </span> by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[http://chrisvaisvil.com/?p=298 Seventeen Selfless Notes blog] <span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/if20101028selfless.mp3 play Seventeen Selfless Notes]'' </span></span> </span> by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[http://chrisvaisvil.com/?p=76 17ET Jazz blog] <span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/60x60-4001-1-ver2.mp3 play 17et Jazz]'' </span></span></span> by[[Chris_Vaisvil| Chris Vaisvil]] (60 x 60 winner)</li><li>[http://chrisvaisvil.com/?p=742 17 Pink Tuxedos blog] <span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/daily20110416-17_pink-tuxes.mp3 play 17 Pink Tuxedos]'' </span></span></span> by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[http://chrisvaisvil.com/?p=706 Devil in the Deep Blue Sea blog] <span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/daily20110401-17-2-devil_in_the_deep_blue_sea.mp3 play Devil in the Deep Blue sea]'' </span></span></span> blues collaboration between The Two Regs (vocals / lyrics) and Norm Harris (percussion) and Chris Vaisvil (17 note per octave electric guitar and fretless bass).</li><li>[http://chrisvaisvil.com/?p=585 Seventeen Years in the Sixties blog] <span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/daily20110318-seventeen-years-in-the-sixties.mp3 play Seventeen Years in the Sixties]'' </span></span></span> by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[http://chrisvaisvil.com/?p=699 CT Scan blog] <span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=ZEEuytYwtbo play video CT Scan]'' </span></span></span> by Chris Vaisvil</li><li>[http://chrisvaisvil.com/?p=470 Fish and a Grenade blog] <span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/daily20110208-17-a-fish-and-a-grenade.mp3 play Fish and a Grenade]'' </span></span></span> (parental advisory language) by Chris Vaisvil</li><li>[http://chrisvaisvil.com/?p=550 Seventeen Unsteady Hands blog] <span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=rAKHCqBNhfc play video of performance of Seventeen Unsteady Hands]'' </span></span></span> by Chris Vaisvil</li><li>[http://chrisvaisvil.com/?p=38 The Pond blog] by Chris Vaisvil and <span style=""><span style=""><span style="">''[http://www.youtube.com/watch?v=25sC3_uheyA video version of The Pond with Japanese Garden]'' </span></span></span></li><li><span style="">[http://chrisvaisvil.com/?p=305 Graveyard blog] </span><span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/if20101028-graveyard.mp3 play Graveyard]'' </span></span> </span> by Chris Vaisvil</li><li><span style=""><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/daily20110713_q49_17_brass_voice_choirs.mp3 For Brass and Voice Choirs in 17 edo]'' </span></span></span> by [http://chrisvaisvil.com/ Chris Vaisvil] [http://chrisvaisvil.com/?p=1039 more on the composition]</li><li><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/daily20111015-fungi.mp3 And I Became One With My Per Fungi]'' </span></span> by [http://chrisvaisvil.com/ Chris Vaisvil] =&gt; [http://chrisvaisvil.com/?p=1439 more on composition]</li><li><span style=""><span style="">''[http://micro.soonlabel.com/17-ET/20111014-STE-002-counter-intuitive.mp3 counterintuitive (17 edo guitar solo)]'' </span></span> by [http://www.chrisvaisvil.com/ Chris Vaisvil] =&gt; [http://chrisvaisvil.com/?p=1427 more on composition]</li><li>[http://micro.soonlabel.com/gene_ward_smith/Others/Sanchez/No_Love_by_Gregory_Sanchez_1_on_SoundCloud___Hear_the_world_s_sounds.mp3 No Love] by [https://soundcloud.com/gregory-sanchez-2/no-love Gregory Sanchez]</li><li>[http://webzoom.freewebs.com/ralphjarzombek/micro7(17tet).mp3 Micro7] by Ralph Jarzombek</li><li>[http://micro.soonlabel.com/17-ET/20161125_linn_speactral_17.MP3 Flying Into O'Hare] by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[https://drive.google.com/drive/folders/0BwsXD8q2VCYUM0tRdDNhVzNhcHM Helas, pitié] by Alex Ness (in 17edo with stretched octaves)</li></ul>


=Instruments=
[[File:Strisoboard_piano2a_s.jpg|frameless]]
<ul><li>'''[http://www.microtonalismo.com/proyecto-xvii Guitar Heptadecatonic from Peruvian - Charles Loli and Antonio Huamani]'''


<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
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]]:
External image: http://sphotos.ak.fbcdn.net/hphotos-ak-snc4/hs883.snc4/71639_167001659983806_100000219181856_601995_1526184_n.jpg<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]


</li></ul>
* [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)


<ul><li>'''[http://www.microtonalismo.com/proyecto-xvii Bass Heptadecatonic from Peruvian - Charles Loli and Antonio Huamani]'''
== Music ==
{{Main| 17edo/Music }}
{{Catrel|17edo tracks}}


<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
; [https://www.youtube.com/playlist?list=PLWB50RFxjvduT6F1Mwu0CmPa41LBRdXf5 YouTube playlist of 17edo pieces]
External image: http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs382.ash2/66019_167001006650538_100000219181856_601987_48585_n.jpg<br>
; [https://www.youtube.com/results?search_query=17edo&search=tag YouTube videos tagged with 17edo]
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]


</li></ul>
; Compositions from the [[SeventeenTonePianoProject|Seventeen Tone Piano Project]]
* [https://www.archive.org/details/seventeenTPP_01 seventeen-tone piano project phase I]
* [[SeventeenTPPPhaseTwo|Seventeen-tone piano project phase II]]
* [[SeventeenTPPPhaseThree|Seventeen-tone piano project phase III]]
<!-- currently redundant:
* [http://www.archive.org/search.php?query=subject%3A%2217-edo%22 17edo] - 17edo-tagged compositions on www.archive.org
-->


<ul><li>[http://chrisvaisvil.com/?p=436 17 note per octave conversion from a "standard" stratocastor copy] - conversion by Brad Smith</li></ul>
== Introductory Materials ==
* [[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.
* [[17edo tetrachords]]
* [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú] {{forbidden}}


[[File:17P1050829r.JPG|alt=17P1050829r.JPG|17P1050829r.JPG]]     
[[Category:Teentuning]]
[[Category:17edo]]
[[Category:edo]]
[[Category:listen]]
[[Category:prime_edo]]
[[Category:pythagorean]]
[[Category:teentuning]]

Latest revision as of 09:14, 21 May 2026

← 16edo 17edo 18edo →
Prime factorization 17 (prime)
Step size 70.5882 ¢ 
Fifth 10\17 (705.882 ¢)
(semiconvergent)
Semitones (A1:m2) 2:1 (141.2 ¢ : 70.59 ¢)
Consistency limit 3
Distinct consistency limit 3

17 equal divisions of the octave (abbreviated 17edo or 17ed2), also called 17-tone equal temperament (17tet) or 17 equal temperament (17et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 17 equal parts of about 70.6 ¢ each. Each step represents a frequency ratio of 21/17, or the 17th root of 2.

English Wikipedia has an article on:

Theory

17edo is the next smallest edo to have a diatonic 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 hardness of Pythagorean tuning rather than mellowing it out as in meantone. It completely misses harmonic 5, with 5/4 and 6/5 both being about halfway between its steps, but it approximates harmonics 7, 11, 13, and 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 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 superpythagorean 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 extension.

As a means of extending harmony

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 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.

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

Odd harmonics

Approximation of odd harmonics in 17edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +3.9 -33.4 +19.4 +7.9 +13.4 +6.5 -29.4 -34.4 -15.2 +23.3 +7.0
Relative (%) +5.6 -47.3 +27.5 +11.1 +19.0 +9.3 -41.7 -48.7 -21.5 +33.1 +9.9
Steps
(reduced) 		27
(10) 		39
(5) 		48
(14) 		54
(3) 		59
(8) 		63
(12) 		66
(15) 		69
(1) 		72
(4) 		75
(7) 		77
(9)
Approximation of odd harmonics in 17edo (continued)
Harmonic 25 27 29 31 33 35 37 39 41 43 45 47
Error Absolute (¢) +3.8 +11.8 +29.2 -15.6 +17.3 -14.0 +31.0 +10.5 -5.5 -17.4 -25.5 -30.2
Relative (%) +5.4 +16.7 +41.4 -22.1 +24.5 -19.8 +43.9 +14.8 -7.8 -24.7 -36.2 -42.8
Steps
(reduced) 		79
(11) 		81
(13) 		83
(15) 		84
(16) 		86
(1) 		87
(2) 		89
(4) 		90
(5) 		91
(6) 		92
(7) 		93
(8) 		94
(9)

Subsets and supersets

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.

Intervals

See also: 17edo solfege
# Cents Approximate ratios[note 1] Circle-of-fifths notation[note 2] Ups and downs notation
(EUs: vvA1 and ^d2) 		SKULO notation (U = 1)
0 0.0 1/1 Unison D unison P1 D unison P1 D
1 70.6 24/23, 25/24, 26/25, 27/26, 28/27 Minor 2nd
(Semiaugmented 1sn) 		Eb
(Dt) 		up unison,
minor 2nd 		^1, m2 Eb uber unison,
minor 2nd 		U1, m2 UD, Eb
2 141.2 12/11, 13/12, 14/13, 25/23 Augmented 1sn
(Neutral 2nd) 		D#
(Ed) 		augmented 1sn,
mid 2nd 		A1, ~2 vE neutral 2nd N2 UEb, uE
3 211.8 8/7, 9/8, 17/15, 25/22, 26/23 Major 2nd E major 2nd M2 E major 2nd M2 E
4 282.4 7/6, 13/11, 20/17 Minor 3rd F minor 3rd m3 F minor 3rd m3 F
5 352.9 11/9, 27/22, 16/13, 39/32 Diminished 4th
(Neutral 3rd) 		Gb
(Ft) 		mid 3rd ~3 ^F neutral 3rd N3 UF, uF#
6 423.5 9/7, 14/11, 23/18, 32/25, 51/40 Major 3rd
(Semidiminished 4th) 		F#
(Gd) 		major 3rd M3 F# major 3rd M3 F#
7 494.1 4/3, 21/16, 85/64 Perfect 4th G perfect 4th P4 G perfect 4th P4 G
8 564.7 11/8, 18/13, 25/18, 32/23 Diminished 5th
(Semiaugmented 4th) 		Ab
(Gt) 		mid 4th,
diminished 5th 		~4, d5 ^G, Ab uber 4th/
neutral 4th 		U4/N4 UG
9 635.3 13/9, 16/11, 23/16, 36/25 Augmented 4th
(Semidiminished 5th) 		G#
(Ad) 		augmented 4th,
mid 5th 		A4, ~5 G#, vA unter 5th/
neutral 5th 		u5/N5 uA
10 705.9 3/2, 32/21, 128/85 Perfect 5th A perfect 5th P5 A perfect 5th P5 A
11 776.5 11/7, 14/9, 25/16, 36/23, 80/51 Minor 6th
(Semiaugmented 5th) 		Bb
(At) 		minor 6th m6 Bb minor 6th m6 Bb
12 847.1 13/8, 18/11, 44/27, 64/39 Augmented 5th
(Neutral 6th) 		A#
(Bd) 		mid 6th ~6 vB neutral 6th N6 UBb, uB
13 917.6 12/7, 17/10, 22/13 Major 6th B major 6th M6 B major 6th M6 B
14 988.2 7/4, 16/9, 23/13, 30/17, 44/25 Minor 7th C minor 7th m7 C minor 7th m7 C
15 1058.8 11/6, 13/7, 24/13, 46/25 Diminished 8ve
(Neutral 7th) 		Db
(Ct) 		mid 7th ~7 ^C neutral 7th N7 UC, uC#
16 1129.4 23/12, 25/13, 27/14, 48/25, 52/27 Major 7th
(Semidiminished 8ve) 		C#
(Dd) 		major 7th,
down 8ve 		M7, v8 C# major 7th,
unter octave 		M7, u8 C#, uD
17 1200.0 2/1 Octave D octave P8 D octave P8 D
  1. ↑ Based on treating 17edo as a 2.3.25.7.11.13.85.23 subgroup temperament; other approaches are also possible.
  2. ↑ 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.

Interval quality and chord names in color notation

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

Quality Color Monzo format Examples
minor zo (a, b, 0, 1) 7/6, 7/4
fourthward wa (a, b), b < -1 32/27, 16/9
mid ilo (a, b, 0, 0, 1) 11/9, 11/6
lu (a, b, 0, 0, -1) 12/11, 18/11
major fifthward wa (a, b), b > 1 9/8, 27/16
ru (a, b, 0, -1) 9/7, 12/7

All 17edo chords can be named using ups and downs. Here are the zo, ilo and ru triads:

Color of the 3rd JI chord Notes as edosteps Notes of C chord Written name Spoken name
zo 6:7:9 0-4-10 C Eb G Cm C minor
ilo 18:22:27 0-5-10 C vE G C~ C mid
ru 14:18:21 0-6-10 C E G C C major or C

Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

0-4-9 = C Eb vG = Cm(v5) = C minor down-five

0-5-9 = C vE vG = C~(v5) = C mid down-five

0-6-11 = C E ^G = C(^5) = C up-five

0-4-10-14 = C Eb G Bb = Cm7 = C minor seven

0-5-10-14 = C vE G Bb = C~,7 = C mid add seven

0-6-10-15 = C E G vB = C,~7 = C add mid-seven

0-5-10-15 = C vE G vB = C~7 = C mid-seven

For a more complete list, see Ups and downs notation #Chords and chord progressions.

Notation

Ups and downs notation

Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp. The gamut runs D, ^D/Eb, D#/vE, E, F etc.

Step offset 0 1 2 3 4 5
Sharp symbol   
 
  
   
  
   
Flat symbol
 
  
   
  
   

Quarter tone notation

Since a sharp raises by 2 steps, 17edo can be notated using quarter-tone accidentals.

Step offset −4 −3 −2 −1 0 +1 +2 +3 +4
Symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

Sagittal notation

This notation uses the same sagittal sequence as edos 24, 31, and 38, and is a subset of the notation for 34edo.

Evo and Revo flavors

17-EDO__Sagittal.svg

Alternative Evo flavor

17-EDO_Alternative_Evo_Sagittal.svg

Evo-SZ flavor

17-EDO_Evo-SZ_Sagittal.svg

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to the Stein-Zimmerman notation.

Sagittal songbook diagram

From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 17edo in the Revo flavor of Sagittal:

3L 4s (mosh) notation

The notation of Neutral[7]. The generator is the perfect 3rd. Notes are denoted as sLsLsLs = DEFGABCD, and raising and lowering by a chroma (L − s), 1 edostep in this instance, is denoted by ♯ and ♭.

# Cents Note Name Associated ratios
0 0.0 D Perfect 1sn 1/1
1 70.6 D# Augmented 1sn 33/32
2 141.2 Eb Minor 2nd 12/11
3 211.8 E Major 2nd 9/8
4 282.4 Fb Diminished 3rd 32/27
5 352.9 F Perfect 3rd 11/9, 27/22
6 423.5 F# Augmented 3rd 81/64
7 494.1 G Minor 4th 4/3
8 564.7 G# Major 4th 11/8
9 635.3 Ab Minor 5th 16/11
10 705.9 A Major 5th 3/2
11 776.5 Bb Diminished 6th 128/81
12 847.1 B Perfect 6th 18/11, 44/27
13 917.6 B# Augmented 6th 27/16
14 988.2 Cb Minor 7th 16/9
15 1058.8 C Major 7th 11/6
16 1129.4 Db Diminished 8ve 64/33
17 1200.0 D Perfect 8ve 2/1

Approximation to JI

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 17edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 17edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/9, 18/13 1.324 1.9
13/12, 24/13 2.604 3.7
3/2, 4/3 3.927 5.6
11/9, 18/11 5.533 7.8
11/7, 14/11 6.021 8.5
13/8, 16/13 6.531 9.3
13/11, 22/13 6.857 9.7
9/8, 16/9 7.855 11.1
11/6, 12/11 9.461 13.4
9/7, 14/9 11.555 16.4
13/7, 14/13 12.878 18.2
11/8, 16/11 13.388 19.0
7/6, 12/7 15.482 21.9
7/5, 10/7 17.806 25.2
7/4, 8/7 19.409 27.5
15/14, 28/15 21.734 30.8
11/10, 20/11 23.828 33.8
15/11, 22/15 27.755 39.3
9/5, 10/9 29.361 41.6
15/8, 16/15 29.445 41.7
13/10, 20/13 30.685 43.5
5/3, 6/5 33.288 47.2
5/4, 8/5 33.373 47.3
15/13, 26/15 34.612 49.0
15-odd-limit intervals in 17edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/9, 18/13 1.324 1.9
13/12, 24/13 2.604 3.7
3/2, 4/3 3.927 5.6
11/9, 18/11 5.533 7.8
11/7, 14/11 6.021 8.5
13/8, 16/13 6.531 9.3
13/11, 22/13 6.857 9.7
9/8, 16/9 7.855 11.1
11/6, 12/11 9.461 13.4
9/7, 14/9 11.555 16.4
13/7, 14/13 12.878 18.2
11/8, 16/11 13.388 19.0
7/6, 12/7 15.482 21.9
7/4, 8/7 19.409 27.5
15/8, 16/15 29.445 41.7
5/4, 8/5 33.373 47.3
15/13, 26/15 35.976 51.0
5/3, 6/5 37.300 52.8
13/10, 20/13 39.904 56.5
9/5, 10/9 41.227 58.4
15/11, 22/15 42.833 60.7
11/10, 20/11 46.760 66.2
15/14, 28/15 48.855 69.2
7/5, 10/7 52.782 74.8
15-odd-limit intervals by 17c val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/9, 18/13 1.324 1.9
13/12, 24/13 2.604 3.7
3/2, 4/3 3.927 5.6
11/9, 18/11 5.533 7.8
11/7, 14/11 6.021 8.5
13/8, 16/13 6.531 9.3
13/11, 22/13 6.857 9.7
9/8, 16/9 7.855 11.1
11/6, 12/11 9.461 13.4
9/7, 14/9 11.555 16.4
13/7, 14/13 12.878 18.2
11/8, 16/11 13.388 19.0
7/6, 12/7 15.482 21.9
7/5, 10/7 17.806 25.2
7/4, 8/7 19.409 27.5
15/14, 28/15 21.734 30.8
11/10, 20/11 23.828 33.8
15/11, 22/15 27.755 39.3
9/5, 10/9 29.361 41.6
13/10, 20/13 30.685 43.5
5/3, 6/5 33.288 47.2
15/13, 26/15 34.612 49.0
5/4, 8/5 37.216 52.7
15/8, 16/15 41.143 58.3

Selected 13-limit intervals

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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.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [27 -17⟩ [⟨17 27]] −1.24 1.24 1.76
2.3.7 64/63, 17496/16807 [⟨17 27 48]] −3.13 2.85 4.05
2.3.7.11 64/63, 99/98, 243/242 [⟨17 27 48 59]] −3.31 2.49 3.54
2.3.7.11.13 64/63, 78/77, 99/98, 144/143 [⟨17 27 48 59 63]] −3.00 2.31 3.28
  • 17et is lower in relative error than any previous equal temperaments in the no-5 11- and 13-limit. The next equal temperaments doing better in these subgroups are 41 and 207, respectively.

Uniform maps

13-limit uniform maps between 16.8 and 17.2
Min. size Max. size Wart notation Map
16.7196 16.8899 17def ⟨17 27 39 47 58 62]
16.8899 16.9103 17de ⟨17 27 39 47 58 63]
16.9103 16.9198 17d ⟨17 27 39 47 59 63]
16.9198 17.0117 17 ⟨17 27 39 48 59 63]
17.0117 17.1601 17c ⟨17 27 40 48 59 63]
17.1601 17.1994 17cff ⟨17 27 40 48 59 64]
17.1994 17.2760 17ceeff ⟨17 27 40 48 60 64]

Commas

17et tempers out the following commas. (Note: This assumes patent val ⟨17 27 39 48 59 63 69 72 77], cent values ​​rounded to 1/100 of a cent.)

Prime
limit 		Ratio[note 1] Monzo Cents Color name Name
3 (18 digits) [27 -17⟩ 66.76 Sasawa Gothic comma
5 25/24 [-3 -1 2⟩ 70.76 Yoyo Dicot comma
5 32805/32768 [-15 8 1⟩ 1.95 Layo Schisma
7 64/63 [6 -2 0 -1⟩ 27.26 Ru Septimal comma
7 525/512 [-9 1 2 1⟩ 43.41 Lazoyoyo Avicennma
7 245/243 [0 -5 1 2⟩ 14.19 Zozoyo Sensamagic comma
7 1728/1715 [6 3 -1 -3⟩ 13.07 Triru-agu Orwellisma
7 17496/16807 [3 7 0 -5⟩ 69.56 Quinru Bleu comma
7 19683/19208 [-3 9 0 -4⟩ 42.29 Laquadru Skwares comma
7 (12 digits) [-6 -8 2 5⟩ 1.12 Quinzo-ayoyo Wizma
11 45/44 [-2 2 1 0 -1⟩ 38.91 Luyo Cake comma
11 99/98 [-1 2 0 -2 1⟩ 17.58 Loruru Mothwellsma
11 896/891 [7 -4 0 1 -1⟩ 9.69 Saluzo Pentacircle comma
11 243/242 [-1 5 0 0 -2⟩ 7.14 Lulu Rastma, neutral thirds comma
11 385/384 [-7 -1 1 1 1⟩ 4.50 Lozoyo Keenanisma
13 40/39 [3 -1 1 0 0 -1⟩ 43.83 Thuyo Unintendo comma
13 65/64 [-6 0 1 0 0 1⟩ 26.84 Thoyo Wilsorma
13 78/77 [1 1 0 -1 -1 1⟩ 22.34 Tholuru Negustma
13 144/143 [4 2 0 0 -1 -1⟩ 12.06 Thulu Grossma
13 169/168 [-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 [2 -1 0 1 -2 1⟩ 4.76 Tholuluzo Minor minthma
13 512/507 [9 -1 0 0 0 -2⟩ 16.99 Thuthu Tridecimal neutral thirds comma
13 1352/1331 [3 0 0 0 -3 2⟩ 27.10 Bithotrilu Lovecraft comma
13 2197/2187 [0 -7 0 0 0 3⟩ 7.90 Satritho Threedie
23 162/161 [1 4 0 -1 0 0 0 0 -1⟩ 10.72 Twethuru Minor kirnbergerisma
23 208/207 [4 -2 0 0 0 1 0 0 -1⟩ 8.34 Twethutho Vicetone comma
23 253/252 [-2 -2 0 -1 1 0 0 0 1⟩ 6.86 Twetholoru Middle neutravicema
23 529/528 [-4 -1 0 0 -1 0 0 0 2⟩ 3.28 Bitwetho-alu Preziosisma
23 736/729 [5 -6 0 0 0 0 0 0 1⟩ 16.54 Satwetho 23-limit Tenney/Cage comma (HEJI)
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

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

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator Cents Associated
ratio 		Temperament
1 2\17 141.18 13/12 Bleu / progression (17c) / glacier
1 3\17 211.76 8/7~9/8 Machine
1 3\17 211.76 26/23 Shoal (trivial tuning)
1 4\17 282.35 13/11 Huxley / lovecraft / subklei (17c)
1 5\17 352.94 11/9 Suhajira / neutrominant (17c) / beatles (17c) / dichotic (17)
Hemif / mohamaq (17c) / salsa (17)
1 6\17 423.53 9/7 Skwares / squares (17c) / sentinel (17) / sidi (17)
1 7\17 494.12 4/3 Archy / supra / quasisuper (17c) / dominant (17c) / superpyth (17) / schism (17)
Fiventeen
1 8\17 564.71 7/5 Lee / liese (17c) / pycnic (17)
Progress (17c)

Octave stretch or compression

17edo's approximations of harmonics 3, 7, 11, and 13 are all tempered sharp, so 17edo adapts well to slightly compressing the octave, if that is acceptable. 44ed6, 27edt and 56zpi are good demonstrations of this, where the octaves are flattened by about 1.5, 2.5 cents and 3 cents respectively.

Scales

MOS scales

Main article: MOS scales of 17edo

Well temperaments

Instruments

Fretted String Instruments

17P1050829r.JPG

  • 17edo soprano Harmony ukulele with a 3D printed fretboard - conversion by Tristan Bay

Keyboards

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 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:

Music

Main article: 17edo/Music
See also: Category:17edo tracks
YouTube playlist of 17edo pieces
YouTube videos tagged with 17edo
Compositions from the Seventeen Tone Piano Project

Introductory Materials

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