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}}
}}
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|17}}
{{ED intro}}
{{Wikipedia|17 equal temperament}}
{{Wikipedia|17 equal temperament}}


== Theory ==
== Theory ==
17edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to an expressive [[diatonic scale]]. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it 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 these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].
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.  


Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.  
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 standard major triad is quite dissonant as the major third is closer to [[9/7]] than the traditional [[5/4]]. Instead, the tonic chords of 17edo could be considered to be the tetrad 6:7:8:9 and its utonal inversion (representing 14:16:18:21 as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of Steely Dan fame). These are realized in 17edo as 0-4-7-10 and 0-3-6-10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0-3-7-10 (which is a sus4 with added second, or sus2 with added fourth). These three chords comprise the three ways to divide the 17edo perfect fifth into two whole tones and one subminor third. Chromatic alterations of them also exist, for example, the 0-3-7-10 chord may be altered to 0-2-7-10 (which approximates 12:13:16:18) or 0-3-8-10 (which approximates 8:9:11:12). The 0-3-8-10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0-3-6-10.  
=== 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.


Another construction of septimal chords involves 4:7:12 and its inversion 7:12:21. These triads span a twelfth, realized in 17edo as 0-14-27 and 0-13-27, respectively. To this we may add 0-12-14-27, representing 8:13:14:24, or 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.


In the no-5 [[13-odd-limit]], 17edo maintains the smallest relative error{{clarify}} of any edo until [[166edo]]. <!-- explain relative error in an odd limit -->
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 ===
=== Odd harmonics ===
{{Harmonics in equal|17|intervals=odd}}
{{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)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. [[34edo]], which doubles it, provides a good correction to harmonics 5 and 17.
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 ==
== Intervals ==
{{See also|17edo solfege}}
{{See also| 17edo solfege }}
 
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! &#35;
! #
! Cents
! Cents
! Approximate ratios<ref group="note">{{sg|limit=2.3.25.7.11.13.85.23&nbsp;subgroup}}</ref>
! Approximate ratios<ref group="note">{{sg|limit=2.3.25.7.11.13.85.23&nbsp;subgroup}}</ref>
! colspan="2" | [[Circle-of-fifths notation]]<sup>&dagger;</sup>
! 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>
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and ^d2)
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and ^d2)
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U {{=}} 1)}}
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U {{=}} 1)}}
! colspan="2" | [[3L 4s]] notation
|-
|-
| 0
| 0
| 0.00
| 0.0
| 1/1
| [[1/1]]
| Unison
| Unison
| D
| D
Line 48: Line 54:
| unison
| unison
| P1
| P1
| D
| unison
| D
| D
|-
|-
| 1
| 1
| 70.59
| 70.6
| [[24/23]], [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| [[24/23]], [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| Minor 2nd<br />(Semiaugmented 1sn)
| Minor 2nd<br>(Semiaugmented 1sn)
| Eb<br />(Dt)
| Eb<br>(Dt)
| up unison, <br />minor 2nd
| up unison, <br>minor 2nd
| ^1, m2
| ^1, m2
| Eb
| Eb
| uber unison, <br />minor 2nd
| uber unison, <br>minor 2nd
| U1, m2
| U1, m2
| UD, Eb
| UD, Eb
| augmented 1sn
| D#
|-
|-
| 2
| 2
| 141.18
| 141.2
| [[12/11]], [[13/12]], [[14/13]], [[25/23]]
| [[12/11]], [[13/12]], [[14/13]], [[25/23]]
| Augmented 1sn<br />(Neutral 2nd)
| Augmented 1sn<br>(Neutral 2nd)
| D#<br />(Ed)
| D#<br>(Ed)
| augmented 1sn, <br />mid 2nd
| augmented 1sn, <br>mid 2nd
| A1, ~2
| A1, ~2
| vE
| vE
Line 77: Line 79:
| N2
| N2
| UEb, uE
| UEb, uE
| minor 2nd
| Eb
|-
|-
| 3
| 3
| 211.76
| 211.8
| [[8/7]], [[9/8]], [[17/15]], [[25/22]], [[26/23]]
| [[8/7]], [[9/8]], [[17/15]], [[25/22]], [[26/23]]
| Major 2nd
| Major 2nd
Line 90: Line 90:
| major 2nd
| major 2nd
| M2
| M2
| E
| major 2nd
| E
| E
|-
|-
| 4
| 4
| 282.35
| 282.4
| [[7/6]], [[13/11]], [[20/17]]
| [[7/6]], [[13/11]], [[20/17]]
| Minor 3rd
| Minor 3rd
Line 105: Line 103:
| m3
| m3
| F
| F
| diminished 3rd
| Fb
|-
|-
| 5
| 5
| 352.94
| 352.9
| [[11/9]], [[27/22]], [[16/13]], [[39/32]]
| [[11/9]], [[27/22]], [[16/13]], [[39/32]]
| Diminished 4th<br />(Neutral 3rd)
| Diminished 4th<br>(Neutral 3rd)
| Gb<br />(Ft)
| Gb<br>(Ft)
| mid 3rd
| mid 3rd
| ~3
| ~3
Line 119: Line 115:
| N3
| N3
| UF, uF#
| UF, uF#
| perfect 3rd
| F
|-
|-
| 6
| 6
| 423.53
| 423.5
| [[9/7]], [[14/11]], [[23/18]], [[32/25]], [[51/40]]
| [[9/7]], [[14/11]], [[23/18]], [[32/25]], [[51/40]]
| Major 3rd<br />(Semidiminished 4th)
| Major 3rd<br>(Semidiminished 4th)
| F#<br />(Gd)
| F#<br>(Gd)
| major 3rd
| major 3rd
| M3
| M3
Line 132: Line 126:
| major 3rd
| major 3rd
| M3
| M3
| F#
| augmented 3rd
| F#
| F#
|-
|-
| 7
| 7
| 494.12
| 494.1
| [[4/3]], [[21/16]], [[85/64]]
| [[4/3]], [[21/16]], [[85/64]]
| Perfect 4th
| Perfect 4th
Line 146: Line 138:
| perfect 4th
| perfect 4th
| P4
| P4
| G
| minor 4th
| G
| G
|-
|-
| 8
| 8
| 564.71
| 564.7
| [[11/8]], [[18/13]], [[25/18]], [[32/23]]
| [[11/8]], [[18/13]], [[25/18]], [[32/23]]
| Diminished 5th<br />(Semiaugmented 4th)
| Diminished 5th<br>(Semiaugmented 4th)
| Ab<br />(Gt)
| Ab<br>(Gt)
| mid 4th, <br />diminished 5th
| mid 4th, <br>diminished 5th
| ~4, d5
| ~4, d5
| ^G, Ab
| ^G, Ab
| uber 4th/<br />neutral 4th
| uber 4th/<br>neutral 4th
| U4/N4
| U4/N4
| UG
| UG
| major 4th
| G#
|-
|-
| 9
| 9
| 635.29
| 635.3
| [[13/9]], [[16/11]], [[23/16]], [[36/25]]
| [[13/9]], [[16/11]], [[23/16]], [[36/25]]
| Augmented 4th<br />(Semidiminished 5th)
| Augmented 4th<br>(Semidiminished 5th)
| G#<br />(Ad)
| G#<br>(Ad)
| augmented 4th, <br />mid 5th
| augmented 4th, <br>mid 5th
| A4, ~5
| A4, ~5
| G#, vA
| G#, vA
| unter 5th/<br />neutral 5th
| unter 5th/<br>neutral 5th
| u5/N5
| u5/N5
| uA
| uA
| minor 5th
| Ab
|-
|-
| 10
| 10
| 705.88
| 705.9
| [[3/2]], [[32/21]], [[128/85]]
| [[3/2]], [[32/21]], [[128/85]]
| Perfect 5th
| Perfect 5th
Line 188: Line 174:
| perfect 5th
| perfect 5th
| P5
| P5
| A
| major 5th
| A
| A
|-
|-
| 11
| 11
| 776.47
| 776.5
| [[11/7]], [[14/9]], [[25/16]], [[36/23]], [[80/51]]
| [[11/7]], [[14/9]], [[25/16]], [[36/23]], [[80/51]]
| Minor 6th<br />(Semiaugmented 5th)
| Minor 6th<br>(Semiaugmented 5th)
| Bb<br />(At)
| Bb<br>(At)
| minor 6th
| minor 6th
| m6
| m6
Line 202: Line 186:
| minor 6th
| minor 6th
| m6
| m6
| Bb
| diminished 6th
| Bb
| Bb
|-
|-
| 12
| 12
| 847.06
| 847.1
| [[13/8]], [[18/11]], [[44/27]], [[64/39]]
| [[13/8]], [[18/11]], [[44/27]], [[64/39]]
| Augmented 5th<br />(Neutral 6th)
| Augmented 5th<br>(Neutral 6th)
| A#<br />(Bd)
| A#<br>(Bd)
| mid 6th
| mid 6th
| ~6
| ~6
Line 217: Line 199:
| N6
| N6
| UBb, uB
| UBb, uB
| perfect 6th
| B
|-
|-
| 13
| 13
| 917.65
| 917.6
| [[12/7]], [[17/10]], [[22/13]]
| [[12/7]], [[17/10]], [[22/13]]
| Major 6th
| Major 6th
Line 231: Line 211:
| M6
| M6
| B
| B
| augmented 6th
| B#
|-
|-
| 14
| 14
| 988.24
| 988.2
| [[7/4]], [[16/9]], [[23/13]], [[30/17]], [[44/25]]
| [[7/4]], [[16/9]], [[23/13]], [[30/17]], [[44/25]]
| Minor 7th
| Minor 7th
Line 245: Line 223:
| m7
| m7
| C
| C
| minor 7th
| Cb
|-
|-
| 15
| 15
| 1058.82
| 1058.8
| [[11/6]], [[13/7]], [[24/13]], [[46/25]]
| [[11/6]], [[13/7]], [[24/13]], [[46/25]]
| Diminished 8ve<br />(Neutral 7th)
| Diminished 8ve<br>(Neutral 7th)
| Db<br />(Ct)
| Db<br>(Ct)
| mid 7th
| mid 7th
| ~7
| ~7
Line 259: Line 235:
| N7
| N7
| UC, uC#
| UC, uC#
| major 7th
| C
|-
|-
| 16
| 16
| 1129.41
| 1129.4
| [[23/12]], [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| [[23/12]], [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| Major 7th<br />(Semidiminished 8ve)
| Major 7th<br>(Semidiminished 8ve)
| C#<br />(Dd)
| C#<br>(Dd)
| major 7th,<br />down 8ve
| major 7th,<br>down 8ve
| M7, v8
| M7, v8
| C#
| C#
| major 7th,<br />unter octave
| major 7th,<br>unter octave
| M7, u8
| M7, u8
| C#, uD
| C#, uD
| diminished 8ve
| Db
|-
|-
| 17
| 17
| 1200.00
| 1200.0
| [[2/1]]
| [[2/1]]
| Octave
| Octave
Line 286: Line 258:
| octave
| octave
| P8
| P8
| D
| octave
| D
| D
|}
|}
<sup>&dagger;</sup> 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.
<references group="note" />


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 299: Line 269:
! Quality
! Quality
! Color
! Color
! Monzo Format
! Monzo format
! Examples
! Examples
|-
|-
Line 308: Line 278:
|-
|-
| fourthward wa
| fourthward wa
| (a, b), b &lt; -1
| (a, b), b < -1
| 32/27, 16/9
| 32/27, 16/9
|-
|-
Line 322: Line 292:
| rowspan="2" | major
| rowspan="2" | major
| fifthward wa
| fifthward wa
| (a, b), b &gt; 1
| (a, b), b > 1
| 9/8, 27/16
| 9/8, 27/16
|-
|-
Line 334: Line 304:
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! [[Kite's color notation|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
|-
|-
| zo
| zo
Line 379: Line 349:
0-5-10-15 = C vE G vB = C~7 = C 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]].
For a more complete list, see [[Ups and downs notation #Chords and chord progressions]].


== Notation ==
== 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 ===
=== Quarter tone notation ===
Since a sharp raises by 2 steps, 17edo can be notated using quarter-tone accidentals.
Since a sharp raises by 2 steps, 17edo can be notated using quarter-tone accidentals.
 
{{Sharpness-sharp2}}
{{sharpness-sharp2}}


=== Sagittal notation ===
=== 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]].


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|34-EDO]].
==== Evo and Revo flavors ====
{{Sagittal chart|}}


====Evo and Revo flavors====
==== Alternative Evo flavor ====
{{Sagittal chart|Alternative_Evo}}


<imagemap>
==== Evo-SZ flavor ====
File:17-EDO_Sagittal.svg
{{Sagittal chart|Evo-SZ}}
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 559 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:17-EDO_Sagittal.svg]]
</imagemap>


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


<imagemap>
==== Sagittal songbook diagram ====
File:17-EDO_Alternative_Evo_Sagittal.svg
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:
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 559 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:17-EDO_Alternative_Evo_Sagittal.svg]]
</imagemap>


====Evo-SZ flavor====
[[File:17edo Sagittal.png|800px]]


<imagemap>
=== 3L 4s (mosh) notation ===
File:17-EDO_Evo-SZ_Sagittal.svg
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 ♭.  
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 627 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:17-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>


Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.
{| class="wikitable center-all right-2 left-4 left-5 mw-collapsible mw-collapsed"
 
|-
====Sagittal songbook diagram====
! #
 
! Cents
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:
! Note
 
! Name
[[File:17edo Sagittal.png|800px]]
! 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 ==
== Approximation to JI ==
=== 15-odd-limit interval mappings ===
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|17}}
{{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 ===
=== Selected 13-limit intervals ===
Line 440: Line 508:


== Tuning by ear ==
== Tuning by ear ==
17edo is very close to a circle of seventeen [[25/24]] chromatic semitones: (25/24)^17 is only 1.43131 cents sharp of an octave. This means that if you can tune seventeen 25/24's accurately (by say, tuning 5/4 up, 3/2 down and 5/4 up, taking care to minimize the error at each step), you have a shot at approximating 17edo within melodic just noticeable difference.
17edo is very close to a circle of seventeen [[25/24]] chromatic semitones: (25/24)<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 ==
== Regular temperament properties ==
Line 454: Line 522:
|-
|-
| 2.3
| 2.3
| {{monzo| 27 -17 }}
| {{Monzo| 27 -17 }}
| {{mapping| 17 27 }}
| {{Mapping| 17 27 }}
| &minus;1.24
| −1.24
| 1.24
| 1.24
| 1.76
| 1.76
Line 462: Line 530:
| 2.3.7
| 2.3.7
| 64/63, 17496/16807
| 64/63, 17496/16807
| {{mapping| 17 27 48 }}
| {{Mapping| 17 27 48 }}
| &minus;3.13
| −3.13
| 2.85
| 2.85
| 4.05
| 4.05
Line 469: Line 537:
| 2.3.7.11
| 2.3.7.11
| 64/63, 99/98, 243/242
| 64/63, 99/98, 243/242
| {{mapping| 17 27 48 59 }}
| {{Mapping| 17 27 48 59 }}
| &minus;3.31
| −3.31
| 2.49
| 2.49
| 3.54
| 3.54
Line 476: Line 544:
| 2.3.7.11.13
| 2.3.7.11.13
| 64/63, 78/77, 99/98, 144/143
| 64/63, 78/77, 99/98, 144/143
| {{mapping| 17 27 48 59 63 }}
| {{Mapping| 17 27 48 59 63 }}
| &minus;3.00
| −3.00
| 2.31
| 2.31
| 3.28
| 3.28
Line 484: Line 552:


=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|16.5|17.5}}
{{Uniform map|edo=17}}


=== Commas ===
=== Commas ===
17et [[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>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints.</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
Line 499: Line 567:
|-
|-
| 3
| 3
| [[134217728/129140163|(18 digits)]]
| <abbr title="134217728/129140163">(18 digits)</abbr>
| {{Monzo| 27 -17 }}
| {{Monzo| 27 -17 }}
| 66.765
| 66.76
| Sasawa
| Sasawa
| [[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
Line 515: Line 583:
| [[32805/32768]]
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| {{Monzo| -15 8 1 }}
| 1.9537
| 1.95
| Layo
| Layo
| Schisma
| Schisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
|-
| 7
| 7
| [[525/512]]
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| {{Monzo| -9 1 2 1 }}
| 43.408
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicennma
| Avicennma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.264
| Ru
| Septimal comma
|-
|-
| 7
| 7
| [[245/243]]
| [[245/243]]
| {{Monzo| 0 -5 1 2 }}
| {{Monzo| 0 -5 1 2 }}
| 14.191
| 14.19
| Zozoyo
| Zozoyo
| Sensamagic comma
| Sensamagic comma
Line 543: Line 611:
| [[1728/1715]]
| [[1728/1715]]
| {{Monzo| 6 3 -1 -3 }}
| {{Monzo| 6 3 -1 -3 }}
| 13.074
| 13.07
| Triru-agu
| Triru-agu
| Orwellisma
| Orwellisma
|-
| 7
| [[17496/16807]]
| {{Monzo| 3 7 0 -5 }}
| 69.56
| Quinru
| Bleu comma
|-
| 7
| [[19683/19208]]
| {{Monzo| -3 9 0 -4 }}
| 42.29
| Laquadru
| Skwares comma
|-
|-
| 7
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| <abbr title="420175/419904">(12 digits)</abbr>
| {{Monzo| -6 -8 2 5 }}
| {{Monzo| -6 -8 2 5 }}
| 1.1170
| 1.12
| Quinzo-ayoyo
| Quinzo-ayoyo
| [[Wizma]]
| [[Wizma]]
|-
| 11
| [[45/44]]
| {{Monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Cake comma
|-
|-
| 11
| 11
| [[99/98]]
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| {{Monzo| -1 2 0 -2 1 }}
| 17.576
| 17.58
| Loruru
| Loruru
| Mothwellsma
| Mothwellsma
Line 564: Line 653:
| [[896/891]]
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| {{Monzo| 7 -4 0 1 -1 }}
| 9.6880
| 9.69
| Saluzo
| Saluzo
| Pentacircle comma
| Pentacircle comma
Line 571: Line 660:
| [[243/242]]
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| {{Monzo| -1 5 0 0 -2 }}
| 7.1391
| 7.14
| Lulu
| Lulu
| Rastma
| Rastma, neutral thirds comma
|-
|-
| 11
| 11
| [[385/384]]
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| {{Monzo| -7 -1 1 1 1 }}
| 4.5026
| 4.50
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|-
| 13
| [[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)
|}
|}
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.
<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 ===
=== Rank-2 temperaments ===
Line 631: Line 805:
| 8/7~9/8
| 8/7~9/8
| [[Machine]]
| [[Machine]]
|-
| 1
| 3\17
| 211.76
| 26/23
| [[Shoal|Shoal (trivial tuning)]]
|-
|-
| 1
| 1
Line 642: Line 822:
| 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 662: Line 842:
| [[Lee]] / [[liese]] (17c) / [[pycnic]] (17)<br>[[Progress]] (17c)
| [[Lee]] / [[liese]] (17c) / [[pycnic]] (17)<br>[[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 [[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 ==
== Scales ==
Line 671: Line 854:
* [[Scorp]]: 3 2 3 1 3 2 3
* [[Scorp]]: 3 2 3 1 3 2 3
* [[Screamapillar]]: 3 3 2 2 3 3 1
* [[Screamapillar]]: 3 3 2 2 3 3 1
* sLmLs: 2 5 3 5 2


=== MOS scales ===
=== MOS scales ===
{{Main| MOS scales of 17edo }}
{{Main| MOS scales of 17edo }}


* diatonic ([[leapfrog]]/[[archy]]) [[5L 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 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 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 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)
* [[squares]] [[3L 8s]] 1 3 1 1 3 1 1 3 (6\17, 1\1)
* [[squares]] [[3L&nbsp;8s]] 1 3 1 1 3 1 1 3 (6\17, 1\1)
* Pathological [[squares]] [[3L 11s]] 1 1 2 1 1 1 2 1 1 1 2 (6\17, 1\1)
* lovecraft [[4L&nbsp;5s]] 3 1 3 1 3 1 3 1 1 (4\17, 1\1)
* lovecraft [[4L 5s]] 3 1 3 1 3 1 3 1 1 (4\17, 1\1)
* Pathological [[1L 13s]] 4 1 1 1 1 1 1 1 1 1 1 1 1 (1\17, 1\1)
* Pathological [[1L 13s|1L 14s]] 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\17, 1\1)
* Pathological [[2L 13s]] 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 (8\17, 1\1)
* Pathological 1L 15s 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\17, 1\1)


=== Well temperaments ===
=== Well temperaments ===
* [[Secor wt17|George Secor’s well temperament of this tuning]]
* [[Secor wt17|George Secor's well temperament]]
* [[User:CritDeathX/Sam's 17-note Well Temperament|Nicolai's 17-note 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]]
* [[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 ==
Line 713: Line 910:
-->
-->


== Instruments ==
== Introductory Materials ==
* '''[http://www.microtonalismo.com/proyecto-xvii Guitar Heptadecatonic from Peruvian - Charles Loli and Antonio Huamani]''' {{forbidden}}
* [[SeventeenTheory]], an introduction to 17edo theory, through the eyes of the [[SeventeenTonePianoProject]].
{{External image| http://sphotos.ak.fbcdn.net/hphotos-ak-snc4/hs883.snc4/71639_167001659983806_100000219181856_601995_1526184_n.jpg {{dead link}} }}
* [http://anaphoria.com/Secor17puzzle.pdf The 17-tone Puzzle] by George Secor, another introduction into 17edo theory.  
 
* [[17edo tetrachords]]
* '''[http://www.microtonalismo.com/proyecto-xvii Bass Heptadecatonic from Peruvian - Charles Loli and Antonio Huamani]''' {{forbidden}}
* [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú] {{forbidden}}
{{External image| http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs382.ash2/66019_167001006650538_100000219181856_601987_48585_n.jpg {{dead link}} }}
 
* [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]]
 
== See also ==
* [[Lumatone mapping for 17edo]]
 
== Notes ==
<references group="note" />


[[Category:17edo| ]] <!-- Main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:3-limit]]
[[Category:Teentuning]]
[[Category:Teentuning]]
Retrieved from "https://en.xen.wiki/w/17edo"