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}}
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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 [[wikipedia: Mu chord|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.
 
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 the harmonic 5.
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"
|-
|-
! #
! #
! Cents
! Cents
! Approximate Ratios*
! Approximate ratios<ref group="note">{{sg|limit=2.3.25.7.11.13.85.23&nbsp;subgroup}}</ref>
! colspan="2" | [[Circle-of-fifths notation|Circle-of-fifths 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>
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and ^d2)
! colspan="2" | [[3L 4s]] Notation
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U {{=}} 1)}}
|-
|-
| 0
| 0
| 0.00
| 0.0
| 1/1
| [[1/1]]
| Unison
| Unison
| D
| D
Line 44: Line 53:
| D
| D
| unison
| unison
| P1
| D
| D
|-
|-
| 1
| 1
| 70.59
| 70.6
| [[22/21]], [[25/24]], [[26/25]], [[28/27]], [[33/32]], [[24/23]]
| [[24/23]], [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| Minor 2nd<br>(Semiaugmented 1sn)
| Minor 2nd<br>(Semiaugmented 1sn)
| Eb<br>(D+)
| Eb<br>(Dt)
| up unison, <br>minor 2nd
| up unison, <br>minor 2nd
| ^1, m2
| ^1, m2
| Eb
| Eb
| augmented 1sn
| uber unison, <br>minor 2nd
| D#
| U1, m2
| UD, Eb
|-
|-
| 2
| 2
| 141.18
| 141.2
| [[13/12]], [[12/11]], [[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)
Line 65: Line 76:
| A1, ~2
| A1, ~2
| vE
| vE
| minor 2nd
| neutral 2nd
| Eb
| N2
| UEb, uE
|-
|-
| 3
| 3
| 211.76
| 211.8
| [[9/8]], [[8/7]], [[28/25]], [[25/22]], [[26/23]]
| [[8/7]], [[9/8]], [[17/15]], [[25/22]], [[26/23]]
| Major 2nd
| Major 2nd
| E
| E
Line 77: Line 89:
| E
| E
| major 2nd
| major 2nd
| M2
| E
| E
|-
|-
| 4
| 4
| 282.35
| 282.4
| [[13/11]], [[7/6]]
| [[7/6]], [[13/11]], [[20/17]]
| Minor 3rd
| Minor 3rd
| F
| F
Line 87: Line 100:
| m3
| m3
| F
| F
| diminished 3rd
| minor 3rd
| Fb
| m3
| F
|-
|-
| 5
| 5
| 352.94
| 352.9
| [[11/9]], [[16/13]], [[28/23]]
| [[11/9]], [[27/22]], [[16/13]], [[39/32]]
| Diminished 4th<br>(Neutral 3rd)
| Diminished 4th<br>(Neutral 3rd)
| Gb<br>(F+)
| Gb<br>(Ft)
| mid 3rd
| mid 3rd
| ~3
| ~3
| ^F
| ^F
| perfect 3rd
| neutral 3rd
| F
| N3
| UF, uF#
|-
|-
| 6
| 6
| 423.53
| 423.5
| [[32/25]], [[9/7]], [[14/11]], [[33/26]], [[23/18]]
| [[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)
Line 109: Line 124:
| M3
| M3
| F#
| F#
| augmented 3rd
| major 3rd
| M3
| F#
| F#
|-
|-
| 7
| 7
| 494.12
| 494.1
| [[4/3]], [[21/16]]
| [[4/3]], [[21/16]], [[85/64]]
| Perfect 4th
| Perfect 4th
| G
| G
Line 120: Line 136:
| P4
| P4
| G
| G
| minor 4th
| perfect 4th
| P4
| G
| G
|-
|-
| 8
| 8
| 564.71
| 564.7
| [[11/8]], [[18/13]], [[32/23]]
| [[11/8]], [[18/13]], [[25/18]], [[32/23]]
| Diminished 5th<br>(Semiaugmented 4th)
| Diminished 5th<br>(Semiaugmented 4th)
| Ab<br>(G+)
| Ab<br>(Gt)
| mid 4th, <br>diminished 5th
| mid 4th, <br>diminished 5th
| ~4, d5
| ~4, d5
| ^G, Ab
| ^G, Ab
| major 4th
| uber 4th/<br>neutral 4th
| G#
| U4/N4
| UG
|-
|-
| 9
| 9
| 635.29
| 635.3
| [[16/11]], [[13/9]], [[23/16]]
| [[13/9]], [[16/11]], [[23/16]], [[36/25]]
| Augmented 4th<br>(Semidiminished 5th)
| Augmented 4th<br>(Semidiminished 5th)
| G#<br>(Ad)
| G#<br>(Ad)
Line 142: Line 160:
| A4, ~5
| A4, ~5
| G#, vA
| G#, vA
| minor 5th
| unter 5th/<br>neutral 5th
| Ab
| u5/N5
| uA
|-
|-
| 10
| 10
| 705.88
| 705.9
| [[3/2]], [[32/21]]
| [[3/2]], [[32/21]], [[128/85]]
| Perfect 5th
| Perfect 5th
| A
| A
Line 153: Line 172:
| P5
| P5
| A
| A
| major 5th
| perfect 5th
| P5
| A
| A
|-
|-
| 11
| 11
| 776.47
| 776.5
| [[25/16]], [[14/9]], [[11/7]], [[52/33]], [[36/23]]
| [[11/7]], [[14/9]], [[25/16]], [[36/23]], [[80/51]]
| Minor 6th<br>(Semiaugmented 5th)
| Minor 6th<br>(Semiaugmented 5th)
| Bb<br>(A+)
| Bb<br>(At)
| minor 6th
| minor 6th
| m6
| m6
| Bb
| Bb
| diminished 6th
| minor 6th
| m6
| Bb
| Bb
|-
|-
| 12
| 12
| 847.06
| 847.1
| [[13/8]], [[18/11]], [[23/14]]
| [[13/8]], [[18/11]], [[44/27]], [[64/39]]
| Augmented 5th<br>(Neutral 6th)
| Augmented 5th<br>(Neutral 6th)
| A#<br>(Bd)
| A#<br>(Bd)
Line 175: Line 196:
| ~6
| ~6
| vB
| vB
| perfect 6th
| neutral 6th
| B
| N6
| UBb, uB
|-
|-
| 13
| 13
| 917.65
| 917.6
| [[17/10]], [[22/13]], [[12/7]]
| [[12/7]], [[17/10]], [[22/13]]
| Major 6th
| Major 6th
| B
| B
Line 186: Line 208:
| M6
| M6
| B
| B
| augmented 6th
| major 6th
| B#
| M6
| B
|-
|-
| 14
| 14
| 988.24
| 988.2
| [[16/9]], [[7/4]], [[25/14]], [[44/25]], [[23/13]]
| [[7/4]], [[16/9]], [[23/13]], [[30/17]], [[44/25]]
| Minor 7th
| Minor 7th
| C
| C
Line 198: Line 221:
| C
| C
| minor 7th
| minor 7th
| Cb
| m7
| C
|-
|-
| 15
| 15
| 1058.82
| 1058.8
| [[11/6]], [[24/13]], [[13/7]], [[46/25]]
| [[11/6]], [[13/7]], [[24/13]], [[46/25]]
| Diminished 8ve<br>(Neutral 7th)
| Diminished 8ve<br>(Neutral 7th)
| Db<br>(C+)
| Db<br>(Ct)
| mid 7th
| mid 7th
| ~7
| ~7
| ^C
| ^C
| major 7th
| neutral 7th
| C
| N7
| UC, uC#
|-
|-
| 16
| 16
| 1129.41
| 1129.4
| [[21/11]], [[25/13]], [[48/25]], [[27/14]], [[64/33]], [[23/12]]
| [[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)
Line 219: Line 244:
| M7, v8
| M7, v8
| C#
| C#
| diminished 8ve
| major 7th,<br>unter octave
| Db
| M7, u8
| C#, uD
|-
|-
| 17
| 17
| 1200.00
| 1200.0
| [[2/1]]
| [[2/1]]
| Octave
| Octave
Line 231: Line 257:
| D
| D
| octave
| octave
| P8
| D
| D
|}
|}
 
<references group="note" />
<nowiki>*</nowiki> Ratios based on treating 17edo as a 2.3.7.11.13.23.25 subgroup temperament.
 
† Half-sharps and half-flats (denoted "+" 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 ===
=== Interval quality and chord names in color notation ===
Line 242: Line 266:


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
! Quality
! Quality
! Color
! Color
! Monzo Format
! Monzo format
! Examples
! Examples
|-
|-
Line 253: Line 278:
|-
|-
| fourthward wa
| fourthward wa
| (a, b), b &lt; -1
| (a, b), b < -1
| 32/27, 16/9
| 32/27, 16/9
|-
|-
Line 267: 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 279: 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 324: 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 ==
=== Sagittal ===
=== 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|}}
 
==== Alternative Evo flavor ====
{{Sagittal chart|Alternative_Evo}}
 
==== Evo-SZ flavor ====
{{Sagittal chart|Evo-SZ}}
 
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 Barton|Jacob A. Barton]], a diagram of how to notate 17edo in the Revo flavor of Sagittal:
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:


[[File:17edo Sagittal.png|800px]]
[[File:17edo Sagittal.png|800px]]


== JI approximation ==
=== 3L 4s (mosh) notation ===
=== 15-odd-limit interval mappings ===
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 ♭.  
The following table shows how [[15-odd-limit intervals]] are represented in 17edo (ordered by absolute error). Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  


{| class="wikitable sortable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all right-2 left-4 left-5 mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
|-
! class="unsortable" | Interval, complement
! #
! Error (abs, [[Cent|¢]])
! Cents
! Error (rel, [[Relative cent|%]])
! Note
! Name
! Associated ratios
|-
|-
| [[18/13]], [[13/9]]
| 0
| 1.324
| 0.0
| 1.9
| D
| Perfect 1sn
| 1/1
|-
|-
| [[13/12]], [[24/13]]
| 1
| 2.604
| 70.6
| 3.7
| D#
| Augmented 1sn
| 33/32
|-
|-
| '''[[4/3]], [[3/2]]'''
| 2
| '''3.927'''
| 141.2
| '''5.6'''
| Eb
| Minor 2nd
| 12/11
|-
|-
| [[11/9]], [[18/11]]
| 3
| 5.533
| 211.8
| 7.8
| E
| Major 2nd
| 9/8
|-
|-
| [[14/11]], [[11/7]]
| 4
| 6.021
| 282.4
| 8.5
| Fb
| Diminished 3rd
| 32/27
|-
|-
| '''[[16/13]], [[13/8]]'''
| 5
| '''6.531'''
| 352.9
| '''9.3'''
| F
| Perfect 3rd
| 11/9, 27/22
|-
|-
| [[13/11]], [[22/13]]
| 6
| 6.857
| 423.5
| 9.7
| F#
| Augmented 3rd
| 81/64
|-
|-
| [[9/8]], [[16/9]]
| 7
| 7.855
| 494.1
| 11.1
| G
| Minor 4th
| 4/3
|-
|-
| [[12/11]], [[11/6]]
| 8
| 9.461
| 564.7
| 13.4
| G#
| Major 4th
| 11/8
|-
|-
| [[9/7]], [[14/9]]
| 9
| 11.555
| 635.3
| 16.4
| Ab
| Minor 5th
| 16/11
|-
|-
| [[14/13]], [[13/7]]
| 10
| 12.878
| 705.9
| 18.2
| A
| Major 5th
| 3/2
|-
|-
| '''[[11/8]], [[16/11]]'''
| 11
| '''13.388'''
| 776.5
| '''19.0'''
| Bb
| Diminished 6th
| 128/81
|-
|-
| [[7/6]], [[12/7]]
| 12
| 15.482
| 847.1
| 21.0
| B
| Perfect 6th
| 18/11, 44/27
|-
|-
| ''[[7/5]], [[10/7]]''
| 13
| ''17.806''
| 917.6
| ''25.2''
| B#
| Augmented 6th
| 27/16
|-
|-
| '''[[8/7]], [[7/4]]'''
| 14
| '''19.409'''
| 988.2
| '''27.5'''
| Cb
| Minor 7th
| 16/9
|-
|-
| ''[[15/14]], [[28/15]]''
| 15
| ''21.734''
| 1058.8
| ''30.8''
| C
| Major 7th
| 11/6
|-
|-
| ''[[11/10]], [[20/11]]''
| 16
| ''23.828''
| 1129.4
| ''33.8''
| Db
| Diminished 8ve
| 64/33
|-
|-
| ''[[15/11]], [[22/15]]''
| 17
| ''27.755''
| 1200.0
| ''39.3''
| D
|-
| Perfect 8ve
| ''[[10/9]], [[9/5]]''
| 2/1
| ''29.361''
| ''41.6''
|-
| [[16/15]], [[15/8]]
| 29.445
| 41.7
|-
| ''[[13/10]], [[20/13]]''
| ''30.685''
| ''43.5''
|-
| ''[[6/5]], [[5/3]]''
| ''33.288''
| ''47.2''
|-
| '''[[5/4]], [[8/5]]'''
| '''33.373'''
| '''47.3'''
|-
| ''[[15/13]], [[26/15]]''
| ''34.612''
| ''49.0''
|}
|}
{{15-odd-limit|17}}
 
== 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 ===
=== Selected 13-limit intervals ===
Line 445: 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 ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 459: Line 522:
|-
|-
| 2.3
| 2.3
| {{monzo| 27 -17 }}
| {{Monzo| 27 -17 }}
| [{{val| 17 27 }}]
| {{Mapping| 17 27 }}
| -1.24
| −1.24
| 1.24
| 1.24
| 1.76
| 1.76
Line 467: Line 530:
| 2.3.7
| 2.3.7
| 64/63, 17496/16807
| 64/63, 17496/16807
| [{{val| 17 27 48 }}]
| {{Mapping| 17 27 48 }}
| -3.13
| −3.13
| 2.85
| 2.85
| 4.05
| 4.05
Line 474: Line 537:
| 2.3.7.11
| 2.3.7.11
| 64/63, 99/98, 243/242
| 64/63, 99/98, 243/242
| [{{val| 17 27 48 59 }}]
| {{Mapping| 17 27 48 59 }}
| -3.31
| −3.31
| 2.49
| 2.49
| 3.54
| 3.54
Line 481: 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
| [{{val| 17 27 48 59 63 }}]
| {{Mapping| 17 27 48 59 63 }}
| -3.00
| −3.00
| 2.31
| 2.31
| 3.28
| 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.  
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.  


=== 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 }}, 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 505: 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 521: 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 549: 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 570: 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
| Pentacircle comma
|-
|-
| 11
| 11
| [[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
| Gentle comma
| Threedie
|-
| 23
| [[162/161]]
| {{Monzo| 1 4 0 -1 0 0 0 0 -1 }}
| 10.72
| Twethuru
| Minor kirnbergerisma
|-
| 23
| [[208/207]]
| {{Monzo| 4 -2 0 0 0 1 0 0 -1 }}
| 8.34
| Twethutho
| Vicetone comma
|-
| 23
| [[253/252]]
| {{Monzo| -2 -2 0 -1 1 0 0 0 1 }}
| 6.86
| Twetholoru
| Middle neutravicema
|-
| 23
| [[529/528]]
| {{Monzo| -4 -1 0 0 -1 0 0 0 2 }}
| 3.28
| Bitwetho-alu
| Preziosisma
|-
| 23
| [[736/729]]
| {{Monzo| 5 -6 0 0 0 0 0 0 1 }}
| 16.54
| Satwetho
| 23-limit Tenney/Cage comma (HEJI)
|}
|}
<references/>
<references group="note" />
Note that despite their relatively large size, the 17-comma, the avicennma and the chromatic semitone are all tempered out by the 13-limit patent val, as stated.
Note that due to the inaccurate prime 5, the rather large commas [[25/24]], [[525/512]], [[45/44]], and [[40/39]] are all tempered out by 17edo's patent val.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 612: Line 786:


{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all right-3 left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator
! Generator
! Cents
! Cents
! Associated<br>Ratio
! Associated<br>ratio
! Temperament
! Temperament
|-
|-
Line 630: 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 635: Line 816:
| 282.35
| 282.35
| 13/11
| 13/11
| [[Huxley]] / [[Lovecraft]] {{clarify}}
| [[Huxley]] / [[lovecraft]] / [[subklei]] (17c)
|-
|-
| 1
| 1
Line 641: 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 653: Line 834:
| 494.12
| 494.12
| 4/3
| 4/3
| [[Archy]] / [[supra]] / [[quasisuper]] (17c) / [[dominant]] (17c) / [[superpyth]] (17) / [[schism]] (17)
| [[Archy]] / [[supra]] / [[quasisuper]] (17c) / [[dominant (temperament)|dominant]] (17c) / [[superpyth]] (17) / [[schism]] (17)<br>[[Fiventeen]]
|-
|-
| 1
| 1
Line 661: 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 670: 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]]) 5L2s 3331331 (10\17, 1\1)
* diatonic ([[leapfrog]]/[[archy]]) [[5L&nbsp;2s]] 3 3 3 1 3 3 1 (10\17, 1\1)
* [[maqamic]] 3L4s 3232322 (5\17, 1\1)
* [[neutrominant]] [[3L&nbsp;4s]] 3 2 3 2 3 2 2 (5\17, 1\1) (''dedicated article: [[17edo neutral scale]]'')
* [[maqamic]] 7L3s 2221221221 (5\17, 1\1)
* [[neutrominant]] [[7L&nbsp;3s]] 2 2 2 1 2 2 1 2 2 1 (5\17, 1\1)
* [[squares]] 3L5s 1141414 (6\17, 1\1)
* [[squares]] [[3L&nbsp;5s]] 1 1 4 1 4 1 4 (6\17, 1\1)
* [[squares]] 3L8s 13113113 (6\17, 1\1)
* [[squares]] [[3L&nbsp;8s]] 1 3 1 1 3 1 1 3 (6\17, 1\1)
* Pathological [[squares]] 3L11s 11211121112 (6\17, 1\1)
* lovecraft [[4L&nbsp;5s]] 3 1 3 1 3 1 3 1 1 (4\17, 1\1)
* [[lovecraft]] 4L5s 313131311 (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 712: 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]]
 
== See also ==
* [[Lumatone mapping for 17edo]]


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