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{{interwiki
{{interwiki
| de = 15edo
| de = 15-EDO
| en = 15edo
| en = 15edo
| es =  
| es =  
| ja = 15平均律
| ja = 15平均律
}}
}}
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3 × 5
{{Wikipedia|15 equal temperament}}
| Step size = 80¢
{{ED intro}}
| Fifth = 9\15 = 720¢ (→[[5edo|3\5]])
 
| Major 2nd = 3\15 = 240¢
== Theory ==
| Minor 2nd = 0\15 = 0¢
15edo can be thought of as three sets of [[5edo]] which do not connect by [[3/2|fifths]]. The fifth at 720 cents is quite wide yet still usable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. 15edo contains 3 circles of five [[3/2]]'s (supporting [[blackwood]], which tempers out the [[Pythagorean limma]]), and 5 circles of three [[5/4]]'s (supporting [[augmented (temperament)|augmented]] temperament). This is radically different than a meantone system, and has a variety of ramifications for chord progressions based on diatonic {{w|Function (music)|functional harmony}}, because if you use the equipentatonic as your "diatonic scale", the same interval can have multiple functions.
| Augmented 1sn = 3\15 = 240¢
 
}}
A useful way to visualize the pitches and intervals of 15edo is to arrange the notes in a grid, with 3/2s or 7/4s on one axis and 5/4s on the other, to create a 3x5 rectangle of notes which tiles the plane.
 
15edo shares 5edo's 2.3.7 subgroup tuning (and thus supports [[superpyth]], [[slendric]], and [[semaphore]], like 5edo). However, by splitting each 5edo step into three parts, reasonable approximations to [[5/4]] and [[11/8]] are obtained (as per [[valentine]] temperament), so 15edo can reasonably be described as an 11-limit temperament, and is generally considered to be the first edo to work as an 11-limit system; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to represent JI with 15edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity, for instance, only using 9/8 when it is being made up of two 3/2s to make its identity clear). 15edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, 6/5, and their octave inverses) that has a positive [[syntonic comma]].
 
In the 15edo system, major thirds cannot be divided perfectly into two, while minor thirds, fourths, wide tritones, subminor sevenths, and supermajor sevenths can. Similarly, supermajor seconds, fourths, fifths, and subminor sevenths can all be divided into 3 equal parts, while minor thirds, tritones, and major sixths cannot.
 
This gives 15edo a whole new set of pitch symmetries and modes of limited transposition. Coupled with the lack of a [[5L 2s|5L 2s diatonic scale]] and of a standard tritone, this tuning can be disorienting at first. Nonetheless, 15edo is notable for being the next-smallest edo after 9edo, 12edo and 14edo that contains recognizable major and minor triads. Under a stricter definition excluding 9edo and 14edo, this is a property noted in the works of theorists like [[Ivor Darreg]] and [[Easley Blackwood]]. In addition, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modulated anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in xenharmony, due to its manageable number of tones and for containing the relatively popular 5edo.
 
A possible analogue to the diatonic scale in 15edo is the [[Zarlino]] diatonic, which flattens one fifth to a large tritone in order to make all 7 notes distinct (and close to corresponding JI intervals, especially if you use the left-handed version). The fact that 15edo supports [[porcupine]] temperament is equivalent to the fact that both accidentals generally required to notate zarlino collapse to a single chromatic step. For a moment-of-symmetry scale, the [[1L 6s]] (onyx) and [[5L 5s]] (pentawood) scales are also an option.
 
15edo is also the second-smallest edo (after [[10edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].
 
=== Prime harmonics ===
{{Harmonics in equal|15}}
 
=== Composition theory ===
* [[User:Unque/15edo Composition Theory|Unque's approach]] - covers scales, chords, intervals, and functional harmony.
* [[15edo/Vector's compositional guides|Vector's guides]] - covers the construction of scales, the kinds of chords found in 15edo, and a possible notation system.
* [[User:Astaryuu/15edo Notes|Astaryuu's notes]] - covers notation, scales, modes, intervals, and chords so far.
* [[Metallic harmony]] - harmony involving stacking sevenths instead of thirds; 15edo is one of the systems it is intended for.


'''15 equal temperament''' or 15-EDO is a tuning which divides the octave into 15 equally spaced pitches.  
== Intervals ==
{{See also|15edo-interval names}}
Relative to 12edo, 15edo maintains some categorically-similar intervals, particularly the 3rds, 4ths, 5ths, and 6ths, but is quite different in the categories of 2nds and 7ths. The closest intervals it has to a 12edo [[whole tone]] are both 40 cents sharp or flat of the 200-cent 12edo whole tone. This makes it rather difficult to translate traditional diatonic melodic approaches into 15edo, and also means that things like 7th, 9th, and 11th chords will behave very differently, even though major and minor triads are still relatively familiar-sounding. One step of 15edo almost exactly equals the reduced 67th harmonic, [[67/64]].


== Theory ==
{| class="wikitable center-all left-8"
{| class="wikitable center-all"
|-
!
! [[Degree]]
! prime 2
! [[Cent]]s
! prime 3
! [[Interval region]]
! prime 5
! Approximate Ratios<ref group="note">{{sg|limit=11-limit}}</ref>
! prime 7
! Audio
! prime 11
! prime 13
! prime 17
! prime 19
|-
|-
! Error (¢)
| 0
| 0
| +18.0
| 0
| +13.7
| Unison<br>(prime)
| -8.8
| 1/1
| +8.7
| [[File:piano_0_1edo.mp3]]
| +39.5
|-
| -25.0
| 1
| +22.5
| 80
| Minor second
| 25/24, 21/20, 16/15, 22/21
| [[File:piano_1_15edo.mp3]]
|-
|-
! Error (%)
| 2
| 0
| 160
| +23
| Submajor second
| +17
| 11/10, 12/11, 10/9
| -11
| [[File:piano_2_15edo.mp3]]
| +11
|-
| +49
| 3
| -31
| 240
| +28
| Supermajor second
| 8/7, 7/6, 9/8
| [[File:piano_1_5edo.mp3]]
|-
| 4
| 320
| Minor third
| 6/5, 11/9
| [[File:piano_4_15edo.mp3]]
|-
|-
! [[Nearest edomapping]]
| 15
| 9
| 5
| 5
| 12
| 400
| Major third
| 5/4, 14/11
| [[File:piano_1_3edo.mp3]]
|-
| 6
| 480
| Perfect fourth/<br>subfourth
| 4/3, ''9/7'', 21/16
| [[File:piano_2_5edo.mp3]]
|-
| 7
| 7
| 560
| Narrow tritone
| 11/8, 7/5
| [[File:piano_7_15edo.mp3]]
|-
| 8
| 640
| Wide tritone
| 16/11, 10/7
| [[File:piano_8_15edo.mp3]]
|-
| 9
| 720
| Perfect fifth/<br>superfifth
| 3/2, ''14/9'', 32/21
| [[File:piano_3_5edo.mp3]]
|-
| 10
| 800
| Minor sixth
| 8/5, 11/7
| [[File:piano_2_3edo.mp3]]
|-
| 11
| 11
| 1
| 880
| 4
| Major sixth
| 5/3, 18/11
| [[File:piano_11_15edo.mp3]]
|-
| 12
| 960
| Subminor seventh
| 7/4, 12/7, 16/9
| [[File:piano_4_5edo.mp3]]
|-
| 13
| 1040
| Supraminor seventh
| 20/11, 11/6, 9/5
| [[File:piano_13_15edo.mp3]]
|-
| 14
| 1120
| Major seventh
| 48/25, 40/21, 15/8, 21/11
| [[File:piano_14_15edo.mp3]]
|-
| 15
| 1200
| Octave
| 2/1
|[[File:piano_1_1edo.mp3]]
|}
|}
<references group="note" />


15-edo can be thought of as three sets of 5-EDO which do not connect by fifths. The fifth at 720 cents is quite wide yet still useable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. The perfect fifth of 15 EDO returns to the octave if stacked five times which is radically different than a meantone system.
== Notation ==
 
There are many ways to notate 15edo, and the choice of notation depends heavily on which temperament or scale one wishes to focus on.
From [[Wikipedia:15_equal_temperament|Wikipedia]]:
 
"In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the [[octave]] into 15 equal steps. Each step represents a frequency ratio of 2<sup>1/15</sup>, or 80 [[cent|cents]]. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of [[5edo|5 equal divisions of the octave]] (or five scales of [[3edo]])."
 
15-edo can be seen as a [[7-limit]] temperament because of its ability to approximate some septimal intervals, but it also contains some fairly obvious approximations to [[11-limit]] intervals, so it can reasonably be described as an 11-limit temperament; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to approximate JI with 15-edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity). 15-edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, 6/5, and their octave inverses) that has a positive [[81/80|syntonic comma]].
 
In the 15-edo system, major thirds cannot be divided perfectly into two, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modulated anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in xenharmony, due to its manageable number of tones and for containing the relatively popular 5-edo.


A recommended method to the notation of 15-edo by some is a system based on porcupine[8] in which eight nominals form the base diatonic scale. In this sense, the "quill" is the name given to the two step interval (160¢) of 15-edo while the "small quill" (80¢) is the chroma of 15-edo. This produces a very consistent notation for both porcupine[8] and Blackwood[10] and seems to work much better than attempting to put 15-edo into a seven nominal based framework.
{| class="wikitable center-all left-8 mw-collapsible"
 
|+ style="font-size: 105%; white-space: nowrap;" | Overview of notation systems for 15edo
[[File:15_tone_keyboard.png|left|thumb|400x400px|Porcupine layout for 15edo]]
[[File:Screen Shot 2020-04-23 at 11.59.17 PM.png|left|thumb|Hanson layout for 15edo]]
<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
== Intervals ==
 
{| class="wikitable center-all left-8"
|-
|-
! Degree
! Degree
! Cents
! Cents
! Solfege<br>(porcupine-based)
! [[Solfege]]<br>(porcupine-based)
! Porcupine[7]<br>(traditional)
! Porcupine[8]<br>(Greek)
! Porcupine[8]<br>(Greek)
! Zarlino diatonic notation
! Blackwood<br>"guitar notation"
! Blackwood<br>"guitar notation"
! Porcupine[7]<br>(traditional)
! Blackwood<br>Decimal
! Blackwood<br>Decimal
! Approximate Ratios*
|-
|-
| 0
| 0
| 0
| 0
| do
| do
| D
| α
| α
| C
| E
| E
| G
| 1
| 1
| 1/1
|-
|-
| 1
| 1
| 80
| 80
| di
| di
| D# / Eb
| α/ β\
| α/ β\
| Db / C#
| E#
| E#
| G# / Abb
| 1# / 2b
| 1# / 2b
| 25/24, 21/20, 16/15
|-
|-
| 2
| 2
| 160
| 160
| ru
| ru
| E
| β
| β
| D
| Gb
| Gb
| Gx / Ab
| 2
| 2
| 11/10, 12/11, 10/9
|-
|-
| 3
| 3
| 240
| 240
| re
| re
| E# / Fb
| β/ χ\
| β/ χ\
| D#
| G
| G
| A
| 3
| 3
| 8/7, 7/6, 9/8
|-
|-
| 4
| 4
| 320
| 320
| me
| me
| F
| χ
| χ
| Eb
| G#
| G#
| A# / Bb
| 3# / 4b
| 3# / 4b
| 6/5, 11/9
|-
|-
| 5
| 5
| 400
| 400
| mi
| mi
| F# / Gb
| χ/ δ\
| χ/ δ\
| E
| Ab
| Ab
| B
| 4
| 4
| 5/4, 14/11
|-
|-
| 6
| 6
| 480
| 480
| fa
| fa
| G
| δ
| δ
| F
| A
| A
| B# / Cb
| 5
| 5
| 4/3, 9/7, 21/16
|-
|-
| 7
| 7
| 560
| 560
| fu
| fu
| G#
| δ/ ε\
| δ/ ε\
| F#
| A#
| A#
| C
| 5# / 6b
| 5# / 6b
| 11/8, 7/5
|-
|-
| 8
| 8
| 640
| 640
| su
| su
| Ab
| ε
| ε
| Gb
| Bb
| Bb
| C# / Db
| 6
| 6
| 16/11, 10/7
|-
|-
| 9
| 9
| 720
| 720
| sol
| sol
| A
| ε/ φ\
| ε/ φ\
| G
| B
| B
| D
| 7
| 7
| 3/2, 14/9, 32/21
|-
|-
| 10
| 10
| 800
| 800
| le
| le
| A# / Bb
| φ
| φ
| Ab / G#
| B#
| B#
| D# / Eb
| 7# / 8b
| 7# / 8b
| 8/5, 11/7
|-
|-
| 11
| 11
| 880
| 880
| la
| la
| B
| φ/ γ\
| φ/ γ\
| A
| Db
| Db
| E
| 8
| 8
| 5/3, 18/11
|-
|-
| 12
| 12
| 960
| 960
| ta
| ta
| B# / Cb
| γ
| γ
| A# / Bbb
| D
| D
| E# / Fb
| 9
| 9
| 7/4, 12/7, 16/9
|-
|-
| 13
| 13
| 1040
| 1040
| tu
| tu
| C
| γ/ η\
| γ/ η\
| Bb
| D#
| D#
| F
| 9# / 0b
| 9# / 0b
| 20/11, 11/6, 9/5
|-
|-
| 14
| 14
| 1120
| 1120
| ti
| ti
| C# / Db
| η
| η
| B
| Eb
| Eb
| F# /Gb
| 0
| 0
| 48/25, 40/21, 15/8
|-
|-
| 15
| 15
| 1200
| 1200
| do
| do
| D
| α
| α
| C
| E
| E
| G
| 1
| 1
| 2/1
|}
|}
<nowiki>*</nowiki> based on treating 15-EDO as an 11-limit temperament; other approaches are possible
In [[Ups_and_Downs_Notation# Summary of EDO notation-"Pentatonic" EDOs|ups and downs notation]], which is fifth-generated, every 15edo note has at least three names. 15edo can also be notated using the [[Ups_and_Downs_Notation #Natural Generators|natural generator]], which is not the 9\15 5th but the 2\15 2nd. For 15edo, this is also known as porcupine notation. The 15edo porcupine genchain in both absolute and relative notation:
…Fx - Gx - A# - B# - C# - D# - E# - F# - G# --- A --- B --- C -- D --- E --- F --- G -- Ab -- Bb - Cb - Db - Eb - Fb - Gb - Abb - Bbb…
…A3 - A4 - A5 - A6 - A7 - A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 -- d8 - d2 - d3 -- d4 - d5 -- d6…


=== Alternative interval names ===
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! step
! Step
! cents
! Cents
! colspan="2" | ups and downs relative notation<br>(partial list, e.g. M2 is also A1 and d4)
! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and m2)<br>(partial list, e.g. M2/m3 is also A1 and d4)
! ups and downs<br>absolute notation
! colspan="2" | Porcupine notation<br>([[Enharmonic unison|EU]]: dd2)
! porcupine<br>relative notation
! porcupine<br>absolute notation
|-
|-
| 0
| 0
Line 377: Line 439:
| D
| D
|}
|}
The 15edo porcupine genchain in both absolute and relative notation:
* …{{dash|Fx, Gx, A#, B#, C#, D#, E#, F#, G#, A, B, C, D, E, F, G, Ab, Bb, Cb, Db, Eb, Fb, Gb, Abb, Bbb|long}}…
* …{{dash|A3, A4, A5, A6, A7, A1, A2, M3, M4, M5, M6, P7, P1, P2, m3, m4, m5, m6, d7, d8, d2, d3, d4, d5, d6|long}}…


All 15edo chords can be named using ups and downs. Because many intervals have several names, many chords do too. Alterations are always enclosed in parentheses, additions never are. An up or down 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).
All 15edo chords can be named using ups and downs. Because many intervals have several names, many chords do too. Alterations are always enclosed in parentheses, additions never are. An up or down 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).
Line 400: Line 467:
0-4-9-13 = D ^F A ^C = D^m7 = "D upminor-seven", or D ^F A ^B = D^m6 = "D upminor-six"
0-4-9-13 = D ^F A ^C = D^m7 = "D upminor-seven", or D ^F A ^B = D^m6 = "D upminor-six"


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]].


== Just approximation ==
=== Notations generated by the fifth ===
15-EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a 5L5s MOS scale wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as Blackwood temperament, named after Easley Blackwood, Jr., who is the first to document its existence. It has also been written on extensively by [[IgliashonJones|Igliashon Jones]] in the paper "[http://www.cityoftheasleep.com/etc/5nEDOs.pdf Five is Not an Odd Number]". For an in-depth treatment of harmony in 15-edo based on this temperament (and its 7- and 11-limit extensions), see [[Harmony_in_15edo_Blacksmith|Harmony in 15edo Blacksmith[10]]].
In these notations, the nominals form a circle of perfect fifths. The other notes are notated using accidentals that raise or lower by one edostep.


=== Selected just intervals ===
==== Stein–Zimmermann–Gould notation (heptatonic) ====
==== 15-odd-limit mappings ====
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
The following table shows how [[15-odd-limit intervals]] are represented in 15edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
{{Sharpness-sharp3-szg}}


{| class="wikitable center-all"
==== Kite's ups and downs notation (heptatonic) ====
|+Direct mapping (even if inconsistent)
15edo can be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that downsharp is equivalent to dup (double-up) and upflat is equivalent to dud (double-down).
{{Sharpness-sharp3a}}
 
==== Sagittal notation (heptatonic) ====
This notation uses the same sagittal sequence as edos [[22edo #Sagittal notation|22]] and [[29edo #Sagittal notation|29]], is a subset of the notation for [[30edo #Sagittal notation|30edo]], and is a superset of the notation for [[5edo #Sagittal notation|5edo]].
 
{{Sagittal chart|}}
 
==== "Eef" notation (pentatonic) ====
[[Kite Giedraitis]] proposes pentatonic (as opposed to heptatonic) note names that omit B and merge E and F into a new letter "eef" that rhymes with "leaf". Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ꘙ (unicode A619) or ⊧ (unicode 22A7) or 𐐆 (unicode 10406). The circle of 5ths is C G D A ꘙ C. All intervals are either perfect, upperfect or downperfect (never major or minor). This is similar to heptatonic interval names in 7edo, 14edo, 21edo, etc.
 
{| class="wikitable"
| C || ^C || vD || D || ^D || vꘙ || ꘙ || ^ꘙ || vG || G || ^G || vA || A || ^A || vC || C
|-
|-
! Interval, complement
| P1 || ^1 || v2 || P2 || ^2 || v3 || P3 || ^3 || v4 || P4 || ^4 || v5 || P5 || ^5 || v6 || P6
! Error (abs, [[cent|¢]])
|}
 
==== Blackwood guitar notation ====
On a 15edo guitar, because the "perfect fourth" comes from 5edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the [[circle of fourths]] on B—B–E–A–D–G-(B)—then the open strings of the guitar can be notated as usual (E–A–D–G–B–E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15edo, it is necessary to use accidentals to notate intervals on the other two chains of 5edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15edo on the guitar, since 5edo provides a useful set of 3-limit landmarks (or "perfect fourths" and "perfect fifths") that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.
 
=== Blackwood decatonic notation ===
Using the nominals 1-X, 0-9 or 1-0 (with X, 9 or 0 representing the tenth degree respectively), one of the three circles of 5edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats, or the converse for 0-9).
 
One could name the nominals with letters instead of numbers, such as ABC... or JKL...
 
=== Notations generated by the second ===
In these notations, the nominals form a chain of perfect 2nds, each of which are two edosteps wide. From the last note of the chain up to the first there is an augmented 2nd of three edosteps. Accidentals raise or lower by one edostep.
 
==== Porcupine notation (heptatonic) ====
Porcupine notation can be based on the Porcupine[7] Lssssss scale. By representing the 3|3 mode (sssLsss) with a chain of seconds (D E F G A B C D) and using sharps and flats (#/b) to denote an edostep up or down respectively, 15edo can be notated using standard notation. Its intervals are here named with respect to diatonic intervals, i.e., as if fifth-generated. Thus the 4th and 5th are called perfect even though they are not generators, and the 2nd and 7th are not called perfect even though they are generators.
{| class="wikitable"
!Cents
!Interval Name(s)
!Note name(s)
!Diamond-mos (on symmetric mode)
|-
|-
| [[18/13]], [[13/9]]
|0
| 3.382
|Unison
|D
|J
|-
|-
| [[6/5]], [[5/3]]
|80
| 4.359
|Augmented Unison / Minor Second
|D# / Eb
|J&/K@
|-
|-
| [[11/10]], [[20/11]]
|160
| 5.004
|Major Second
|E
|K
|-
|-
| [[15/13]], [[26/15]]
|240
| 7.741
|Augmented Second / Diminished Third
|E# / Fb
|K&/L@
|-
|-
| '''[[11/8]], [[16/11]]'''
|320
| '''8.682'''
|Minor Third
|F
|L
|-
|-
| '''[[8/7]], [[7/4]]'''
|400
| '''8.826'''
|Major Third / Diminished Fourth
|F# / Gb
|L&/M@
|-
|-
| [[12/11]], [[11/6]]
|480
| 9.363
|Perfect Fourth
|G
|M
|-
|-
| '''[[5/4]], [[8/5]]'''
|560
| '''13.686'''
|Augmented Fourth
|G#
|M&
|-
|-
| [[14/11]], [[11/7]]
|640
| 17.508
|Diminished Fifth
|Ab
|N@
|-
|-
| '''[[4/3]], [[3/2]]'''
|720
| '''18.045'''
|Perfect Fifth
|A
|N
|-
|-
| [[13/12]], [[24/13]]
|800
| 21.427
|Augmented Fifth / Minor Sixth
|A# / Bb
|N&/O@
|-
|-
| [[10/9]], [[9/5]]
|880
| 22.404
|Major Sixth
|B
|O
|-
|-
| [[7/5]], [[10/7]]
|960
| 22.512
|Augmented Sixth / Diminished Seventh
|B# / Cb
|O&/P@
|-
|-
| [[15/11]], [[22/15]]
|1040
| 23.049
|Minor Seventh
|C
|P
|-
|-
| [[13/10]], [[20/13]]
|1120
| 25.786
|Major Seventh / Diminished Octave
|C# / Db
|P&/J@
|-
|-
| [[7/6]], [[12/7]]
|1200
| 26.871
|Octave
|D
|J
|}
One advantage of this notation is that its notated D major scale, D E F# G A B C# D, directly corresponds to 15edo’s zarlino LH Ionian scale. However, this only holds true for the key of D. Furthermore, the perfect 4th and/or 5th of most other keys is notated the same way a diminished or augmented fourth or fifth is in standard diatonic. For example, in the key of A the perfect fifth is E#.
 
==== Zarlino notation (heptatonic) ====
15edo's zarlino scale can also be treated as the primary scale, analogously to diatonic.
{| class="wikitable"
!Cents
!Note name(s)
|-
|-
| [[11/9]], [[18/11]]
|0
| 27.408
|D
|-
|-
| [[13/11]], [[22/13]]
|80
| 30.790
|D#
|-
|-
| ''[[14/13]], [[13/7]]''
|160
| ''31.702''
|Eb
|-
|-
| [[16/15]], [[15/8]]
|240
| 31.731
|E
|-
|-
| ''[[9/7]], [[14/9]]''
|320
| ''35.084''
|F
|-
|-
| [[9/8]], [[16/9]]
|400
| 36.090
|F#
|-
|-
| ''[[15/14]], [[28/15]]''
|480
| ''39.443''
|Gb
|-
|-
| '''[[16/13]], [[13/8]]'''
|560
| '''39.472'''
|G
|}
 
{| class="wikitable center-all"
|+Patent val mapping
|-
|-
! Interval, complement
|640
! Error (abs, [[cent|cents]])
|G# / Ab
|-
|-
| [[18/13]], [[13/9]]
|720
| 3.382
|A
|-
|-
| [[6/5]], [[5/3]]
|800
| 4.359
|A#
|-
|-
| [[11/10]], [[20/11]]
|880
| 5.004
|Bb
|-
|-
| [[15/13]], [[26/15]]
|960
| 7.741
|B
|-
|-
| '''[[11/8]], [[16/11]]'''
|1040
| '''8.682'''
|C
|-
|-
| '''[[8/7]], [[7/4]]'''
|1120
| '''8.826'''
|C# / Db
|-
|-
| [[12/11]], [[11/6]]
|1200
| 9.363
|D
|-
|}
| '''[[5/4]], [[8/5]]'''
 
| '''13.686'''
==== Porcupine "quill" notation (octatonic) ====
Porcupine notation can also be based on the Porcupine[8] LLLLLLLs scale using eight nominals: either α β χ δ ε φ γ η or A B C D E F G H. Latin letters are easier to type and more generalizable, but they have the downside of conflicts with standard notation. Thus, Greek letters can be used in their place with a close resemblance to the spelling of ABCDEFGHA. The letters are not in greek alphabetic order.
 
The eight nominals form the base diatonic scale. In the "quill name" column, the "quill" is the name given to the two-edostep interval (160¢) of 15edo while the "small quill" (80¢) is the chroma of 15edo. This produces a very consistent notation for both Porcupine[8] and Blackwood[10], moreso than putting 15edo into a 5L 2s framework.
{| class="wikitable"
!Cents
!Quill Name
!MOSstep Name
!Note names (Greek)
!Note names (Latin)
|-
|-
| [[14/11]], [[11/7]]
|0
| 17.508
|Zeroquill
|Perfect 0-step
|α - α
|A - A
|-
|-
| '''[[4/3]], [[3/2]]'''
|80
| '''18.045'''
|Small Quill / Half Quill
|Diminished 1-step
|α - β\
|A - Bb
|-
|-
| [[13/12]], [[24/13]]
|160
| 21.427
|Quill
|Perfect 1-step
|α - β
|A - B
|-
|-
| [[10/9]], [[9/5]]
|240
| 22.404
|Small Diquill
|Minor 2-step
|α - χ\
|A - Cb
|-
|-
| [[7/5]], [[10/7]]
|320
| 22.512
|Large Diquill
|Major 2-step
|α - χ
|A - C
|-
|-
| [[15/11]], [[22/15]]
|400
| 23.049
|Small Triquill
|Minor 3-step
|α - δ\
|A - Db
|-
|-
| [[13/10]], [[20/13]]
|480
| 25.786
|Large Triquill
|Major 3-step
|α - δ
|A - D
|-
|-
| [[7/6]], [[12/7]]
|560
| 26.871
|Small Fourquill
|Minor 4-step
|α - ε\
|A - Eb
|-
|-
| [[11/9]], [[18/11]]
|640
| 27.408
|Large Fourquill
|Major 4-step
|α - ε
|A - E
|-
|-
| [[13/11]], [[22/13]]
|720
| 30.790
|Small Fivequill
|Minor 5-step
|α - φ\
|A - Fb
|-
|-
| [[16/15]], [[15/8]]
|800
| 31.731
|Large Fivequill
|Major 5-step
|α - φ
|A - F
|-
|-
| [[9/8]], [[16/9]]
|880
| 36.090
|Small Sixquill
|Minor 6-step
|α - γ\
|A - Gb
|-
|-
| '''[[16/13]], [[13/8]]'''
|960
| '''39.472'''
|Large Sixquill
|Major 6-step
|α - γ
|A - G
|-
|-
| ''[[15/14]], [[28/15]]''
|1040
| ''40.557''
|Small Sevenquill
|Perfect 7-step
|α - η\
|A - Hb
|-
|-
| ''[[9/7]], [[14/9]]''
|1120
| ''44.916''
|Large Sevenquill
|Augmented 7-step
|α - η
|A - H
|-
|-
| ''[[14/13]], [[13/7]]''
|1200
| ''48.298''
|Octoquill
|Perfect 8-step
|α - α
|A - A
|}
|}
A regular keyboard can be designed using this system by placing 7 black keys as Porcupine[7] and 8 whites as Porcupine[8]. In fact, [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] has already done this with his pink Halberstadt keyboard.


==== Selected 13-limit intervals ====
== Approximation to JI ==
[[File:15ed2-001.svg|alt=alt : Your browser has no SVG support.]]
[[File:15ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 13-limit intervals]]
15edo offers some minor improvements over 12et in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a [[5L 5s]] [[mos scale]] wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as the [[blackwood]] temperament, named after [[Easley Blackwood Jr.]], who is the first to document its existence. It has also been written on extensively by [[Igliashon Jones]] in the paper [http://www.cityoftheasleep.com/etc/5nEDOs.pdf ''Five is Not an Odd Number'']. For an in-depth treatment of harmony in 15edo based on this temperament (and its 7- and 11-limit extensions), see [[Blackwood temperament modal harmony (in 15edo)]].


== Notation ==
=== 15-odd-limit interval mappings ===
There are a variety of other ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15-edo composition.
{{Q-odd-limit intervals}}


=== Blackwood Notation ===
== Regular temperament properties ==
* '''Decimal Version:''' Using the nominals 1-0 (with 0 representing "10"), one of the three chains of 5-edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).
{| class="wikitable center-4 center-5 center-6"
* '''Guitar Version:''' On a 15-edo guitar, because the "perfect fourth" comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or "perfect fourths" and "perfect fifths") that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.
|-
 
! rowspan="2" | [[Subgroup]]
=== Porcupine Notation ===
! rowspan="2" | [[Comma list]]
* See the main [[Porcupine Notation|porcupine notation]] page.
! rowspan="2" | [[Mapping]]
* Porcupine notation bases porcupine[8] LLLLLLLs scale using eight nominals α β χ δ ε φ γ η. Others have proposed ABCDEFGHA but conflicts with european notation have caused many to reject this approach. Thus greek letters can be used in place with a close resemblance to the spelling of ABCDEFGHA.
! rowspan="2" | Optimal<br>8ve stretch (¢)
 
! colspan="2" | Tuning error
=== Interval names for Porcupine[8] ===
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 128/125, 250/243
| {{Mapping| 15 24 35 }}
| −5.75
| 4.63
| 5.81
|-
| 2.3.5.7
| 28/27, 49/48, 126/125
| {{Mapping| 15 24 35 42 }}
| −3.55
| 5.56
| 6.97
|-
| 2.3.5.7.11
| 28/27, 49/48, 55/54, 77/75
| {{Mapping| 15 24 35 42 52 }}
| −3.34
| 4.99
| 6.25
|}


{| class="wikitable"
=== Errors by subgroup ===
{| class="wikitable center-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Errors by subgroup
|-
|-
| | Cents
! Subgroup
| | Interval Name
! Mapping
| | Note names
! Adjusted error (¢)
|-
|-
| | 80
| 2.3
| | Half Quill
| {{Mapping| 15 24 }}
| | α - β\
| 8.979801
|-
|-
| | 160
| 2.5
| | Quill
| {{Mapping| 15 35 }}
| | α - β
| 6.826357
|-
|-
| | 240
| 2.7
| | Small Diquill
| {{Mapping| 15 42 }}
| | α - χ\
| 4.418738
|-
|-
| | 320
| 2.11
| | Large Diquill
| {{Mapping| 15 52 }}
| | α - χ
| 4.336492
|-
|-
| | 400
| 2.3.5
| | Small Triquill
| {{Mapping| 15 24 35 }}
| | α - δ\
| 10.742841
|-
|-
| | 480
| 2.3.7
| | Large Triquill
| {{Mapping| 15 24 42 }}
| | α - δ
| 17.481581
|-
|-
| | 560
| 2.3.11
| | Small Fourquill
| {{Mapping| 15 24 52 }}
| | α - ε\
| 16.831238
|-
|-
| | 640
| 2.5.7
| | Large Fourquill
| {{Mapping| 15 35 42 }}
| | α - ε
| 10.509269
|-
|-
| | 720
| 2.5.11
| | Small Fivequill
| {{Mapping| 15 35 52 }}
| | α - φ\
| 8.335693
|-
|-
| | 800
| 2.7.11
| | Large Fivequill
| {{Mapping| 15 42 52 }}
| | α - φ
| 8.002641
|-
|-
| | 880
| 2.3.5.7
| | Small Sixquill
| {{Mapping| 15 24 35 42 }}
| | α - γ\
| 15.603114
|-
|-
| | 960
| 2.3.5.11
| | Large Sixquill
| {{Mapping| 15 24 35 52 }}
| | α - γ
| 14.693746
|-
|-
| | 1040
| 2.3.7.11
| | Small Sevenquill
| {{Mapping| 15 24 42 52 }}
| | α - η\
| 18.660367
|-
|-
| | 1120
| 2.5.7.11
| | Large Sevenquill
| {{Mapping| 15 35 42 52 }}
| | α - η
| 11.462127
|-
|-
| | 1200
| 2.3.5.7.11
| | Octoquill
| {{Mapping| 15 24 35 42 52 }}
| | α - α
| 17.258371
|}
|}
A regular keyboard can be designed using this system placing 7 black keys as porcupine[7] and 8 whites as porcupine[8], and in fact, [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] has already done this with his pink Halberstadt keyboard.


== Rank two temperaments (with horagrams for octave-periodic MOS) ==
=== Uniform maps ===
{{Uniform map|edo=15}}
 
=== Rank-2 temperaments ===
* [[List of 15et rank two temperaments by badness]]
* [[List of 15et rank two temperaments by badness]]
* [[List of edo-distinct 15et rank two temperaments]]
* [[List of edo-distinct 15et rank two temperaments]]
Important MOSes include:
* [[blackwood]] 5L5s 2121212121 (2\15, 1\5)
* [[porcupine]] 7L1s 12222222 (2\15, 1\1)
* [[orgone]]/[[hanson]] 4L3s 3313131 (4\15, 1\1)
* [[augmented]] ([[augene]]) 3L6s 311311311 (1\15, 1\3)
* Pathological [[augmented]] ([[augene]]) 3L9s 211121112111 (1\15, 1\3)
* [[triforce]] 6L3s 221221221 (2\15, 1\3)
* Pathological [[1L 11s]] 4 1 1 1 1 1 1 1 1 1 1 (1\15, 1\1)
* Pathological [[2L 11s]] 2 1 1 1 1 1 1 2 1 1 1 1 1 (7\15, 1\1)
* Pathological [[1L 12s]] 3 1 1 1 1 1 1 1 1 1 1 1 (1\15, 1\1)
* Pathological [[1L 13s]] 2 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\15, 1\1)


{| class="wikitable"
{| class="wikitable center-all left-4 left-5"
|-
|-
! | Periods<br>per octave
! Periods<br>per 8ve
! | Period
! Generator
! | Generator
! Associated<br>ratio
! | Temperaments
! Temperaments
! Mos scales
|-
|-
| | 1
| 1
| | 15\15
| 1\15
| | 1\15
| 21/20
| | [[Nautilus]]/[[valentine]]
| [[Nautilus]]<br>[[Valentine]]
|
|-
|-
| | 1
| 1
| | 15\15
| 2\15
| | 2\15
| 11/10
| | [[Porcupine]]/[[opossum]]
| [[Porcupine]] / [[opossum]]
| [[1L 6s]], [[7L 1s]]
|-
|-
| | 1
| 1
| | 15\15
| 4\15
| | 4\15
| 6/5<br>77/64
| | [[Hanson]]/[[keemun]]/[[orgone]]
| [[Cata]] / [[keemun]] / [[catalan]]<br>[[Orgone]] / [[superkleismic]]
| [[3L 1s]], [[4L 3s]], [[4L 7s]]
|-
|-
| | 1
| 1
| | 15\15
| 7\15
| | 7\15
| 7/5
| | [[Progress]]
| [[Progress]]<br>[[Parakangaroo]]
| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]] [[2L 9s]], [[2L 11s]]
|-
|-
| | 3
| 3
| | 5\15
| 1\15
| | 1\15
| 16/15
| | [[Augmented]]/[[augene]]
| [[Augmented (temperament)|Augmented]] / [[augene]]
| [[3L 3s]], [[3L 6s]], [[3L 9s]]
|-
|-
| | 3
| 3
| | 5\15
| 2\15
| | 2\15
| 7/6
| | [[Triforce]]
| [[Triforce]]
| [[3L 3s]], [[6L 3s]]
|-
|-
| | 5
| 5
| | 3\15
| 1\15
| | 1\15
| 16/15
| | [[Blackwood]]/[[blacksmith]]
| [[Blackwood]]
| [[5L 5s]]
|}
|}
[[File:Screen Shot 2020-04-24 at 12.04.03 AM.png|none|thumb|986x986px|2\15 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 7L 1s]]
[[File:Screen Shot 2020-04-24 at 12.04.45 AM.png|none|thumb|735x735px|4\15 MOS using 1L 1s, 1L 2s, 3L 1s, 4L 3s, 4L 7s]]
[[File:Screen Shot 2020-04-24 at 12.05.29 AM.png|none|thumb|927x927px|7\15 MOS using 1L 1s, 2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s ]]


== Commas ==
===Commas===
 
15et [[tempering out|tempers out]] the following [[comma]]s using the [[patent val]] {{val| 15 24 35 42 52 56 }}.  
15 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val|15 24 35 42 52 56}}.)


{| 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>{{rd}}</ref>
! [[Monzo]]
![[Monzo]]
! [[Cent]]s
![[Cent]]s
! [[Color name]]
![[Color name]]
! Name(s)
!Name
|-
|-
| 3
|3
| [[256/243]]
|[[256/243]]
| {{monzo| 8 -5 }}
| {{monzo| 8 -5 }}
| 90.22
|90.225
| Sawa
|Sawa
| Limma, Pythagorean minor 2nd
|Blackwood comma, Pythagorean limma
|-
|5
|<abbr title="254803968/244140625">(18 digits)</abbr>
|{{monzo| 20 5 -12 }}
|74.01
|Saquadtrigu
| [[Hypovishnuzma]]
|-
|-
| 5
|5
| [[250/243]]
|[[250/243]]
| {{monzo| 1 -5 3 }}
|{{monzo| 1 -5 3 }}
| 49.17
| 49.166
| Triyo
|Triyo
| Maximal diesis, Porcupine comma
|Porcupine comma, maximal
|-
|-
| 5
|5
| [[128/125]]
|[[128/125]]
| {{monzo| 7 0 -3 }}
|{{monzo| 7 0 -3 }}
| 41.06
|41.059
| Trigu
|Trigu
| Diesis, Augmented comma
| Augmented comma, lesser diesis
|-
|-
| 5
|5
| [[15625/15552]]
|[[15625/15552]]
| {{monzo| -6 -5 6 }}
|{{monzo| -6 -5 6 }}
| 8.11
|8.107
| Tribiyo
|Tribiyo
| Kleisma, semicomma majeur
|Kleisma
|-
|-
| 7
|7
| [[28/27]]
|[[28/27]]
| {{monzo| 2 -3 0 1 }}
|{{monzo| 2 -3 0 1 }}
| 62.96
|62.961
| Zo
|Zo
| Septimal third tone, Small septimal chroma
| Septimal third-tone, trienstonic comma
|-
|-
| 7
|7
| [[1029/1000]]
|[[1029/1000]]
| {{monzo| -3 1 -3 3 }}
|{{monzo| -3 1 -3 3 }}
| 49.49
|49.492
| Trizogu
| Trizogu
| Keega
|Keega
|-
|-
| 7
|7
| [[49/48]]
|[[49/48]]
| {{monzo| -4 -1 0 2 }}
|{{monzo| -4 -1 0 2 }}
| 35.70
|35.697
| Zozo
|Zozo
| Slendro diesis
|Semaphoresma, Slendro diesis
|-
|-
| 7
|7
| [[64/63]]
|[[64/63]]
| {{monzo| 6 -2 0 -1 }}
|{{monzo| 6 -2 0 -1 }}
| 27.26
|27.264
| Ru
|Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|Archytas' comma, septimal comma
|-
|-
| 7
|7
| [[64827/64000]]
| [[64827/64000]]
| {{monzo| -9 3 -3 4 }}
|{{monzo| -9 3 -3 4 }}
| 22.23
|22.227
| Laquadzo-atrigu
|Laquadzo-atrigu
| Squalentine
|Squalentine comma
|-
|-
| 7
|7
| [[875/864]]
|
| {{monzo| -5 -3 3 1 }}
[[875/864]]
| 21.90
|{{monzo| -5 -3 3 1 }}
| Zotriyo
|21.902
| Keema
|Zotriyo
|Keema
|-
|-
| 7
|7
| [[126/125]]
|[[126/125]]
| {{monzo| 1 2 -3 1 }}
|{{monzo| 1 2 -3 1 }}
| 13.79
| 13.795
| Zotrigu
| Zotrigu
| Septimal semicomma, Starling comma
| Starling comma
|-
|-
| 7
|7
| [[4000/3969]]
|[[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
|
| 13.47
{{monzo| 5 -4 3 -2 }}
|13.469
| Rurutriyo
| Rurutriyo
| Octagar
|Octagar comma
|-
|-
| 7
|7
| [[1029/1024]]
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
|{{monzo| -10 1 0 3 }}
| 8.43
|8.433
| Latrizo
|Latrizo
| Gamelisma
|Gamelisma
|-
|-
| 7
|7
| [[6144/6125]]
|[[6144/6125]]
| {{monzo|  11 1 -3 -2 }}
|{{monzo|  11 1 -3 -2 }}
| 5.36
|5.362
| Saruru-atrigu
|Saruru-atrigu
| Porwell
|Porwell comma
|-
|-
| 7
|7
| <abbr title="250047/250000">(12 digits)</abbr>
|<abbr title="250047/250000">(12 digits)</abbr>
| {{monzo| -4 6 -6 3 }}
|{{monzo| -4 6 -6 3 }}
| 0.33
|0.325
| Trizogugu
| Trizogugu
| [[Landscape comma]]
|[[Landscape comma]]
|-
|-
| 11
|11
| [[100/99]]
|[[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
|{{monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.399
| Luyoyo
|Luyoyo
| Ptolemisma
|Ptolemisma
|-
|-
| 11
| 11
| [[121/120]]
|[[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
|
| 14.37
{{monzo| -3 -1 -1 0 2 }}
|14.367
| Lologu
| Lologu
| Biyatisma
|Biyatisma
|-
|-
| 11
|11
| [[176/175]]
|[[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
|{{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.865
| Lorugugu
|Lorugugu
| Valinorsma
| Valinorsma
|-
|-
| 11
|11
| [[65536/65219]]
|[[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
|{{monzo| 16 0 0 -2 -3 }}
| 8.39
|8.394
| Satrilu-aruru
|Satrilu-aruru
| Orgonisma
|Orgonisma
|-
|-
| 11
| 11
| [[385/384]]
|[[385/384]]
| {{monzo| -7 -1 1 1 1 }}
|{{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.503
| Lozoyo
|Lozoyo
| Keenanisma
| Keenanisma
|-
|-
| 11
|11
| [[441/440]]
|[[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
|3.930
| Luzozogu
|Luzozogu
| Werckisma
|Werckisma
|-
|-
| 11
|11
| [[4000/3993]]
|[[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
|{{monzo| 5 -1 3 0 -3 }}
| 3.03
|3.032
| Triluyo
|Triluyo
| Wizardharry
| Wizardharry
|-
|-
| 11
|11
| [[3025/3024]]
|
| {{monzo| -4 -3 2 -1 2 }}
[[3025/3024]]
| 0.57
|{{monzo| -4 -3 2 -1 2 }}
| Loloruyoyo
|0.572
|Loloruyoyo
| Lehmerisma
| Lehmerisma
|-
|-
| 13
|13
| [[91/90]]
|[[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
|{{monzo| -1 -2 -1 1 0 1 }}
| 19.13
|19.130
| Thozogu
| Thozogu
| Superleap
|Superleap comma, biome comma
|-
|-
| 13
|13
| [[676/675]]
|[[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
|{{monzo| 2 -3 -2 0 0 2 }}
| 2.56
|2.563
| Bithogu
|Bithogu
| Parizeksma
|Island comma
|}
|}
<references/>
<references />


== Books ==
== Octave stretch or compression ==
=== Theory ===
15edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]]. Shrunk-octaves versions of 15edo include [[equal tuning|50ed10]], [[47zpi]] and [[ed12|54ed12]].
* [http://tonalsoft.com/sonic-arts/darreg/dar35.htm The 15-Tone Scale System] by [[Ivor Darreg]] (Mirrored at http://www.webcitation.org/5xZyzKBEW. Originally at [http://sonic-arts.org/darreg/dar35.htm], now broken)
* [https://web.archive.org/web/20110713044141/http://www.inteas.com/Penta01.htm The Pentadecaphonic System] (Originally at [http://www.inteas.com/Penta01.htm], now broken)
* [http://www.webcitation.org/5xeJYBsDg 15-EDO Tutorial] by [[Brent Carson]] (Originally at [http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html], now broken)


=== Guitar ===
== Scales ==
[http://www.swordguitars.com Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.] - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.
Some scales commonly used in 15edo, written in a common mode, in steps of 15edo:
 
=== MOS scales ===
* Augene[6] [[3L 3s]] (period = 5\15, gen = 1\15): 4 1 4 1 4 1
*Augene[9] [[3L 6s]] (period = 5\15, gen = 1\15): 3 1 1 3 1 1 3 1 1
*Augene[12] [[3L 9s]] (period = 5\15, gen = 1\15): 2 1 1 1 2 1 1 1 2 1 1 1
*Triforce[6] [[3L 3s]] (period = 5\15, gen = 2\15): 3 2 3 2 3 2
*Triforce[9] [[6L 3s]] (period = 5\15, gen = 2\15): 2 1 2 2 1 2 2 1 2
*Porcupine[7] [[1L 6s]] (gen = 2\15): 3 2 2 2 2 2 2
*Porcupine[8] [[7L 1s]] (gen = 2\15): 2 1 2 2 2 2 2 2
*Hanson/Keemun/Orgone[7] [[4L 3s]] (gen = 4\15): 1 3 1 3 1 3 3
*Hanson/Keemun/Orgone[11] [[4L 7s]] (gen = 4\15): 1 2 1 1 2 1 1 2 1 2 1
*Blackwood[10] [[5L 5s]] (period = 3\15, gen = 1\15): 2 1 2 1 2 1 2 1 2 1 (Blackwood Decatonic)
 
[[File:BlackwoodMajor 15edo.mp3]]
Blackwood decatonic, major mode, in 15edo
 
=== Other scales ===
*[[Zarlino]]/Ptolemy diatonic, "just" major (Porcupine[7] 6|0 b4 #7): 3 2 1 3 2 3 1[[File:15edo nicetone.mp3|thumb|Zarlino/Ptolemy diatonic/Nicetone major scale 3 2 1 3 2 3 1 in 15edo tuning]]
*inverse of [[Zarlino]]/Ptolemy diatonic, natural minor (Porcupine[7] 3|3 #2 b6): 3 1 2 3 1 3 2
*tetrachordal major: 3 2 1 3 3 2 1
*inverse of tetrachordal major, tetrachordal minor: 3 1 2 3 1 2 3
*"just"/[[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Porcupine[7]): 3 2 4 2 4
* "just"/[[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (inverse of "just" major pentatonic, subset of Porcupine[7]): 4 2 3 4 2
*Porcupine bright major #7 (Porcupine harmonic major) - Porcupine[7] 6|0 #7: 3 2 2 2 2 3 1
*Porcupine bright major #6 #7 (Porcupine melodic major) - Porcupine[7] 6|0 #7: 3 2 2 2 3 2 1
*Porcupine bright minor #2 (Porcupine harmonic minor) - Porcupine[7] 4|2 #2: 3 1 3 2 2 2 2 (mode of bright major #7)
*Porcupine dark minor #2 (Porcupine melodic minor) - Porcupine[7] 3|3 #2: 3 1 2 3 2 2 2 (inverse of bright major #6 #7)
*Porcupine bright harmonic 11th mode - Porcupine[7] 6|0 b7: 3 2 2 2 2 1 3
* [[The Pinetone System#Pinetone harmonic diminished octatonic|Pinetone diminished octatonic]] / Porcupine[8] bright minor #2 - Porcupine[8] 2|5 #5: 2 2 1 3 1 2 2 2
*"just" harmonic minor: 3 1 2 3 1 4 1
*"just" harmonic major: 3 2 1 3 1 4 1
* "just" melodic minor ascending: 3 1 2 3 2 3 1
* Marvel double harmonic hexatonic (Augene[6] [[4M]]): 1 4 1 4 4 1, 1 4 4 1 4 1
*[[Marvel double harmonic major]]: 1 4 1 3 1 4 1
*Marvel double harmonic nonatonic (Augene[9] [[4M]]): 1 3 1 1 3 1 3 1 1, 1 1 3 1 3 1 1 3 1
*Marvel double harmonic decatonic: 1 3 1 1 2 1 1 3 1 1
*enharmonic trichord octave species: 1 5 3 1 5 , 5 1 3 5 1
*chromatic tetrachord octave species: 1 1 4 3 1 1 4, 4 1 1 3 4 1 1, 1 4 1 3 1 4 1
*[[Chopsticks]] double octave scale: 4 2 4 2 4 2 4 2 4 2
*[[User:BudjarnLambeth/Antechinus|antechinus scale]] (''nonoctave period'')
*[[5- to 10-tone scales in 47zpi]] (slightly stretched 15edo)
 
=== Horagrams ===
[[File:Screen Shot 2020-04-24 at 12.04.03 AM.png|none|thumb|986x986px|2\15 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 7L 1s]]
[[File:Screen Shot 2020-04-24 at 12.04.45 AM.png|none|thumb|735x735px|4\15 MOS using 1L 1s, 1L 2s, 3L 1s, 4L 3s, 4L 7s]]
[[File:Screen Shot 2020-04-24 at 12.05.29 AM.png|none|thumb|927x927px|7\15 MOS using 1L 1s, 2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s ]]
 
== Diagrams ==
[[File:15edo_wheel.png|alt=15edo wheel.png|225x225px|15edo wheel.png]] [[File:15edo_wheel_02.png|alt=15edo wheel 02.png|250x250px|15edo wheel 02.png]] [[File:15edo_wheel_03.png|alt=15edo wheel 03.png|220x220px|15edo wheel 03.png]]


=== Keyboard ===
=== Porcupine (Halberstadt-Inspired) Keyboard Chord Shapes ===
The following chord shapes use the porcupine layout for 15edo (quasi-Halberstadt) (see [[#Halberstadt-Inspired_Keyboards|Halberstadt-Inspired Keyboards]] below).
[[File:Porcupine keyboard major triad shapes.png|none|thumb|500x500px|Major chord shapes for a Porcupine keyboard in 15edo, using Blackwood logic (i.e. native fifth notation) for the letters.]]
[[File:Porcupine keyboard major triad shapes.png|none|thumb|500x500px|Major chord shapes for a Porcupine keyboard in 15edo, using Blackwood logic (i.e. native fifth notation) for the letters.]]


== Images ==
== Instruments ==
=== Halberstadt-Inspired Keyboards ===
<gallery widths="300px">
File:15_tone_keyboard.png|Porcupine layout for 15edo
File:Screen Shot 2020-04-23 at 11.59.17 PM.png|Hanson layout for 15edo
File:15edo kb3.png|Zarlino layout for 15edo
</gallery>


[[File:15edo_wheel.png|alt=15edo wheel.png|225x225px|15edo wheel.png]] [[File:15edo_wheel_02.png|alt=15edo wheel 02.png|250x250px|15edo wheel 02.png]] [[File:15edo_wheel_03.png|alt=15edo wheel 03.png|220x220px|15edo wheel 03.png]]
=== Lumatone ===
''See [[Lumatone mapping for 15edo]]''
 
== Guitars ==
[[Benjamin Strange]] built a 15edo Hello Kitty guitar and documented the process on his blog: https://www.strangeguitarworks.com/benjamins-fender-hello-kitty-microtonal-strat-of-doom/
 
== Music ==
{{Main| 15edo/Music }}
{{Catrel|15edo tracks}}
; [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]{{dead link}}


== Compositions ==
== Further reading ==
=== Theory===
* Carson, Brent. [http://web.archive.org/web/20121025054304/http://home.comcast.net/~brentishere/15noteequaltempermenttutorial.html Fifteen Note Equal Temperment]
*[[Darreg, Ivor]]. ''[http://tonalsoft.com/sonic-arts/darreg/dar35.htm 15-Tone Scale System]''. 1991. (Originally at [http://sonic-arts.org/darreg/dar35.htm], now broken)
*InTeAS. ''[https://web.archive.org/web/20110713044141/http://www.inteas.com/Penta01.htm The Pentadecaphonic System]'' (2001, archived)


* [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]
=== Guitar ===
* [https://xenharmonicgod.bandcamp.com/album/weigel-family-christmas-xenharmonic-chocolate WEIGEL FAMILY CHRISTMAS (xenharmonic chocolate)], an album of xenharmonic Christmas covers played by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel], many are in 15edo
*[[Sword, Ron]]. [http://www.metatonalmusic.com/books.html Pentadecaphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for Fifteen Equal Divisions of the Octave]''. (A large repository of all known scales and temperament families in the 15edo system. 300+ examples with chord-scale progressions.)''
* [https://soundcloud.com/overtoneshock/keep-out-of-my-psyche-15-edo Keep out of my Psyche] by Stephen Weigel
* [https://sites.google.com/site/teamouse/home Mizarian Porcupine Overture] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/MizarianPorcupineOverture.mp3 play]'' by [[Herman_Miller|Herman Miller]] ([http://teamouse.googlepages.com/home Herman Miller]) ([[Regular_Temperaments#porcupine|porcupine]] chord progressions; this is the song that 'porcupine' is named after)
* ''[http://www.microtonalmusic.net/audio/15edostudy.mp3 Study for Bells]'' by [[Daniel_Thompson|Daniel Thompson]] ([http://danielthompson.blogspot.com/ Daniel Thompson]) (Jan. 2007)
* [http://www.soundclick.com/bands/songInfo.cfm?bandID=145852&songID=2920478 Hyperimprovisation 3.3] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/Hyperimprovisation3.3.mp3 play]'' by [[Jacob Barton]] (2003)
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=145852&songID=5483130+OFOIOB OFOIOB] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/OFOIOB.mp3 play]'' by Jacob Barton
* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Oldani/15%20tone%20E.T.Improvisationn.mp3 15 Tone ET Improvisationn]'' by [[Norbert Oldani]]
* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/15ET.mp3 Elegy in 15ET]'' by [[Aaron Andrew Hunt]]
* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/15ETa3fugue2.mp3 Fugue a3 in 15ET]'' by Aaron Andrew Hunt
* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/12%20-%2012.%2015%20octave.mp3 Comets Over Flatland 12]'' by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/13%20-%2013.%2015%20octave.mp3 Comets Over Flatland 13] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/16%20-%2016.%2015%20octave.mp3 Comets Over Flatland 16] by [[Randy Winchester]]
* Study for Kyle Gann by [http://www.akjmusic.com/works.html Aaron K. Johnson] (12-out-of-15)
* [http://azuma-asobi.com/Music/index.html Rick McGowan]: [http://azuma-asobi.com/Music/Music-FullWorks.html Four Ballet Scenes]
* "[http://micro.soonlabel.com/15-ET/daily20110619_millers_porcupine_7a.mp3 Gently Playing With Miller's Porcupine]" by [http://www.chrisvaisvil.com Chris Vaisvil] (uses Miller's Porcupine-7 mode 2 2 2 3 2 2 2)
* [http://micro.soonlabel.com/15-ET/daily20110619_15edo_after_dark_in_the_pedway.mp3 After Dark on the Pedway] by [http://chrisvaisvil.com/?p=969 Chris Vaisvil]
* [http://micro.soonlabel.com/15-ET/15_sandles.mp3 15 Sandles] by [http://chrisvaisvil.com/?p=722 Chris Vaisvil] [http://micro.soonlabel.com/15-ET/15_sandles.mid scordatura midi file]
* [http://micro.soonlabel.com/15-ET/Improv_on_15edo/Improv_on_15.mp3 Improv on 15 EDO] by [http://www.chrisvaisvil.com Chris Vaisvil] [http://micro.soonlabel.com/15-ET/Improv_on_15edo/Improv_on_15.mid scordatura midi file] and [http://micro.soonlabel.com/15-ET/Improv_on_15edo/Improv_on_15.pdf scordatura PDF score]
* [http://micro.soonlabel.com/15-ET/20120317-synth-15edo-trolls.mp3 15 edo Trolls] by [[Chris_Vaisvil|Chris Vaisvil]] - [http://chrisvaisvil.com/?p=2206 details]
* ''[http://micro.soonlabel.com/15-ET/20120903_15edo_through_the_fire_of_the_sun.mp3 Through the Fire of the Sun (15 edo rock band)]'' by [http://chrisvaisvil.com Chris Vaisvil]
* [http://chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/ 2-2-1-2-2-1-2-2-1 mode of 15 edo] [http://micro.soonlabel.com/15-ET/20130831_221of15.mp3 play] by [[Chris_Vaisvil|Chris Vaisvil]]
* [http://youtu.be/6TyQ9kDm2fk Ode For Ada] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html blog entry])
* [http://www.seraph.it/dep/det/Sahara.mp3 Sahara] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/152ffe3684d7a11a91e5bc6c92a5bc9a-271.html blog entry])
* Suite in 15-Note Equal Tuning, opus 33 by [[Easley Blackwood]] (as well as one of the Twelve Microtonal Etudes, opus 28)
* [[:File:15edo-Chords.ogg|15edo-Chords.ogg]] Some nice sounds I found in 15 EDO
* [[:File:happenstance15.ogg|happenstance15.ogg]] Sonic experiment in 15. Somewhat familiar tonality.
* [http://www.youtube.com/watch?v=kQECU5ecCd4 Portrait of insects with 15-tone equal tempered guitar music]
* [http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et PentadecafoniCoda (15et)] by F.F.F. Fiale
* [http://soonlabel.com/xenharmonic/archives/1946 Cuckoo-Rag Fugue by Claudi Meneghin]
* [http://soonlabel.com/xenharmonic/archives/2313 15-penny jingle, by Claudi Meneghin]
* [http://soonlabel.com/xenharmonic/archives/2710 Tocada in 15edo, by Claudi Meneghin]
* [https://soundcloud.com/andrew-j-milne/a-broken-stern-2012 A Broken Stern (2012) by Andrew J Milne on SoundCloud]
* [https://youtu.be/wrLfJ8j2Kt0 Canada Geese by Diamond Doll (punk song)]
* [https://youtu.be/P4NXb2-LJZw Mask On by Diamond Doll (punk song)]
* [https://soundcloud.com/user-455564099/qryzglulaemledaid 15 Somethings' Alien Disco] by Jutomi
* [https://soundcloud.com/user-455564099/march-of-the-lichens March of the Lichens] by Jutomi
* [https://youtu.be/Y0oSyQqFNmc The Wish To Wish The Wish] by [[User:Userminusone|Userminusone]]


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