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== Theory ==
== Theory ==
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]]<nowiki/>s (supporting [[blackwood]], which tempers out the [[Pythagorean limma]]), and 5 circles of three [[5/4]]<nowiki/>s (supporting [[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.


=== Composition theory ===
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.
 
* [[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.
* [[Metallic harmony]] - harmony involving stacking sevenths instead of thirds; 15edo is one of the systems it is intended for.


=== Tuning theory ===
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]].
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. The perfect fifth of 15edo returns to the octave if stacked five times. In regular temperament terms, this  means the [[Pythagorean limma]] is tempered out, which is radically different than a meantone system. This has a variety of ramifications for chord progressions based on {{w|Function (music)|functional harmony}}, because if you use the equipentatonic as your "diatonic scale", the same interval can have multiple functions. Additionally, 15 being equal to {{nowrap|3 × 5}} also implies that 15edo contains five sets of [[3edo]].
 
15edo 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, 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.  
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.  
{| class="wikitable"
|+Logarithmic divisions of intervals in 15edo
!Interval
!2
!3
!4
!5
!6
!7
|-
|2 (Minor tone)
|★
|
|
|
|
|
|-
|3 (Major tone)
|
|★
|
|
|
|
|-
|4 (Minor third)
|★
|
|★
|
|
|
|-
|5 (Major third)
|
|
|
|★
|
|
|-
|6 (Perfect fourth)
|★
|★
|
|
|★
|
|-
|7 (Small tritone)
|
|
|
|
|
|★
|-
|8 (Large tritone)
|★
|
|★
|
|
|
|-
|9 (Perfect fifth)
|
|★
|
|
|
|
|-
|10 (Minor sixth)
|★
|
|
|★
|
|
|-
|11 (Major sixth)
|
|
|
|
|
|
|-
|12 (Small minor seventh)
|★
|★
|★
|
|★
|
|-
|13 (Large minor seventh)
|
|
|
|
|
|
|-
|14 (Major seventh)
|★
|
|
|
|
|★
|-
|15 (Octave)
|
|★
|
|★
|
|
|}


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.
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]] (blackwood) scales are also an option.
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&nbsp;6s]] (onyx) and [[5L&nbsp;5s]] (blackwood) 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]].
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 ====
=== Prime harmonics ===
{{Harmonics in equal|15}}
{{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.


==Intervals==
==Intervals==
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|}
|}


===Alternate interval names===
===Alternative interval names===
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
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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 ==
There are a variety of ways to notate 15edo, and the choice of notation depends heavily on which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15edo composition.
There are many ways to notate 15edo, and the choice of notation depends heavily on which temperament or scale one wishes to focus on.


===Ups and downs notation===
=== Notations generated by the fifth ===
15edo can be notated with [[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).
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.
{{Sharpness-sharp3a}}


[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
==== Ups and downs notation (heptatonic) ====
{{Sharpness-sharp3}}
15edo can be notated with [[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}}[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:{{Sharpness-sharp3}}
 
=== Sagittal notation===
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|30-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].
 
<imagemap>
File:15-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 439 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 439 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:15-EDO_Sagittal.svg]]
</imagemap>
 
===Blackwood Notation (Pentatonic)===
For note names, [[Kite Giedraitis]] proposes a possible alternative to heptatonic names, pentatonic names that omit B and merge E and F into a new letter, "eef" (it rhymes with leaf). Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. The circle of 5ths is C G D A Eef 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).


==== "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 dowperfect (never major or minor). This is similar to heptatonic interval names in 7edo, 14edo, 21edo, etc.
{| class="wikitable"
{| class="wikitable"
|-
|C
|C
|^C
|^C
Line 552: Line 411:
|vꘙ
|vꘙ
|ꘙ
|ꘙ
| ^ꘙ
|^ꘙ
|vG
|vG
|G
|G
|^G
|^G
| vA
|vA
|A
|A
|^A
|^A
Line 575: Line 434:
|v5
|v5
|P5
|P5
| ^5
|^5
|v6
|v6
|P6
|P6
|}
|}
=== Blackwood Notation (Decatonic)===
*'''Decimal Version:''' Using the nominals 1-0 (with 0 representing "10"), one of the three chains 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).
*'''Guitar Version:''' 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.


===Porcupine Notation (Heptatonic) ===
==== Sagittal 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 can likewise be named with respect to diatonic intervals.
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|30-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].<imagemap>
File:15-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 439 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 439 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:15-EDO_Sagittal.svg]]
</imagemap>
 
==== 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 1-0 (with 0 representing "10"), 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).
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"
{| class="wikitable"
|-
!Cents
!Cents
! Interval Name(s)
!Interval Name(s)
! Note name(s)
!Note name(s)
!Diamond-mos (on symmetric mode)
|-
|-
|0
|0
| Unison
|Unison
|D
|D
|J
|-
|-
| 80
|80
|Augmented Unison / Minor Second
|Augmented Unison / Minor Second
| D# / Eb
|D# / Eb
|J&/K@
|-
|-
| 160
|160
|Major Second
|Major Second
|E
|E
|K
|-
|-
| 240
|240
|Augmented Second / Diminished Third
|Augmented Second / Diminished Third
|E# / Fb
|E# / Fb
|K&/L@
|-
|-
| 320
|320
| Minor Third
|Minor Third
|F
|F
|L
|-
|-
|400
|400
| Major Third / Diminished Fourth
|Major Third / Diminished Fourth
|F# / Gb
|F# / Gb
|L&/M@
|-
|-
|480
|480
| Perfect Fourth
|Perfect Fourth
| G
|G
|M
|-
|-
|560
|560
|Augmented Fourth
|Augmented Fourth
|G#
|G#
|M&
|-
|-
|640
|640
|Diminished Fifth
|Diminished Fifth
|Ab
|Ab
|N@
|-
|-
|720
|720
|Perfect Fifth
|Perfect Fifth
| A
|A
|N
|-
|-
| 800
|800
| Augmented Fifth / Minor Sixth
|Augmented Fifth / Minor Sixth
|A# / Bb
|A# / Bb
|N&/O@
|-
|-
|880
|880
| Major Sixth
|Major Sixth
| B
|B
|O
|-
|-
|960
|960
|Augmented Sixth / Diminished Seventh
|Augmented Sixth / Diminished Seventh
|B# / Cb
|B# / Cb
|O&/P@
|-
|-
|1040
|1040
|Minor Seventh
|Minor Seventh
|C
|C
|P
|-
|-
|1120
|1120
|Major Seventh / Diminished Octave
|Major Seventh / Diminished Octave
|C# / Db
|C# / Db
|P&/J@
|-
|-
| 1200
|1200
|Octave
|Octave
| D
|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#.


One neat thing about using this notation system is that its notated major scale, D E F# G A B C# D, directly corresponds to 15edo’s LH Nice-Ionian scale.
==== Zarlino notation (heptatonic) ====
 
15edo's zarlino scale can also be treated as the primary scale, analogously to diatonic.
==== Zarlino notation ====
When 15edo zarlino is treated as a MODMOS of porcupine, it can also be used as nominals. Note that each interval is given a ''functional name'', since MOS-based names act unexpectedly with this as a scale:
{| class="wikitable"
{| class="wikitable"
|-
!Cents
!Cents
!Functional name
!Note name(s)
! Note name(s)
|-
|-
|0
|0
|Unison
|D
|D
|-
|-
| 80
|80
|Minor Second / Chromatic Semitone / Semitone
|D#
| D#
|-
|-
| 160
|160
|Small Wholetone
|Eb
|Eb
|-
|-
| 240
|240
|Large Wholetone / Wolf Third
|E
|E
|-
|-
| 320
|320
|Minor Third
|F
|F
|-
|-
|400
|400
|Major Third
|F#
|F#
|-
|-
|480
|480
|Augmented Third / Perfect Fourth
|Gb
| Gb
|-
|-
|560
|560
|Wolf Fourth / Diminished Fifth
|G
|G
|-
|-
|640
|640
|Wolf Fifth / Augmented Fourth
|G# / Ab
|G# / Ab
|-
|-
|720
|720
|Perfect Fifth / Diminished Sixth
|A
| A
|-
|-
| 800
|800
|Minor Sixth
|A#
|A#
|-
|-
|880
|880
|Major Sixth
|Bb
| Bb
|-
|-
|960
|960
|Harmonic Seventh / Small Minor Seventh / Wolf Sixth
|B
|B
|-
|-
|1040
|1040
|Large Minor Seventh
|C
|C
|-
|-
|1120
|1120
|Major Seventh / Diminished Octave
|C# / Db
|C# / Db
|-
|-
| 1200
|1200
|Octave
|D
| D
|}
|}


===Porcupine Notation (Octatonic)===
==== 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.
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.  
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"
{| class="wikitable"
|-
!Cents
!Cents
!Quill Name
!Quill Name
!MOSstep Name
!MOSstep Name
! Note names (Greek)
!Note names (Greek)
!Note names (Latin)
!Note names (Latin)
|-
|-
Line 781: Line 653:
|-
|-
|480
|480
| Large Triquill
|Large Triquill
|Major 3-step
|Major 3-step
|α - δ
|α - δ
Line 793: Line 665:
|-
|-
|640
|640
| Large Fourquill
|Large Fourquill
|Major 4-step
|Major 4-step
|α - ε
|α - ε
Line 829: Line 701:
|-
|-
|1120
|1120
| Large Sevenquill
|Large Sevenquill
|Augmented 7-step
|Augmented 7-step
| α - η
|α - η
|A - H
|A - H
|-
|-
Line 840: Line 712:
|A - A
|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.
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.


Line 849: Line 720:
===15-odd-limit interval mappings===
===15-odd-limit interval mappings===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals}}
===Zeta peak index===
{{ZPI
| zpi = 47
| steps = 15.0534898676781
| step size = 79.7157343943591
| tempered height = 5.050324
| pure height = 4.390681
| integral = 1.104057
| gap = 14.918297
| octave = 1195.73601591539
| consistent = 8
| distinct = 7
}}


==Regular temperament properties==
==Regular temperament properties==
Line 1,244: Line 1,101:
*Blackwood[10] [[5L 5s]] (period = 3\15, gen = 1\15): 2 1 2 1 2 1 2 1 2 1 (Blackwood Decatonic)
*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]] [[:BlackwoodMajor 15edo.mp3|BlackwoodMajor 15edo.mp3]]
[[File:BlackwoodMajor 15edo.mp3]]
 
Blackwood decatonic, major mode, in 15edo
Blackwood decatonic, major mode, in 15edo


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