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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 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.


===Ups and downs notation===
Additional notation schemes can be found at [[15edo/Notation]].
 
=== 5edo-based notations ===
These notations use the notes of one or two chains of 5edo as the nominals.
 
==== Ups and downs notation ====
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).
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}}
{{Sharpness-sharp3a}}
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{{Sharpness-sharp3}}
{{Sharpness-sharp3}}


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


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</imagemap>
</imagemap>


===Blackwood Notation (Pentatonic)===
==== Blackwood guitar notation ====
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).
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.


{| class="wikitable"
==== Blackwood decatonic notation ====
|-
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).
|C
 
|^C
One can also use other systematic ways of naming nominals, such as ABC..., or use diamond-mos.
|vD
|D
|^D
|vꘙ
|ꘙ
| ^ꘙ
|vG
|G
|^G
| vA
|A
|^A
|vC
|C
|-
|P1
|^1
|v2
|P2
|^2
|v3
|P3
|^3
|v4
|P4
|^4
|v5
|P5
| ^5
|v6
|P6
|}
Alternatively, one can just use the letter E for this purpose, for consistency with the major pentatonic scale.
{| class="wikitable"
|-
|C
|^C
|vD
|D
|^D
|vE
|E
| ^E
|vG
|G
|^G
| vA
|A
|^A
|vC
|C
|-
|P1
|^1
|v2
|P2
|^2
|v3
|P3
|^3
|v4
|P4
|^4
|v5
|P5
| ^5
|v6
|P6
|}


=== Blackwood Notation (Decatonic)===
=== Heptatonic notations ===
*'''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) ===
==== 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 can likewise be named with respect to diatonic intervals.
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.


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! 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 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.
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 zarlino LH Ionian scale.


==== Zarlino notation ====
==== Zarlino notation (Heptatonic) ====
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:
15edo's zarlino scale can also be treated as the primary scale, analogously to diatonic. 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
!Functional name
! Note name(s)
!Note name(s)
|-
|-
|0
|0
Line 709: Line 659:
|D
|D
|-
|-
| 80
|80
|Minor Second / Chromatic Semitone / Semitone
|Minor Second / Chromatic Semitone / Semitone
| D#
|D#
|-
|-
| 160
|160
|Small Wholetone
|Small Wholetone
|Eb
|Eb
|-
|-
| 240
|240
|Large Wholetone / Wolf Third
|Large Wholetone / Wolf Third
|E
|E
|-
|-
| 320
|320
|Minor Third
|Minor Third
|F
|F
Line 731: Line 681:
|480
|480
|Augmented Third / Perfect Fourth
|Augmented Third / Perfect Fourth
| Gb
|Gb
|-
|-
|560
|560
Line 743: Line 693:
|720
|720
|Perfect Fifth / Diminished Sixth
|Perfect Fifth / Diminished Sixth
| A
|A
|-
|-
| 800
|800
|Minor Sixth
|Minor Sixth
|A#
|A#
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|880
|880
|Major Sixth
|Major Sixth
| Bb
|Bb
|-
|-
|960
|960
Line 765: Line 715:
|C# / Dd
|C# / Dd
|-
|-
| 1200
|1200
|Octave
|Octave
| D
|D
|}
|}
===Porcupine 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)
|-
|0
|Zeroquill
|Perfect 0-step
|α - α
|A - A
|-
|80
|Small Quill / Half Quill
|Diminished 1-step
|α - β\
|A - Bb
|-
|160
|Quill
|Perfect 1-step
|α - β
|A - B
|-
|240
|Small Diquill
|Minor 2-step
|α - χ\
|A - Cb
|-
|320
|Large Diquill
|Major 2-step
|α - χ
|A - C
|-
|400
|Small Triquill
|Minor 3-step
|α - δ\
|A - Db
|-
|480
| Large Triquill
|Major 3-step
|α - δ
|A - D
|-
|560
|Small Fourquill
|Minor 4-step
|α - ε\
|A - Eb
|-
|640
| Large Fourquill
|Major 4-step
|α - ε
|A - E
|-
|720
|Small Fivequill
|Minor 5-step
|α - φ\
|A - Fb
|-
|800
|Large Fivequill
|Major 5-step
|α - φ
|A - F
|-
|880
|Small Sixquill
|Minor 6-step
|α - γ\
|A - Gb
|-
|960
|Large Sixquill
|Major 6-step
|α - γ
|A - G
|-
|1040
|Small Sevenquill
|Perfect 7-step
|α - η\
|A - Hb
|-
|1120
| Large Sevenquill
|Augmented 7-step
| α - η
|A - H
|-
|1200
|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.


==Approximation to JI==
==Approximation to JI==
[[File:15ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 13-limit intervals]]
[[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 [[Blacksmith temperament modal harmony (in 15edo)]].
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 [[Blacksmith temperament modal harmony (in 15edo)|Blackwood temperament modal harmony (in 15edo)]].


===15-odd-limit interval mappings===
===15-odd-limit interval mappings===
Retrieved from "https://en.xen.wiki/w/15edo"