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→Notation: categorized notations more accurately. expanded on the disadvantages of porcupine notation. expanded on interval names in eef notation. cleaned up the writing.
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===Alternate interval names===
===Alternative interval names===
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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 temperament or 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.


Additional notation schemes can be found at [[15edo/Notation]].
Additional notation schemes can be found at [[15edo/Notation]].


=== 5edo-based notations ===
=== Notations generated by the fifth ===
These notations use the notes of one or two chains of 5edo as the nominals. They (except Blackwood decatonic notation) function basically the same as each other, but with notes and accidentals relabelled.
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.


====Ups and downs notation====
==== Ups and downs notation (heptatonic) ====
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}}
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}}


==== Blackwood "Eef" Notation ====
==== "Eef" notation (pentatonic) ====
For note names, [[Kite Giedraitis]] proposes a possible alternative to heptatonic names. This uses pentatonic note names as opposed to heptatonic note names, and other than the intervals, is identical to ups and downs, except for replacing E/F with a letter called "eef" (it rhymes with leaf) and omitting B. 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, like E, is a 5th above A. Eef, like F, is a 4th above C. The circle of 5ths is C G D A ꘙ C.  
[[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.  
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==== Sagittal notation====
==== 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|30-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].<imagemap>
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
File:15-EDO_Sagittal.svg
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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.
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 ====
=== 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).
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 can also use other systematic ways of naming nominals, such as ABC..., or use diamond-mos.
One could name the nominals with letters instead of numbers, such as ABC... or JKL...


=== Heptatonic notations ===
=== 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 (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 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.
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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.
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 misleadingly appears to be augmented or diminished. For example, in the key of A the perfect fifth is E#.


==== Zarlino notation (Heptatonic) ====
==== Zarlino notation (heptatonic) ====
15edo's zarlino scale can also be treated as the primary scale, analogously to diatonic.
15edo's zarlino scale can also be treated as the primary scale, analogously to diatonic.
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=== Porcupine "Quill" 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.


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