15edo: Difference between revisions

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In the 15edo system, major thirds cannot be divided perfectly into two, while minor 3rds, 4ths, wide tritones, subminor 7ths, and supermajor 7ths can. Similarly, 4ths, 5ths, and subminor 7ths can all be divided into 3 equal parts, while minor 3rds, tritones, and major 6ths cannot. This gives 15edo a whole new set of pitch symmetries and modes of limited transposition. Coupled with the lack of a [[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 12edo that contains recognizable major and minor triads (unless the 14edo supermajor triad is considered as a "recognizable major triad"), 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 recommended method to the notation of 15edo 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 15edo while the "small quill" (80¢) is the chroma of 15edo. This produces a very consistent notation for both porcupine[8] and Blackwood[10] and seems to work much better than attempting to put 15edo into a 5L 2s based framework.

=== Prime harmonics ===

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There are a variety of other ways to notate 15edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15edo composition.

=== Blackwood Notation ===

=== 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 ===

=== Porcupine Notation (Octotonic) ===

* See the main [[Porcupine Notation|porcupine notation]] page.

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 their place with a close resemblance to the spelling of ABCDEFGHA. The letters are not in greek alphabetic order.

* 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.

~~=== Interval names~~ for porcupine[8] ~~===~~

The eight nominals form the base diatonic scale. In this sense, 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. See the main [[Porcupine Notation|porcupine notation]] page.

{| class="wikitable"

|-

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|}

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.

=== 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" (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).

{| class="wikitable"

|+

|C

|^C

|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

|}

== Regular temperament properties ==

@@ Line 15: / Line 15: @@
 In the 15edo system, major thirds cannot be divided perfectly into two, while minor 3rds, 4ths, wide tritones, subminor 7ths, and supermajor 7ths can. Similarly, 4ths, 5ths, and subminor 7ths can all be divided into 3 equal parts, while minor 3rds, tritones, and major 6ths cannot. This gives 15edo a whole new set of pitch symmetries and modes of limited transposition. Coupled with the lack of a [[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 12edo that contains recognizable major and minor triads (unless the 14edo supermajor triad is considered as a "recognizable major triad"), 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 recommended method to the notation of 15edo 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 15edo while the "small quill" (80¢) is the chroma of 15edo. This produces a very consistent notation for both porcupine[8] and Blackwood[10] and seems to work much better than attempting to put 15edo into a 5L 2s based framework.
 === Prime harmonics ===
 {{Harmonics in equal|15}}
@@ Line 488: / Line 486: @@
 There are a variety of other ways to notate 15edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15edo composition.
-=== Blackwood Notation ===
+=== 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 ===
+=== Porcupine Notation (Octotonic) ===
-* See the main [[Porcupine Notation|porcupine notation]] page.
+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 their place with a close resemblance to the spelling of ABCDEFGHA. The letters are not in greek alphabetic order.
-* 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.
-=== Interval names for porcupine[8] ===
+The eight nominals form the base diatonic scale. In this sense, 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. See the main [[Porcupine Notation|porcupine notation]] page.
 {| class="wikitable"
 |-
@@ Line 564: / Line 561: @@
 |}
 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.
+=== 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" (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).
+{| class="wikitable"
+|+
+|C
+|^C
+|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
+|}
 == Regular temperament properties ==