15edo: Difference between revisions

=== Tuning 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. 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]].

 

{| class="wikitable"

|+Logarithmic divisions of intervals in 15edo

|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.

 

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.

 

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 ====

 

 

 

{{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]].

 

!Degree

!Cents

![[Solfege]]<br>(porcupine-based)

!Porcupine[7]<br>(traditional)

!Porcupine[8]<br>(Greek)

!Zarlino diatonic notation

!Blackwood<br>Decimal

|0

|do

|α

|C

|E

|1

|1

|di

|D# / Eb

|Db / C#

|1# / 2b

|2

|160

|ru

|E

|β

|D

|2

|3

|β/ χ\

|D#

|3

|4

|320

|me

|F

|Eb

|G#

|3# / 4b

|5

|mi

|F# / Gb

|χ/ δ\

|E

|Ab

|4

|480

|fa

|G

|δ

|F

|A

|5

|7

|560

|fu

|G#

|δ/ ε\

|F#

|A#

|5# / 6b

|8

|640

|su

|ε

|Gb

|Bb

|6

|9

|720

|sol

|A

|ε/ φ\

|G

|B

|7

|10

|800

|le

|A# / Bb

|φ

|Ab / G#

|B#

|7# / 8b

|11

|880

|la

|B

|φ/ γ\

|A

|Db

|8

|12

|ta

|B# / Cb

|γ

|A# / Bbb

|D

|13

|tu

|C

|γ/ η\

|Bb

|D#

|9# / 0b

|14

|ti

|C# / Db

|η

|B

|Eb

|0

|1200

|do

|D

|C

===Alternate interval names===

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. They (except Blackwood decatonic notation) function basically the same as each other, but with notes and accidentals relabelled.

====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).{{Sharpness-sharp3a}}[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:{{Sharpness-sharp3}}

|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

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 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).

One can also use other systematic ways of naming nominals, such as ABC..., or use diamond-mos.

=== Heptatonic notations ===

====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.

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 (Heptatonic) ====

=== Porcupine "Quill" Notation (Octatonic) ===

==Approximation to JI==

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)]].

===15-odd-limit interval mappings===

 

==Regular temperament properties==

! rowspan="2" |[[Subgroup]]

! rowspan="2" |[[Mapping]]

! rowspan="2" |Optimal<br>8ve stretch (¢)

! colspan="2" |Tuning error

!

[[TE error|Absolute]] (¢)

|2.3.5

|{{mapping| 15 24 35 }}

|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

|+ style="font-size: 105%; white-space: nowrap;" |Errors by subgroup

| {{mapping| 15 24 }}

| {{mapping| 15 35 }}

| {{mapping| 15 42 }}

| {{mapping| 15 52 }}

|  

{{mapping| 15 24 35 }}

| {{mapping| 15 24 42 }}

|  

{{mapping| 15 24 52 }}

| {{mapping| 15 35 42 }}

| {{mapping| 15 35 52 }}

| {{mapping| 15 42 52 }}

| {{mapping| 15 24 35 42 }}

| {{mapping| 15 24 35 52 }}

| {{mapping| 15 24 42 52 }}

| {{mapping| 15 35 42 52 }}

| {{mapping| 15 24 35 42 52 }}

===Uniform maps===

===Rank-2 temperaments===

*[[List of edo-distinct 15et rank two temperaments]]

!Generator

!Temperaments

!Mos scales

|1\15

|21/20

|[[Nautilus]]<br>[[Valentine]]

|1

|2\15

|11/10

|[[Porcupine]] / [[opossum]]

|[[1L 6s]], [[7L 1s]]

|1

|4\15

|6/5<br>77/64

|[[Cata]] / [[keemun]] / [[catalan]]<br>[[Orgone]] / [[superkleismic]]

|1

|7\15

|7/5

|[[Progress]]<br>[[Parakangaroo]]

|[[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]] [[2L 9s]], [[2L 11s]]

| [[Augmented]] / [[augene]]

|[[3L 3s]], [[3L 6s]], [[3L 9s]]

|3

|2\15

|7/6

|[[Triforce]]

|[[3L 3s]], [[6L 3s]]

|5

|16/15

|[[Blackwood]] / [[blacksmith]]

== Scales==

=== MOS scales===

[[File:BlackwoodMajor 15edo.mp3]] [[:BlackwoodMajor 15edo.mp3|BlackwoodMajor 15edo.mp3]]

 

===Other scales===

*[[Zarlino]]/Ptolemy diatonic, "just" major (Porcupine[7] 6|0 b4 #7): 3 2 1 3 2 3 1

=== Horagrams===

==Diagrams==

=== Keyboard===

===Lumatone ===

==Music==

; [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]

==Further reading==

===Guitar===