15edo: Difference between revisions

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, meaning 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 with a closed circle of five notes, 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 approximate 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, fourths, fifths, and subminor sevenths can all be divided into 3 equal parts, while minor thirds, tritones, and major sixths 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.

=== Keyboard layouts ===

File:15_tone_keyboard.png|Porcupine layout for 15edo

File:Screen Shot 2020-04-23 at 11.59.17 PM.png|Hanson layout for 15edo

File:15edo kb3.png|Zarlino layout for 15edo

==Intervals==

!Degree

!Cents

!Approximate Ratios<ref group="note">{{rd|limit=11-limit}}</ref>

![[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===

!step

! cents

! colspan="3" |[[Ups and downs notation|ups and downs]] notation ([[Enharmonic unisons in ups and downs notation|EUs]]: v³A1 and m2)<br>(partial list, e.g. M2/m3 is also A1 and d4)

! colspan="2" | porcupine notation<br>([[Enharmonic unison|EU]]: dd2)

|0

|0

|unison, min 2nd

|C# / D / Eb

|unison

|D

|1

|80

|^1, ^m2

|up-unison, upminor 2nd

|^C# / ^D / ^Eb

|aug unison, dim 2nd

|2

|160

|vM2

|vD# / vE / vF / vGb

|3

|240

|M2, m3

|major 2nd, minor 3rd

|D# / E / F / Gb

|aug 2nd, dim 3rd

|E# / Fb

|4

|320

|^m3

|upminor 3rd

|^D# / ^E / ^F / ^Gb

|minor 3rd

|F

|vM3

|downmajor 3rd

|major 3rd, dim 4th

|F# / Gb

|6

|480

|major 3rd, perfect 4th, dim 5th

|aug 3rd, minor 4th

|Fx / G

|7

|up 4th, updim 5th

|^F# / ^G / ^Ab

|major 4th, dim 5th

|G# / Abb

|8

|640

|vA4, v5

|downaug 4th, down 5th

|aug 4th, minor 5th

|Gx / Ab

|9

|720

|A4, P5, m6

|aug 4th, perfect 5th, minor 6th

|G# / A / Bb

|major 5th, dim 6th

|A / Bbb

|10

|800

|up 5th, upminor 6th

|aug 5th, minor 6th

|A# / Bb

|11

|vA5, vM6

|downaug 5th, downmajor 6th

|vA# / vB / vC / vDb

|major 6th

|12

|960

|M6, m7

|major 6th, minor 7th

|A# / B / C / Db

|aug 6th, dim 7th

|1040

|^m7

|upminor 7th

|perfect 7th

|C

|14

|downmajor 7th, down octave

|vC# / vD / vEb

|aug 7th, dim 8ve

|C# / Db

|15

|1200

|major 7th, octave

|C# / D / Eb

|8ve

|D

*…Fx - Gx - A# - B# - C# - D# - E# - F# - G# --- A --- B --- C -- D --- E --- F --- G -- Ab -- Bb - Cb - Db - Eb - Fb - Gb - Abb - Bbb…

* …A3 - A4 - A5 - A6 - A7 - A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 -- d8 - d2 - d3 -- d4 - d5 -- d6…

==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.

===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).

 

=== 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]].

rect 20 80 439 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]

===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).

|C

|^C

|^D

|vꘙ

|ꘙ

| ^ꘙ

|vG

|G

|^G

| vA

|A

|^A

|vC

|C

|^1

|v2

|P2

|^2

|v3

|P3

|^3

|P4

|^4

|v5

|P5

| ^5

|v6

|P6

=== Blackwood Notation (Decatonic)===

*'''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 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.

! Interval Name(s)

! Note name(s)

| Unison

| 80

| D# / Eb

| 160

| 240

| 320

| Minor Third

| Major Third / Diminished Fourth

| Perfect Fourth

| G

| A

| 800

| Augmented Fifth / Minor Sixth

| Major Sixth

| B

| 1200

| D

 

==== 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:

! Note name(s)

| 80

| D#

| 160

| 240

| 320

| Gb

| A

| 800

| Bb

|C# / Dd

| 1200

| D

===Porcupine Notation (Octatonic)===

Porcupine notation can also be based on the 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.

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.

!Interval Name

! Note names (Greek)

| Large Triquill

| Large Fourquill

| Large Sevenquill

| α - η

==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===