# 15edo

(Redirected from 15-edo)
 ← 14edo 15edo 16edo →
Prime factorization 3 × 5
Step size 80¢
Fifth 9\15 (720¢) (→3\5)
Semitones (A1:m2) 3:0 (240¢ : 0¢)
Consistency limit 7
Distinct consistency limit 5
English Wikipedia has an article on:

15 equal divisions of the octave (abbreviated 15edo or 15ed2), also called 15-tone equal temperament (15tet) or 15 equal temperament (15et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 15 equal parts of exactly 80 ¢ each. Each step represents a frequency ratio of 21/15, or the 15th root of 2.

## Theory

15edo can be thought of as three sets of 5edo which do not connect by 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 functional harmony, because with a closed circle of five notes, the same interval can have multiple functions. Additionally, 15 being equal to 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 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 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.

### Prime harmonics

Approximation of prime harmonics in 15edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +18.0 +13.7 -8.8 +8.7 +39.5 -25.0 +22.5 +11.7 +10.4 -25.0
Relative (%) +0.0 +22.6 +17.1 -11.0 +10.9 +49.3 -31.2 +28.1 +14.7 +13.0 -31.3
Steps
(reduced)
15
(0)
24
(9)
35
(5)
42
(12)
52
(7)
56
(11)
61
(1)
64
(4)
68
(8)
73
(13)
74
(14)

## Intervals

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 Approximate Ratios* Solfege
(porcupine-based)
Porcupine[7]
Porcupine[8]
(Greek)
Blackwood
"guitar notation"
Blackwood
Decimal
Audio
0 0 1/1 do G α E 1
1 80 25/24, 21/20, 16/15 di G# / Abb α/ β\ E# 1# / 2b
2 160 11/10, 12/11, 10/9 ru Gx / Ab β Gb 2
3 240 8/7, 7/6, 9/8 re A β/ χ\ G 3
4 320 6/5, 11/9 me A# / Bb χ G# 3# / 4b
5 400 5/4, 14/11 mi B χ/ δ\ Ab 4
6 480 4/3, 9/7, 21/16 fa B# / Cb δ A 5
7 560 11/8, 7/5 fu C δ/ ε\ A# 5# / 6b
8 640 16/11, 10/7 su C# / Db ε Bb 6
9 720 3/2, 14/9, 32/21 sol D ε/ φ\ B 7
10 800 8/5, 11/7 le D# / Eb φ B# 7# / 8b
11 880 5/3, 18/11 la E φ/ γ\ Db 8
12 960 7/4, 12/7, 16/9 ta E# / Fb γ D 9
13 1040 20/11, 11/6, 9/5 tu F γ/ η\ D# 9# / 0b
14 1120 48/25, 40/21, 15/8 ti F# /Gb η Eb 0
15 1200 2/1 do G α E 1

* based on treating 15edo as an 11-limit temperament; other approaches are possible

### Alternate interval names

step cents ups and downs relative notation
(partial list, e.g. M2 is also A1 and d4)
ups and downs
absolute notation
porcupine
relative notation
porcupine
absolute notation
0 0 P1, m2 unison, min 2nd C# / D / Eb unison D
1 80 ^1, ^m2 up-unison, upminor 2nd ^C# / ^D / ^Eb aug unison, dim 2nd D# / Eb
2 160 vM2 downmajor 2nd vD# / vE / vF / vGb perfect 2nd E
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
5 400 vM3 downmajor 3rd vF# / vG / vAb major 3rd, dim 4th F# / Gb
6 480 M3, P4, d5 major 3rd, perfect 4th, dim 5th F# / G / Ab aug 3rd, minor 4th Fx / G
7 560 ^4, ^d5 up 4th, updim 5th ^F# / ^G / ^Ab major 4th, dim 5th G# / Abb
8 640 vA4, v5 downaug 4th, down 5th vG# / vA / vBb 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 ^5, ^m6 up 5th, upminor 6th ^G# / ^A / ^Bb aug 5th, minor 6th A# / Bb
11 880 vA5, vM6 downaug 5th, downmajor 6th vA# / vB / vC / vDb major 6th B
12 960 M6, m7 major 6th, minor 7th A# / B / C / Db aug 6th, dim 7th B# / Cb
13 1040 ^m7 upminor 7th ^A# / ^B / ^C / ^Db perfect 7th C
14 1120 vM7, v8 downmajor 7th, down octave vC# / vD / vEb aug 7th, dim 8ve C# / Db
15 1200 M7, P8 major 7th, octave C# / D / Eb 8ve D

The 15edo porcupine genchain in both absolute and relative notation:

• …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…

All 15edo chords can be named using ups and downs. Because many intervals have several names, many chords do too. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

0-3-9 = D E A = D2 = "D sus 2", or D F A = Dm = "D minor" (approximate 6:7:9)

0-4-9 = D ^F A = D^m = "D upminor" (approximate 10:12:15)

0-5-9 = D vF# A = Dv = "D down" or "D downmajor" (approximate 4:5:6)

0-6-9 = D G A = D4, or D F# A = D = "D" or "D major" (approximate 14:18:21)

0-3-9-12 = D F A C = Dm7 = "D minor seven", or D F A B = Dm6 = "D minor six"

0-4-9-12 = D ^F A C = D^m,7 = "D upminor, add seven", or D ^F A B = D^m,6 = "D upminor add-six"

0-5-9-12 = D vF# A C = Dv,7 = "D down add-seven", or D vF# A B = Dv,6 = "D down add-six"

0-6-9-12 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"

0-5-9-14 = D vF# A vC# = DvM7 = "D downmajor seven"

0-4-9-13 = D ^F A ^C = D^m7 = "D upminor-seven", or D ^F A ^B = D^m6 = "D upminor-six"

For a more complete list, see Ups and downs notation#Chords and Chord Progressions.

## Approximation to JI

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 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

The following tables show how 15-odd-limit intervals are represented in 15edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 15edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/9, 18/13 3.382 4.2
5/3, 6/5 4.359 5.4
11/10, 20/11 5.004 6.3
15/13, 26/15 7.741 9.7
11/8, 16/11 8.682 10.9
7/4, 8/7 8.826 11.0
11/6, 12/11 9.363 11.7
5/4, 8/5 13.686 17.1
11/7, 14/11 17.508 21.9
3/2, 4/3 18.045 22.6
13/12, 24/13 21.427 26.8
9/5, 10/9 22.404 28.0
7/5, 10/7 22.512 28.1
15/11, 22/15 23.049 28.8
13/10, 20/13 25.786 32.2
7/6, 12/7 26.871 33.6
11/9, 18/11 27.408 34.3
13/11, 22/13 30.790 38.5
13/7, 14/13 31.702 39.6
15/8, 16/15 31.731 39.7
9/7, 14/9 35.084 43.9
9/8, 16/9 36.090 45.1
15/14, 28/15 39.443 49.3
13/8, 16/13 39.472 49.3
15-odd-limit intervals in 15edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/9, 18/13 3.382 4.2
5/3, 6/5 4.359 5.4
11/10, 20/11 5.004 6.3
15/13, 26/15 7.741 9.7
11/8, 16/11 8.682 10.9
7/4, 8/7 8.826 11.0
11/6, 12/11 9.363 11.7
5/4, 8/5 13.686 17.1
11/7, 14/11 17.508 21.9
3/2, 4/3 18.045 22.6
13/12, 24/13 21.427 26.8
9/5, 10/9 22.404 28.0
7/5, 10/7 22.512 28.1
15/11, 22/15 23.049 28.8
13/10, 20/13 25.786 32.2
7/6, 12/7 26.871 33.6
11/9, 18/11 27.408 34.3
13/11, 22/13 30.790 38.5
15/8, 16/15 31.731 39.7
9/8, 16/9 36.090 45.1
13/8, 16/13 39.472 49.3
15/14, 28/15 40.557 50.7
9/7, 14/9 44.916 56.1
13/7, 14/13 48.298 60.4

## Notation

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 (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 (Octotonic)

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.

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 page.

 Cents Interval Name Note names 80 Half Quill α - β\ 160 Quill α - β 240 Small Diquill α - χ\ 320 Large Diquill α - χ 400 Small Triquill α - δ\ 480 Large Triquill α - δ 560 Small Fourquill α - ε\ 640 Large Fourquill α - ε 720 Small Fivequill α - φ\ 800 Large Fivequill α - φ 880 Small Sixquill α - γ\ 960 Large Sixquill α - γ 1040 Small Sevenquill α - η\ 1120 Large Sevenquill α - η 1200 Octoquill α - α

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, 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).

 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

15edo prime-limit subgroup errors
Subgroup Comma list Mapping Optimal
8ve stretch (¢)
Tuning error
Absolute (¢) Relative (%)
2.3.5 128/125, 250/243 [15 24 35]] -5.75 4.63 5.81
2.3.5.7 28/27, 49/48, 126/125 [15 24 35 42]] -3.55 5.56 6.97
2.3.5.7.11 28/27, 49/48, 55/54, 77/75 [15 24 35 42 52]] -3.34 4.99 6.25
15edo subgroup errors (continued)
2.3 [15 24]] 8.979801
2.5 [15 35]] 6.826357
2.7 [15 42]] 4.418738
2.11 [15 52]] 4.336492
2.3.5 [15 24 35]] 10.742841
2.3.7 [15 24 42]] 17.481581
2.3.11 [15 24 52]] 16.831238
2.5.7 [15 35 42]] 10.509269
2.5.11 [15 35 52]] 8.335693
2.7.11 [15 42 52]] 8.002641
2.3.5.7 [15 24 35 42]] 15.603114
2.3.5.11 [15 24 35 52]] 14.693746
2.3.7.11 [15 24 42 52]] 18.660367
2.5.7.11 [15 35 42 52]] 11.462127
2.3.5.7.11 [15 24 35 42 52]] 17.258371

### Uniform maps

13-limit uniform maps between 14.5 and 15.5
Min. size Max. size Wart notation Map
14.5000 14.5978 15bcddeeefff 15 23 34 41 50 54]
14.5978 14.7280 15bcddefff 15 23 34 41 51 54]
14.7280 14.7826 15bcddef 15 23 34 41 51 55]
14.7826 14.8268 15bcef 15 23 34 42 51 55]
14.8268 14.8583 15cef 15 24 34 42 51 55]
14.8583 14.8868 15ef 15 24 35 42 51 55]
14.8868 14.9982 15f 15 24 35 42 52 55]
14.9982 15.1388 15 15 24 35 42 52 56]
15.1388 15.1759 15d 15 24 35 43 52 56]
15.1759 15.2685 15dee 15 24 35 43 53 56]
15.2685 15.2890 15deeff 15 24 35 43 53 57]
15.2890 15.4578 15ccdeeff 15 24 36 43 53 57]
15.4578 15.4650 15bbccdeeff 15 25 36 43 53 57]
15.4650 15.4950 15bbccdeeeeff 15 25 36 43 54 57]
15.4950 15.5000 15bbccdddeeeeff 15 25 36 44 54 57]

### Rank-2 temperaments

Periods
per octave
Generator Associated
ratio
Temperaments MOS Scales
1 1\15 21/20 Nautilus
Valentine
1 2\15 11/10 Porcupine / opossum 1L 6s, 7L 1s
1 4\15 6/5
77/64
Hanson / keemun
Orgone
3L 1s, 4L 3s, 4L 7s
1 7\15 7/5 Progress 2L 1s, 2L 3s, 2L 5s, 2L 7s 2L 9s, 2L 11s
3 1\15 16/15 Augmented / augene 3L 3s, 3L 6s, 3L 9s
3 2\15 7/6 Triforce 3L 3s, 6L 3s
5 1\15 16/15 Blackwood / blacksmith 5L 5s

### Commas

15edo tempers out the following commas using the patent val 15 24 35 42 52 56].

Prime limit Ratio[1] Monzo Cents Color name Name
3 256/243 [8 -5 90.225 Sawa Pythagorean limma
5 (18 digits) [20 5 -12 74.01 Saquadtrigu Hypovishnuzma
5 250/243 [1 -5 3 49.166 Triyo Porcupine comma
5 128/125 [7 0 -3 41.059 Trigu Diesis
5 15625/15552 [-6 -5 6 8.107 Tribiyo Kleisma
7 28/27 [2 -3 0 1 62.961 Zo Trienstonic comma
7 1029/1000 [-3 1 -3 3 49.492 Trizogu Keega
7 49/48 [-4 -1 0 2 35.697 Zozo Slendro diesis
7 64/63 [6 -2 0 -1 27.264 Ru Septimal comma
7 64827/64000 [-9 3 -3 4 22.227 Laquadzo-atrigu Squalentine
7 875/864 [-5 -3 3 1 21.902 Zotriyo Keema
7 126/125 [1 2 -3 1 13.795 Zotrigu Starling comma
7 4000/3969 [5 -4 3 -2 13.469 Rurutriyo Octagar
7 1029/1024 [-10 1 0 3 8.433 Latrizo Gamelisma
7 6144/6125 [11 1 -3 -2 5.362 Saruru-atrigu Porwell
7 (12 digits) [-4 6 -6 3 0.325 Trizogugu Landscape comma
11 100/99 [2 -2 2 0 -1 17.399 Luyoyo Ptolemisma
11 121/120 [-3 -1 -1 0 2 14.367 Lologu Biyatisma
11 176/175 [4 0 -2 -1 1 9.865 Lorugugu Valinorsma
11 65536/65219 [16 0 0 -2 -3 8.394 Satrilu-aruru Orgonisma
11 385/384 [-7 -1 1 1 1 4.503 Lozoyo Keenanisma
11 441/440 [-3 2 -1 2 -1 3.930 Luzozogu Werckisma
11 4000/3993 [5 -1 3 0 -3 3.032 Triluyo Wizardharry
11 3025/3024 [-4 -3 2 -1 2 0.572 Loloruyoyo Lehmerisma
13 91/90 [-1 -2 -1 1 0 1 19.130 Thozogu Superleap
13 676/675 [2 -3 -2 0 0 2 2.563 Bithogu Island comma
1. Ratios longer than 10 digits are presented by placeholders with informative hints

## Scales

Some scales commonly used in 15edo, written in a common mode, in steps of 15edo:

### MOS scales

• Augene[6] 3L 3s (period = 5\15, gen = 1\15): 4 1 4 1 4 1
• Augene[9] 3L 6s (period = 5\15, gen = 1\15): 3 1 1 3 1 1 3 1 1
• Augene[12] 3L 9s (period = 5\15, gen = 1\15): 2 1 1 1 2 1 1 1 2 1 1 1
• Triforce[6] 3L 3s (period = 5\15, gen = 2\15): 3 2 3 2 3 2
• Triforce[9] 6L 3s (period = 5\15, gen = 2\15): 2 1 2 2 1 2 2 1 2
• Porcupine[7] 1L 6s (gen = 2\15): 3 2 2 2 2 2 2
• Porcupine[8] 7L 1s (gen = 2\15): 2 1 2 2 2 2 2 2
• Hanson/Keemun/Orgone[7] 4L 3s (gen = 4\15): 1 3 1 3 1 3 3
• Hanson/Keemun/Orgone[11] 4L 7s (gen = 4\15): 1 2 1 1 2 1 1 2 1 2 1
• Blackwood[10] 5L 5s (period = 3\15, gen = 1\15): 2 1 2 1 2 1 2 1 2 1 (Blackwood Decatonic)

### Other scales

• Zarlino/Ptolemy diatonic, "just" major (Porcupine[7] 6|0 b4 #7): 3 2 1 3 2 3 1
• inverse of Zarlino/Ptolemy diatonic, natural minor (Porcupine[7] 3|3 #2 b6): 3 1 2 3 1 3 2
• tetrachordal major: 3 2 1 3 3 2 1
• inverse of tetrachordal major, tetrachordal minor: 3 1 2 3 1 2 3
• "just"/Pinetone major pentatonic (subset of Porcupine[7]): 3 2 4 2 4
• "just"/Pinetone minor pentatonic (inverse of "just" major pentatonic, subset of Porcupine[7]): 4 2 3 4 2
• Porcupine bright major #7 (Porcupine harmonic major) - Porcupine[7] 6|0 #7: 3 2 2 2 2 3 1
• Porcupine bright major #6 #7 (Porcupine melodic major) - Porcupine[7] 6|0 #7: 3 2 2 2 3 2 1
• Porcupine bright minor #2 (Porcupine harmonic minor) - Porcupine[7] 4|2 #2: 3 1 3 2 2 2 2 (mode of bright major #7)
• Porcupine dark minor #2 (Porcupine melodic minor) - Porcupine[7] 3|3 #2: 3 1 2 3 2 2 2 (inverse of bright major #6 #7)
• Porcupine bright harmonic 11th mode - Porcupine[7] 6|0 b7: 3 2 2 2 2 1 3
• Pinetone diminished octatonic / Porcupine[8] bright minor #2 - Porcupine[8] 2|5 #5: 2 2 1 3 1 2 2 2
• "just" harmonic minor: 3 1 2 3 1 4 1
• "just" harmonic major: 3 2 1 3 1 4 1
• "just" melodic minor ascending: 3 1 2 3 2 3 1
• Marvel double harmonic hexatonic (Augene[6] 4M): 1 4 1 4 4 1, 1 4 4 1 4 1
• Marvel double harmonic major: 1 4 1 3 1 4 1
• Marvel double harmonic nonatonic (Augene[9] 4M): 1 3 1 1 3 1 3 1 1, 1 1 3 1 3 1 1 3 1
• Marvel double harmonic decatonic: 1 3 1 1 2 1 1 3 1 1
• enharmonic trichord octave species: 1 5 3 1 5 , 5 1 3 5 1
• chromatic tetrachord octave species: 1 1 4 3 1 1 4, 4 1 1 3 4 1 1, 1 4 1 3 1 4 1
• Chopsticks double octave scale: 4 2 4 2 4 2 4 2 4 2
• 5- to 10-tone scales in 47zpi (slightly stretched 15edo)

### Horagrams

2\15 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 7L 1s
4\15 MOS using 1L 1s, 1L 2s, 3L 1s, 4L 3s, 4L 7s
7\15 MOS using 1L 1s, 2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s

## Diagrams

### Keyboard

Major chord shapes for a Porcupine keyboard in 15edo, using Blackwood logic (i.e. native fifth notation) for the letters.