Porcupine temperament modal harmony

Porcupine is a temperament splitting 4/3 into three equal parts representing 10/9 and 27/25, so that two steps represent 6/5. This results in a slightly flat 4/3 and a slightly sharp 6/5, meaning that 5/4 is accurately tuned.

Porcupine scales

Todo: rework Explain in a non-MOS-centric way

Porcupine is one of the few reasonably good 5-limit temperaments, alongside meantone, which possesses a 7-note MOS scale ("porcupine[7]" or "onyx") that is well-supplied with 5-limit major and minor triads (specifically, it has two of each of them). What is perhaps more significant about it is how naturally it extends those triads to involve the 11th harmonic; for instance, the first five consecutive notes of the Lssssss mode approximate the consecutive harmonics 8:9:10:11:12. This makes it a prime candidate for using extended JI-style harmonies in a compact and intuitive scale structure.

It is also significant because it is supported by both 15edo and 22edo, the smallest ETs which improve on the 11-limit damage of 12edo (other than 19edo). In those edos, porcupine is an excellent candidate for providing a 7-nominal notational basis and 7-interval-class categorical framework. This is important, because if you want to use 15edo (or to a lesser extent, 22edo) in a somewhat "heptatonic" fashion, where 5/4 and 6/5 are types of "thirds", 3/2 is a type of "fifth," and 5-limit major and minor triads share the same "1 – 3 – 5" triad shape, thinking of those ETs in terms of porcupine temperament is, in a certain sense, the natural approach. This most transparently leads to notation systems based upon porcupine[7], but 15edo and 22edo also contain a zarlino scale which can be naturally interpreted as a porcupine scale (what this means for zarlino is that the three sizes of step are equidistant). In fact, *any* heptatonic scale in 15edo or 22edo can be viewed as a MODMOS of porcupine[7].

In summary:

1. Porcupine is the simplest 5-limit temperament that has less tuning error than 12edo which 12edo does not support.
2. In a temperament search of 5-limit temperaments, using the standard search parameter of 5 ¢ of error, porcupine is the first temperament in the results that is not supported by 12.

Porcupine remains the "best" such temperament in the 5-limit even if the maximum error is relaxed somewhat. While mavila eventually supersedes it due to being simpler, mavila's accuracy is much lower as its fifths are extremely flat (670 – 680 ¢).

Brief intro to modal harmony and MODMOSes

Porcupine is a temperament completely capable of diverse harmony. Its MOS and its MODMOS scales (including zarlino) allow for some very diverse chordal movement that will still be tonally solid.

Because the generator of porcupine is simultaneously 10/9, 11/10, and 12/11, a chord made up of consecutive generators approximates 9:10:11:12; since -6 generators approximates 9/8, the chord 8:9:10:11:12 can be approximated by a series of consecutive 2nds in porcupine[7]. Varying subsets of 8:9:10:11:12 provide fruitful ground for building harmonic movement in porcupine. Additionally, chords of stacked sevenths of varying sizes may be used to improve voice leading and add variety. Chords without a 'fifth' are recommended for use on the root for modes without a 3/2 from the root. Further, quartal triads, tetrads, and pentads are available in more variety than in Meantone, and make good use of Porcupine's 11-limit consonances.

Stacked 3-step triads may be employed in Porcupine[8] and it's MODMOS, comprising in their root position three 3:4:5 triads (second inversion major triad) as major 3-step, minor 3-step: major[8] triad; three 15:20:24 triads (first inversion minor triad) as minor 3-step, major 3-step: minor[8] triad; and two 9:12:16 triads (second inversion suspended triad) as major 3-step, major 3-step: augmented[8] triad. At least one MODMOS of Porcupine[8] contains the same number of major and minor triads as the MOS, but more spread out across the scale, similar to Porcupine[7] MODMOS.

The MODMOS scales of porcupine can be used in conjunction with the regular MOS in order to build powerful tonal movement. While each useful MODMOS has its own distinct name, they most likely will be useful to use as a particular function in each porcupine mode. They can also be used stand alone but little is known about the very complex system of tonality.

Porcupine[7] is great, but it lumps all the consonant chords together at one end of the scale (see below). It also fails to contain a complete 11-limit otonal hexad (4:5:6:7:9:11), even though porcupine is not excessively complex in the 11-limit. Thankfully, unlike in meantone[7] where the MODMOSs all reduce the number of consonant triads available, there are several MODMOSs of porcupine that contain an equal or greater number of consonant triads to the MOS (these are highlighted in blue text below). These MODMOSs also conveniently alter the distribution of triads within the scale, providing much-needed variety of possible chord progressions. Even better, some single-alteration MODMOSs make at least one full 11-limit hexad available, meaning 6 of the 7 notes of the scale can be sounded simultaneously and produce a consonant otonal harmony. The fact that porcupine temperament provides not just one 7-note scale but several that are equally-rich in harmonic resources makes it extremely fertile for modal harmony.

Porcupine[7] and its MODMOSes

The below mode names are given in UDP notation. For porcupine[7], the bright generator is the 9/5, or the 11/6 if you prefer. All generators below refer to this generator, NOT the 10/9 one. For an explanation of the spellings, see Porcupine Notation.

The functional names are pretty straightforward: there are two major modes, one bright and one dark (differing in the second scale degree), two similarly-named minor modes (differing in the fourth scale degree), and then three modes with a diminished triad on the tonic (differing in the sixth and seventh scale degrees). The brightest two of these diminished modes both have a half-diminished tetrad on the tonic, and so they are named the bright and dark half-diminished modes, whereas the darkest has a fully diminished (or magical seventh tetrad) instead. These are sometimes just written bright/dark diminished and magical seventh modes when clear.

William Lynch's system

William Lynch proposed the fun idea of naming modes and MODMOS's of porcupine by various animal names, which are listed as "William's Name" below. He has named these scales as such as he views none of them to be remotely similar to any existing mode name from anywhere. The initial four modes: Porcupine, Chinchilla, Zebra, and Dingo all contain 3/2 from the root while the latter three: Gazelle, Lemur, and Panda lack 3/2 and instead have a triad similar to a diminished triad on the root. In order to remedy this problem, it may be fruitful to use other methods than triadic tertian harmony to build tonal frameworks in porcupine.

The scale's nominals can be named by the normal 7-tone alphabet starting on G. This has become somewhat of a standard. Some feel however that given Superpyth uses the same nominals in a contrasting way, it may be beneficial to use an entirely different set of letters for Porcupine. William Lynch's suggestion involves using the old-English alphabet, allowing for both porcupine[8] and porcupine[7] to be notated by leaving out "ð" from the scale.

Porcupine[8]: b c ð d e f þ æ

Porcupine[7]: b c d e f þ æ with chinchillian mode on "c"

pronounced "Be - Ce - Eth - De - E - Eff - Thorn - Ash"

An alternative suggestion involves using the letters J, K, L, M, N, O, and P for the nominals, replacing A, B, C, D, E, F, and G respectively, such that the bright major mode starts on the note P.

The Porcupine[7] MOS

Functional name	William's came	22edo step pattern	15edo step pattern	UDP name	Spelling	Alt spelling	Triad qualities	Other notes
Bright major	Chinchillian	4333333	3222222	6|0	G A B C D E F	P J K L M N O	I iiº iiiº ivº v vi VII	Bright, otonal major (VII chord is major)
Dark major	Badgerian	3433333	2322222	5|1	G Av B C D E F	P Jv K L M N O	I II iiiº ivº vº vi vii	Darker major (vii is minor)
Bright minor	Zebrian	3343333	2232222	4|2	G Av Bv C D E F	P Jv Kv L M N O	i II III ivº vº viº vii	Bright minor (II is major)
Dark minor (symmetric)	Dingoian	3334333	2223222	3|3	G Av Bv Cv D E F	P Jv Kv Lv M N O	i ii III IV vº viº viiº	Symmetric minor (ii is minor)
Bright (half-)diminished	Gazellian	3333433	2222322	2|4	G Av Bv Cv Dv E F	P Jv Kv Lv Mv N O	iº ii iii IV V viº viiº	Half-Diminished
Dark (half-)diminished	Lemurian	3333343	2222232	1|5	G Av Bv Cv Dv Ev F	P Jv Kv Lv Mv Nv O	iº iiº iii iv V VI viiº	Half-Diminished
Magical/diminished seventh	Pandian	3333334	2222223	0|6	G Av Bv Cv Dv Ev Fv	P Jv Kv Lv Mv Nv Ov	iº iiº iiiº iv v VI VII	Magical seventh or diminished seventh mode

Single-alteration MODMOS's

MODMOS with 1st note in generator chain flattened: #.......######

Note: Contains a broken 4/3 chain so it has only 3 4/3s

22edo steps	15edo steps	UDP name	Spelling	Other notes
5333332	4222221	6|0 ^2^3^4^5^6^7	G A^ B^ C^ D^ E^ F^
2533333	1422222	5|1 v2	G Avv B C D E F
3253333	2142222	4|2 v3	G Av Bvv C D E F
3325333	2214222	3|3 v4	G Av Bv Cvv D E F
3332533	2221422	2|4 v5	G Av Bv Cv Dvv E F
3333253	2222142	1|5 v6	G Av Bv Cv Dv Evv F
3333325	2222214	0|6 v7	G Av Bv Cv Dv Ev Fvv

MODMOS with 2nd note in generator chain flattened: #.....#.#####

Note: Contains a broken 4/3 chain so it has only 3 4/3s

22edo steps	15edo steps	UDP name	Spelling	Other notes
4333324	3222213	6|0 v7	G A B C D E Fv	8:9:10:11:12:14 hexad on the tonic, in both 15edo and 22edo as G-A-B-C-D-Fv
4433332	3322221	6|0 ^3^4^5^6^7	G A B^ C^ D^ E^ F^
2443333	1332222	4|2 v2	G Avv Bv C D E F
3244333	2133222	3|3 v3	G Av Bvv Cv D E F
3324433	2213322	2|4 v4	G Av Bv Cvv Dv E F
3332443	2221332	1|5 v5	G Av Bv Cv Dvv Ev F
3333244	2222133	0|6 v6	G Av Bv Cv Dv Evv Fv

MODMOS with 3rd note in generator chain flattened: #....##.####

Note: Contains a broken 4/3 chain so it has only 3 4/3s

22edo steps	15edo steps	UDP name	Spelling	Other notes
4333243	3222132	6|0 v6	G A B C D Ev F
3433324	2322213	5|1 v7	G Av B C D E Fv
4343332	3232221	6|0 ^4^5^6^7	G A B C^ D^ E^ F^
2434333	1323222	3|3 v2	G Avv Bv Cv D E F
3243433	2132322	2|4 v3	G Av Bvv Cv Dv E F
3324343	2213232	1|5 v4	G Av Bv Cvv Dv Ev F
3332434	2221323	0|6 v5	G Av Bv Cv Dvv Ev Fv

MODMOS with 4th note in generator chain flattened: #...###.###

Note: Contains a broken 4/3 chain so it has only 3 4/3s

22edo steps	15edo steps	UDP name	Spelling	Other notes
4332433	3221322	6|0 v5	G A B C Dv E F
3433243	2322132	5|1 v6	G Av B C D Ev F
3343324	2232213	4|2 v7	G Av Bv C D E Fv
4334332	3223221	6|0 ^5^6^7	G A B C D^ E^ F^
2433433	1322322	2|4 v2	G Avv Bv Cv Dv E F
3243343	2132232	1|5 v3	G Av Bvv Cv Dv Ev F
3324334	2213223	0|6 v4	G Av Bv Cvv Dv Ev Fv

MODMOS with 5th note in generator chain flattened: #..####.## (melodic major)

Consists of unbroken chains of 4/3s

Functional Name	William's Name (repltiles)	22edo steps	15edo steps	UDP name	Spelling	Other notes
Bright major v4	Lizard	4324333	3213222	6|0 v4	G A B Cv D E F	I, IV, v, vi "mixolydian-like"*
Dark major v5	Iguana	3432433	2321322	5|1 v5	G Av B C Dv E F
Bright minor v6	Serpent	3343243	2232132	4|2 v6	G Av Bv C D Ev F	i, III, VI, vii
Dark minor v7	Iguana	3334324	2223213	3|3 v7	G Av Bv Cv D E Fv	i, ii, IV, VII "dorian-like"
Bright major ^6^7 (melodic major)	Chameleon	4333432	3222321	6|0 ^6^7	G A B C D E^ F^	I, ii, iii, V "ionian-like"; 8:9:10:11:12:14:15 on the tonic in 15edo as G-A-B-C-D-E^-F^ (but not in 22edo) Igliashon finds it the "most grounded", and suggests it as a candidate for "default major"
Dark (half-)diminished v2	Crocodile	2433343	1322232	1|5 v2	G Avv Bv Cv Dv Ev F
Magical seventh v3	Turtle	3243334	2132223	0|6 v3	G Av Bvv Cv Dv Ev Fv

• This kind of description is obviously stupid because these porcupine scales are not really like the diatonic modes "mixolydian", "dorian", etc. As always, feel free to edit if you have some better description.

Symmetrical MODMOS with 2nd note in generator chain sharpened or 6th note flattened: #.#####.# (harmonic)

Consists of unbroken chains of 4/3s

Has the most evenly-distributed 5-limit triads of all the MODMOSs

Functional Name	William's Name (Farm animals)	22edo steps	15edo steps	UDP name #1	UDP name #2	Spelling	Other notes
Bright major ^7 (harmonic major)	Horse	4333342	3222231	6|0 ^7	(1|5 v1)	G A B C D E F^	I ii° iii iv° V vi vii° Major with major V chord (most major-sounding major)
Magical seventh v2	Pig	2433334	1322223	(5|1 ^1)	0|6 v2	G Avv Bv Cv Dv Ev Fv	i° II iii° iv v° VI vii Magical seventh
Bright minor ^2 (harmonic minor)	Sheep	4243333	3132222	4|2 ^2	6|0 v3	G A Bv C D E F	i ii° III iv° v vi° VII Minor with minor v chord (most minor-sounding minor)
Dark major v4	Cow	3424333	2313222	3|3 ^3	5|1 v4	G Av B Cv D E F	I ii iii° IV v° vi vii° Major with major IV chord
Symmetric half-diminished	Llama	3342433	2231322	2|4 ^4	4|2 v5	G Av Bv C Dv E F	i° II iii iv° V vi° vii Symmetric diminished
Dark minor v6	Goat	3334243	2223132	1|5 ^5	3|3 v6	G Av Bv Cv D Ev F	i ii° III iv v° VI vii° Minor with minor iv
Magical seventh ^6	Rooster	3333424	2222313	0|6 ^6	2|4 v7	G Av Bv Cv Dv E Fv	i° ii iii° IV v vi° VII Magical seventh

MODMOS with 3rd note in generator chain sharpened: ##.####..# (melodic minor)

Consists of unbroken chains of 4/3s

Functional Name	William's Name (big cats)	22edo steps	15edo steps	UDP name #1	UDP name #2	Spelling	Other notes
Bright major ^6	Lionian	4333423	3222312	6|0 ^6	(2|4 v1v7)	G A B C D E^ F	I, ii, v, VII "mixolydian-like" 8:9:10:11:12:14 on the tonic in 15edo only
Dark major ^7	Tigonian	3433342	2322231	5|1 ^7	(1|5 v1v2)	G Av B C D E F^	I, II, iii, vi
Bright minor ^1	Cougian	2343334	1232223	(4|2 ^1)	0|6 v2v3	G Avv Bvv Cv Dv Ev Fv
Dark minor ^2 (melodic minor)	Leopardian	4234333	3123222	3|3 ^2	6|0 v3v4	G A Bv Cv D E F	i, III, IV, v "dorian-like" Keenan and Igliashon both find this exceptionally beautiful and suggest it as a candidate for default "minor"
Bright (half-)diminished ^3	Cheetian	3423433	2312322	2|4 ^3	5|1 v4v5	G Av B Cv Dv E F
Dark (half-)diminished ^4	Jaguarian	3342343	2231232	1|5 ^4	4|2 v5v6	G Av Bv C Dv Ev F
Dark minor v6 v7	Pantherian	3334234	2223123	0|6 ^5	3|3 v6v7	G Av Bv Cv D Ev Fv	i, iv, VI, VII "aeolian-like"

MODMOS with 4th note in generator chain sharpened: ###.###...#

Note: Contains a broken 4/3 chain so it has only 3 4/3s

22edo steps	15edo steps	UDP name	Spelling	Other notes
4334233	3223122	6|0 ^5	G A B C D^ E F
3433423	2322312	5|1 ^6	G Av B C D E^ F
3343342	2232231	4|2 ^7	G Av Bv C D E F^
2334334	1223223	0|6 v2v3v4	G Avv Bvv Cvv Dv Ev Fv
4233433	3122322	2|4 ^2	G A Bv Cv Dv E F
3423343	2312232	1|5 ^3	G Av B Cv Dv Ev F
3342334	2231223	0|6 ^4	G Av Bv C Dv Ev Fv

MODMOS with 5th note in generator chain sharpened: ####.##....#

Note: Contains a broken 4/3 chain so it has only 3 4/3s

22edo steps	15edo steps	UDP name	Spelling	Other notes
4342333	3231222	6|0 ^4	G A B C^ D E F
3434233	2323122	5|1 ^5	G Av B C D^ E F
3343423	2232312	4|2 ^6	G Av Bv C D E^ F
3334342	2223231	3|3 ^7	G Av Bv Cv D E F^
2333434	1222323	0|6 v2v3v4v5	G Avv Bvv Cvv Dvv Ev Fv
4233343	3122232	1|5 ^2	G A Bv Cv Dv Ev F
3423334	2312223	0|6 ^3	G Av B Cv Dv Ev Fv

MODMOS with 6th note in generator chain sharpened: #####.#.....#

Note: Contains a broken 4/3 chain so it has only 3 4/3s

22edo steps	15edo steps	UDP name	Spelling	Other notes
4423333	3312222	6|0 ^3	G A B^ C D E F
3442333	2331222	5|1 ^4	G Av B C^ D E F
3344233	2233122	4|2 ^5	G Av Bv C D^ E F
3334423	2223312	3|3 ^6	G Av Bv Cv D E^ F
3333442	2222331	2|4 ^7	G Av Bv Cv Dv E F^
2333344	1222233	0|6 v2v3v4v5v6	G Avv Bvv Cvv Dvv Evv Fv
4233334	3122223	0|6 ^2	G A Bv Cv Dv Ev Fv

MODMOS with 7th note in generator chain sharpened: ######.......#

Note: Contains a broken 4/3 chain so it has only 3 4/3s

22edo steps	15edo steps	UDP name	Spelling	Other notes
5233333	4122222	6|0 ^2	G A^ B C D E F
3523333	2412222	5|1 ^3	G Av B^ C D E F
3352333	2241222	4|2 ^4	G Av Bv C^ D E F
3335233	2224122	3|3 ^5	G Av Bv Cv D^ E F
3333523	2222412	2|4 ^6	G Av Bv Cv Dv E^ F
3333352	2222241	1|5 ^7	G Av Bv Cv Dv Ev F^
2333335	1222224	0|6 v2v3v4v5v6v7	G Avv Bvv Cvv Dvv Evv Fvv

Notable double-alteration MODMOS's

22edo steps	15edo steps	UDP name	Spelling	Other notes
4324342	3213231	6|0 v4 ^7	G A B Cv D E F^	"JI Major" scale a.k.a. Ptolemy/Zarlino
4234243	3123132	3|3 ^2 v6	G A Bv Cv D Ev F	"JI Minor" scale

Porcupine[8] and its MODMOS's

All generators below refer to the porcupine[8] bright generator (the 10/9~11/10~whatever).

Similarly to Porcupine[7], William Lynch has proposed naming the modes off of sea creatures, with Octopus being relevant to the number eight.

Functional mode names are proposed based on the 3-step triads, which may be augmented (~9:12:16), major (~3:4:5) or minor (~12:15:20). However, it can also useful to look at a different functional system in which the octatonic MOS modes extend the heptatonic ones by adding one extra note.

The Porcupine[8] MOS

Functional Name	William's Name	Step pattern	UDP name	Traditional Spelling	Triad qualities (3-step triads)	Other notes
Bright augmented	Octopus	33333331	7|0	A B C D E F G H	I+ II+ III IV V vi vii viii
Dark augmented	Mantis	33333313	6|1	A B C D E F G Hb	I+ II III IV v vi vii VIII+
Bright major	Dolphin	33333133	5|2	A B C D E F Gb Hb	I II III iv v vi VII+ VIII+
Middle major	Crab	33331333	4|3	A B C D E Fb Gb Hb	I II iii iv v VI+ VII+ VII	10:12:15 on the tonic Has both 4/3 and 3/2
Dark major	Tuna	33313333	3|4	A B C D Eb Fb Gb Hb	I ii iii iv V+ VI+ VII VIII	10:12:15 on the tonic Has both 4/3 and 3/2
Bright minor	Salmon	33133333	2|5	A B C Db Eb Fb Gb Hb	i ii iii IV+ V+ VI VII VIII	8:10:11:12 and 10:12:15 on the tonic
Middle minor	Starfish	31333333	1|6	A B Cb Db Eb Fb Gb Hb	i ii III+ IV+ V VI VII viii	8:9:10:11:12 on the tonic
Dark minor	Whale	13333333	0|7	A Bb Cb Db Eb Fb Gb Hb	i II+ III+ IV V VI vii viii	8:9:10:11:12 on the tonic

Single-alteration MODMOSes

Alterations by Porcupine[8] chroma

MODMOS with 2nd note in generator chain flattened: #......'#.'######

Step pattern	UDP name	Spelling	Other Notes
15333331	7|0 b2	A Bb C D E F G H
53333311	7|0 #2#3#4#5#6#7	A B# C# D# E# F# G# H
33333115	5|2 b8	A B C D E F Gb Hbb
33331153	4|3 b7	A B C D E Fb Gbb Hb
33311533	3|4 b6	A B C D Eb Fbb Gb Hb
33115333	2|5 b5	A B C Db Ebb Fb Gb Hb
31153333	1|6 b4	A B Cb Dbb Eb Fb Gb Hb
11533333	0|7 b3	A Bb Cbb Db Eb Fb Gb Hb

MODMOS with 3rd note in generator chain flattened: #.....'##.'#####

Step pattern	UDP name	Spelling	Other Notes
31533331	7|0 b3	A B Cb D E F G H
15333313	6|1 b2	A Bb C D E F G Hb
53333131	7|0 #2#3#4#5#6	A B# C# D# E# F# G H
33331315	4|3 b8	A B C D E Fb Gb Hbb
33313153	3|4 b7	A B C D Eb Fb Gbb Hb
33131533	2|5 b6	A B C Db Eb Fbb Gb Hb
31315333	1|6 b5	A B Cb Db Ebb Fb Gb Hb
13153333	0|7 b4	A Bb Cb Dbb Eb Fb Gb Hb

MODMOS with 4th note in generator chain flattened: #....'##'#.####

Step pattern	UDP name	Spelling	Other Notes
33153331	7|0 b4	A B C Db E F G H
31533313	6|1 b3	A B Cb D E F G Hb
15333133	5|2 b2	A Bb C D E F Gb Hb
53331331	7|0 #2#3#4#5	A B# C# D# E# F G H
33313315	3|4 b8	A B C D Eb Fb Gb Hbb
33133153	2|5 b7	A B C Db Eb Fb Gbb Hb
31331533	1|6 b6	A B Cb Db Eb Fbb Gb Hb
13315333	0|7 b5	A Bb Cb Db Ebb Fb Gb Hb

MODMOS with 5th note in generator chain flattened: #...'##'##.###

Step pattern	UDP name	Spelling	Other Notes
33315331	7|0 b5	A B C D Eb F G H
33153313	6|1 b4	A B C Db E F G Hb
31533133	5|2 b3	A B Cb D E F Gb Hb
15331333	4|3 b2	A Bb C D E Fb Gb Hb
53313331	7|0 #2#3#4	A B# C# D# E F G H
33133315	2|5 b8	A B C Db Eb Fb Gb Hbb
31333153	1|6 b7	A B Cb Db Eb Fb Gbb Hb
13331533	0|7 b6	A Bb Cb Db Eb Fbb Gb Hb

MODMOS with 6th note in generator chain flattened: #..'##'###.##

Functional name	Step pattern	UDP name #1	UDP name #2	Spelling	Triad qualities (3-step triads)	Other Notes
Bright augmented b6	33331531	7|0 b6	4|3 #7#8	A B C D E Fb G H	I+ II+ iii IV V VI+ vii viiiº	10:12:15 on tonic
Dark augmented b5	33315313	6|1 b5	3|4 #6#7	A B C D Eb F G Hb	I+ ii III IV V+ vi viiº VIII+
Bright minor #5#6	33153133	5|2 b4	2|5 #5#6	A B C Db E F Gb Hb	i II III IV+ v viº VII+ VIII+
Middle major b3	31531333	4|3 b3	1|6 #4#5	A B Cb D E Fb Gb Hb	I II III+ iv vº VI+ VII+ viii	Has both 4/3 and 3/2
Dark major b2	15313333	3|4 b2	0|7 #3#4	A Bb C D Eb Fb Gb Hb	I III+ iii ivº V+ VI+ vii VIII	10:12:15 on the tonic Has both 4/3 and 3/2
Bright augmented #2#3	53313331		7|0 #2#3	A B# C# D E F G H	I+ ii iiiº IV+ V+ vi VII VIII+
Middle minor b8	33133315	2|5 b8		A B Cb Db Eb Fb Gb Hbb	i iiº III+ IV+ v VI VII+ VIII+	8:9:10:11:12 on the tonic
Bright diminished	31333153	1|6 b7		A Bb Cb Db Eb Fb Gbb Hb	iº II+ III+ iv V VI+ VII+ viii	8:9:10:11:12 on the tonic

MODMOS with 7th note in generator chain flattened or 2nd sharpened: #.'##'####.#

Functional name	Step pattern	UDP name #1	UDP name #2	Spelling	Triad qualities (3-step triads)	Other Notes
Bright major #8	33333151	7|0 b7	5|2 #8	A B C D E F Gb H	I II+ III iv V vi VII+ viii	Brightest major with all consonant 3-step triads Major triads on 1/1, 6/5, 36/25 (with roots on 4/3, 8/5, 48/25) Minor triads on 4/3, 8/5, 48/25 (with roots on 10/9, 4/3, 8/5)
Dark augmented b6	33331513	6|1 b6	4|3 #7	A B C D E Fb G Hb	I+ II iii IV v VI+ vii VIII	10:12:15 on the tonic Has both 4/3 and 3/2
Bright major b5, Dark major #6	33315133	5|2 b5	3|4 #6	A B C D Eb F Gb Hb	I ii III iv V+ vi VII VIII+	Major triads on 1/1, 6/5, 5/3 (with roots on 4/3, 8/5, 10/9) Minor triads on 10/9, 4/3, 8/5 (with roots on 9/5, 10/9, 4/3)
Bright minor #5	33151333	4|3 b4	2|5 #5	A B C Db E Fb Gb Hb	i II iii IV+ v VI VII+ VIII	Brightest minor with all consonant 3-step triads Major triads on 10/9, 3/2, 9/5 (with roots on 36/25, 1/1, 6/5) Minor triads on 1/1, 6/5, 36/25 (with roots on 5/3, 1/1, 6/5)
Dark major v3	31513333	3|4 b3	1|6 #4	A B Cb D Eb Fb Gb Hb	I ii III+ iv V VI+ VII viii	Major triads on 1/1, 25/18, 5/3 (with roots on 4/3, 9/5, 10/9) Minor triads on 10/9, 4/3, 9/5 (with roots on 9/5, 10/9, 3/2) Has both 4/3 and 3/2
Bright minor b2, Dark minor #3	15133333	2|5 b2	0|7 #3	A Bb C Db Eb Fb Gb Hb	i II+ iii IV V+ VI vii VIII	8:10:11:12 and 10:12:15 on the tonic Major triads on 5/4, 3/2, 9/5 (with roots on 5/3, 1/1, 6/5) Minor triads on 1/1, 6/5, 5/3 (with roots on 5/3, 1/1, 25/18)
Bright augmented #2	51333331		7|0 #2	A B# C D E F G H	I+ ii III IV+ V vi VII viii	Brightest mode with all consonant 3-step triads
Dark minor b8	13333315	0|7 b8		A Bb Cb Db Eb Fb Gb Hbb	i II III+ IV v VI vii VIII+	Darkest mode/minor with all consonant 3-step triads 8:9:10:11:12 on the tonic (8:9:10:11:12:14 in 15edo) Major triads on 25/24, 5/4, 3/2 (with roots on 25/18, 5/3, 1/1) Minor triads on 1/1, 25/18, 5/3 (with roots on 5/3, 9/8, 25/18)

MODMOS with 3rd note in generator chain sharpened: ##.'#'####..#

Functional name	Step pattern	UDP name #1	UDP name #2	Spelling	Triad qualities (3-step triads)	Other Notes
Bright augmented #3	35133331	7|0 #3		A B C# D E F G H	I+ II+ iiiº IV V+ vi vii VIII
Dark augmented #2	51333313	6|1 #2		A B# C D E F G Hb	I+ iiº III IV+ v vi VII VIII+
Dark diminished	13333135		0|7 b7b8	A Bb Cb Db Eb Fb Gbb Hbb	iº II III+ iv v VI VII+ VIII+	8:9:10:11:12 on the tonic 8:9:10:11:12:14 on the tonic in 15edo
Middle major #8	33331351	4|3 #8	7|0 b6b7	A B C D E Fb Gb H	I II+ iii iv V VI+ VII+ viiiº	10:12:15 on the tonic Has both 4/3 and 3/2
Dark augmented b5b6	33313513	3|4 #7	6|1 b5b6	A B C D Eb Fb G Hb	I+ ii iii IV V+ VI+ viiº VIII	10:12:15 on the tonic Has both 4/3 and 3/2
Bright minor #6	33135133	2|5 #6	5|2 b4b5	A B C Db Eb F Gb Hb	i ii III IV+ V+ viº VII VIII+
Middle minor #5	31351333	1|6 #5	4|3 b3b4	A B Cb Db E Fb Gb Hb	i II III+ IV+ vº VI VII+ viii	8:9:10:12 on the tonic
Dark major b2b3	13513333	0|7 #4	3|4 b2b3	A Bb Cb D Eb Fb Gb Hb	I II+ III+ ivº V VI+ vii viii	Has both 4/3 and 3/2

'MODMOS with 4th note in generator chain sharpened: ###.'####...#

Step pattern	UDP name	Spelling	Other Notes
33513331	7|0 #4	A B C D# E F G H
35133331	6|1 #3	A B C# D E F G Hb
51333313	5|2 #2	A B# C D E F Gb Hb
13333135	0|7 b6b7b8	A Bb Cb Db Eb Fbb Gbb Hbb
33331351	3|4 #8	A B C D Eb Fb Gb H
33313513	2|5 #7	A B C Db Eb Fb G Hb
33135133	1|6 #6	A B Cb Db Eb F Gb Hb
31351333	0|7 #5	A Bb Cb Db E Fb Gb Hb

'MODMOS with 5th note in generator chain sharpened: ####.'###....#

Step pattern	UDP name	Spelling	Other Notes
33351331	7|0 #5	A B C D E# F G H
33513331	6|1 #4	A B C D# E F G Hb
35133313	5|2 #3	A B C# D E F Gb Hb
51333133	4|3 #2	A B# C D E Fb Gb Hb
13331335	0|7 b5b6b7b8	A Bb Cb Db Ebb Fbb Gbb Hbb
33313351	2|5 #8	A B C Db Eb Fb Gb H
33133513	1|6 #7	A B Cb Db Eb Fb G Hb
31335133	0|7 #6	A Bb Cb Db Eb F Gb Hb

'MODMOS with 6th note in generator chain sharpened: #####.'##.....#

Step pattern	UDP name	Spelling	Other Notes
33335131	7|0 #6	A B C D E F# G H
33351313	6|1 #5	A B C D E# F G Hb
33513133	5|2 #4	A B C D# E F Gb Hb
35131333	4|3 #3	A B C# D E Fb Gb Hb
51313333	3|4 #2	A B# C D Eb Fb Gb Hb
13133335	0|7 b4b5b6b7b8	A Bb Cb Dbb Ebb Fbb Gbb Hbb
31333351	1|6 #8	A B Cb Db Eb Fb Gb H
13333513	0|7 #7	A Bb Cb Db Eb Fb G Hb

'MODMOS with 7th note in generator chain sharpened: ######.'#......#

Step pattern	UDP name	Spelling	Other Notes
33333511	7|0 #7	A B C D E F G# H
33335113	6|1 #6	A B C D E F# G Hb
33351133	5|2 #5	A B C D E# F Gb Hb
33511333	4|3 #4	A B C D# E Fb Gb Hb
35113333	3|4 #3	A B C# D Eb Fb Gb Hb
51133333	2|5 #2	A B# C Db Eb Fb Gb Hb
11333335	0|7 b3b4b5b6b7b8	A Bb Cbb Dbb Ebb Fbb Gbb Hbb
13333351	0|7 #8	A Bb Cb Db Eb Fb Gb H

Alterations by Porcupine[7] chroma

MODMOS with 2nd note in generator chain flattened

Step pattern	UDP name	Spelling	Other notes
24333331	7|0 v2	A Bv C D E F G H
43333312	6|1 ^2^3^4^5^6^7^8	A B^ C^ D^ E^ F^ G^ Hb^
33333124	5|2 v8	A B C D E F Gb Hbv
33331243	4|3 v7	A B C D E Fb Gbv Hb
33312433	3|4 v6	A B C D Eb Fbv Gb Hb
33124333	2|5 v5	A B C Db Ebv Fb Gb Hb
31243333	1|6 v4	A B Cb Dbv Eb Fb Gb Hb
12433333	0|7 v3	A Bb Cbv Db Eb Fb Gb Hb

MODMOS with 3rd note in generator chain flattened

Step pattern	UDP name	Spelling	Other notes
32433331	7|0 v3	A B Cv D E F G H
24333313	6|1 v2	A Bv C D E F G Hb
43333132	5|2 ^2^3^4^5^6^7^8	A B^ C^ D^ E^ F^ Gb^ Hb^
33331324	4|3 v8	A B C D E Fb Gb Hbv
33313243	3|4 v7	A B C D Eb Fb Gbv Hb
33132433	2|5 v6	A B C Db Eb Fbv Gb Hb
31324333	1|6 v5	A B Cb Db Ebv Fb Gb Hb
13243333	0|7 v4	A Bb Cb Dbv Eb Fb Gb Hb

MODMOS with 4th note in generator chain flattened

Step pattern	UDP name	Spelling	Other notes
33243331	7|0 v4	A B C Dv E F G H
32433313	6|1 v3	A B Cv D E F G Hb
24333133	5|2 v2	A Bv C D E F Gb Hb
43331332	4|3 ^2^3^4^5^6^7^8	A B^ C^ D^ E^ Fb^ Gb^ Hb^
33313324	3|4 v8	A B C D Eb Fb Gb Hbv
33133243	2|5 v7	A B C Db Eb Fb Gbv Hb
31332433	1|6 v6	A B Cb Db Eb Fbv Gb Hb
13324333	0|7 v5	A Bb Cb Db Ebv Fb Gb Hb

MODMOS with 5th note in generator chain flattened

Step pattern	UDP name	Spelling	Other notes
33324331	7|0 v5	A B C D Ev F G H
33243313	6|1 v4	A B C Dv E F G Hb
32433133	5|2 v3	A B Cv D E F Gb Hb
24331333	4|3 v2	A Bv C D E Fb Gb Hb
43313332	3|4 ^2^3^4^5^6^7^8	A B^ C^ D^ Eb^ Fb^ Gb^ Hb^
33133324	2|5 v8	A B C Db Eb Fb Gb Hbv
31333243	1|6 v7	A B Cb Db Eb Fb Gbv Hb
13332433	0|7 v6	A Bb Cb Db Eb Fbv Gb Hb

MODMOS with 6th note in generator chain flattened

Step pattern	UDP name	Spelling	Other notes
33332431	7|0 v6	A B C D E Fv G H
33324313	6|1 v5	A B C D Ev F G Hb
33243133	5|2 v4	A B C Dv E F Gb Hb
32431333	4|3 v3	A B Cv D E Fb Gb Hb
24313333	3|4 v2	A Bv C D Eb Fb Gb Hb
43133332	2|5 ^2^3^4^5^6^7^8	A B^ C^ Db^ Eb^ Fb^ Gb^ Hb^
31333324	1|6 v8	A B Cb Db Eb Fb Gb Hbv
13333243	0|7 v7	A Bb Cb Db Eb Fb Gbv Hb

MODMOS with 7th note in generator chain flattened

Step pattern	UDP name	Spelling	Other notes
33333241	7|0 v7	A B C D E F Gv H
33332413	6|1 v6	A B C D E Fv G Hb
33324133	5|2 v5	A B C D Ev F Gb Hb
33241333	4|3 v4	A B C Dv E Fb Gb Hb
32413333	3|4 v3	A B Cv D Eb Fb Gb Hb
24133333	2|5 v2	A Bv C Db Eb Fb Gb Hb
41333332	1|6 ^2^3^4^5^6^7^8	A B^ Cb^ Db^ Eb^ Fb^ Gb^ Hb^
13333324	0|7 v8	A Bb Cb Db Eb Fb Gb Hbv

MODMOS with 8th note in generator chain flattened

Step pattern	UDP name	Spelling	Other notes
33333322	7|0 v8	A B C D E F G Hv
33333223	6|1 v7	A B C D E F Gv Hb
33332233	5|2 v6	A B C D E Fv Gb Hb
33322333	4|3 v5	A B C D Ev Fb Gb Hb
33223333	3|4 v4	A B C Dv Eb Fb Gb Hb
32233333	2|5 v3	A B Cv Db Eb Fb Gb Hb
22333333	1|6 v2	A Bv Cb Db Eb Fb Gb Hb
23333332	0|7 ^2^3^4^5^6^7^8	A Bb^ Cb^ Db^ Eb^ Fb^ Gb^ Hb^

MODMOS with 1st note in generator chain sharpened (in 22edo this gives same modes as flattening 8th note)

Step pattern	UDP name	Spelling	Other notes
23333332	7|0 v2v3v4v5v6v7v8	A Bv Cv Dv Ev Fv Gv Hv
33333322	6|1 ^8	A B C D E F G Hb^
33333223	5|2 ^7	A B C D E F Gb^ Hb
33332233	4|3 ^6	A B C D E Fb^ Gb Hb
33322333	3|4 ^5	A B C D Eb^ Fb Gb Hb
33223333	2|5 ^4	A B C Db^ Eb Fb Gb Hb
32233333	1|6 ^3	A B Cb^ Db Eb Fb Gb Hb
22333333	0|7 ^2	A Bb^ Cb Db Eb Fb Gb Hb

MODMOS with 2nd note in generator chain sharpened

Step pattern	UDP name	Spelling	Other notes
42333331	7|0 ^2	A B^ C D E F G H
23333314	6|1 v2v3v4v5v6v7v8	A Bv Cv Dv Ev Fv Gv Hbv
33333142	5|2 ^8	A B C D E F Gb Hb^
33331423	4|3 ^7	A B C D E Fb Gb^ Hb
33314233	3|4 ^6	A B C D Eb Fb^ Gb Hb
33142333	2|5 ^5	A B C Db Eb^ Fb Gb Hb
31423333	1|6 ^4	A B Cb Db^ Eb Fb Gb Hb
14233333	0|7 ^3	A Bb Cb^ Db Eb Fb Gb Hb

MODMOS with 3rd note in generator chain sharpened

Step pattern	UDP name	Spelling	Other notes
34233331	7|0 ^3	A B C^ D E F G H
42333313	6|1^2	A B^ C D E F G Hb
23333134	5|2 v2v3v4v5v6v7v8	A Bv Cv Dv Ev Fv Gbv Hbv
33331342	4|3 ^8	A B C D E Fb Gb Hb^
33313423	3|4 ^7	A B C D Eb Fb Gb^ Hb
33134233	2|5 ^6	A B C Db Eb Fb^ Gb Hb
31342333	1|6 ^5	A B Cb Db Eb^ Fb Gb Hb
13423333	0|7 ^4	A Bb Cb Db^ Eb Fb Gb Hb

MODMOS with 4th note in generator chain sharpened

Step pattern	UDP name	Spelling	Other notes
33423331	7|0 ^4	A B C D^ E F G H
34233313	6|1^3	A B C^ D E F G Hb
42333133	5|2 ^2	A B^ C D E F Gb Hb
23331334	4|3 v2v3v4v5v6v7v8	A Bv Cv Dv Ev Fbv Gbv Hbv
33313342	3|4 ^8	A B C D Eb Fb Gb Hb^
33133423	2|5 ^7	A B C Db Eb Fb Gb^ Hb
31334233	1|6 ^6	A B Cb Db Eb Fb^ Gb Hb
13342333	0|7 ^5	A Bb Cb Db Eb^ Fb Gb Hb

MODMOS with 5th note in generator chain sharpened

Step pattern	UDP name	Spelling	Other notes
33342331	7|0 ^5	A B C D E^ F G H
33423313	6|1^4	A B C D^ E F G Hb
34233133	5|2 ^3	A B C^ D E F Gb Hb
42331333	4|3 ^2	A B^ C D E Fb Gb Hb
23313334	3|4 v2v3v4v5v6v7v8	A Bv Cv Dv Ebv Fbv Gbv Hbv
33133342	2|5 ^8	A B C Db Eb Fb Gb Hb^
31333423	1|6 ^7	A B Cb Db Eb Fb Gb^ Hb
13334233	0|7 ^6	A Bb Cb Db Eb Fb^ Gb Hb

MODMOS with 6th note in generator chain sharpened

Step pattern	UDP name	Spelling	Other notes
33334231	7|0 ^6	A B C D E F^ G H
33342313	6|1 ^5	A B C D E^ F G Hb
33423133	5|2 ^4	A B C D^ E F Gb Hb
34231333	4|3 ^3	A B C^ D E Fb Gb Hb
42313333	3|4 ^2	A B^ C D Eb Fb Gb Hb
23133334	2|5 v2v3v4v5v6v7v8	A Bv Cv Dbv Ebv Fbv Gbv Hbv
31333342	1|6 ^8	A B Cb Db Eb Fb Gb Hb^
13333423	0|7 ^7	A Bb Cb Db Eb Fb Gb^ Hb

MODMOS with 7th note in generator chain sharpened

Step pattern	UDP name	Spelling	Other notes
33333421	7|0 ^7	A B C D E F G^ H
33334213	6|1 ^6	A B C D E F^ G Hb
33342133	5|2 ^5	A B C D E^ F Gb Hb
33421333	4|3 ^4	A B C D^ E Fb Gb Hb
34213333	3|4 ^3	A B C^ D Eb Fb Gb Hb
42133333	2|5 ^2	A B^ C Db Eb Fb Gb Hb
21333334	1|6 v2v3v4v5v6v7v8	A Bv Cbv Dbv Ebv Fbv Gbv Hbv
13333342	0|7 ^8	A Bb Cb Db Eb Fb Gb Hb^