Porcupine

Porcupine, [15 & 22] (usually defined in 2.3.5.11 or 2.3.5.7.11), is a temperament that splits 4/3 into three submajor seconds (approximately 11/10), representing 10/9~11/10~12/11. In the 5-limit, it equates 81/80 with 25/24. This makes it an excellent compromise between accuracy and simplicity on the side of simplicity while at the same time not fully exotempering (the intervals it detunes significantly, such as 10/9, can be seen as "connective" intervals rather than distinct harmonic identities, with the notable exception of 11/9). Porcupine is also notable for being inherently "un-Meantone" in the sense that rather than tempering out 81/80, it equates 81/80 to a fundamental 5-limit structural interval 25/24, the difference between 5/4 and 6/5; in fact, 7edo (essentially a trivial tuning of both) is the unique edo that is both Porcupine and Meantone.

The simplest Porcupine edo join is [7 & 8], and surprisingly this correctly defines 11-limit porcupine (implying 8edo technically supports archy) - however this results in an inaccurate extension to higher primes than 11. Also note that 29 agrees with 15 & 22 in 2.3.5.11 but not in 2.3.5.7.11, thus [22 & 29] represents a separate extension from [15 & 22] in the full 11-limit.

The Porcupine generator generates 1L6s, 7L1s, and 7L8s.

Interval chain

In the following table, odd harmonics 1–15 and their inverses are in bold. Interpretations in parentheses are only found in the Septimal Porcupine (2.3.5.7.11.17 [15 & 22]) extension.

#	Cents*	Approximate ratios
0	0.0	1/1
1	162.8	10/9, 11/10, 12/11
2	325.6	6/5, 11/9, (17/14)
3	488.4	4/3
4	651.3	16/11
5	814.1	8/5
6	976.9	(7/4), 16/9
7	1139.7	160/81, 64/33, 48/25
8	102.5	16/15, (17/16)
9	265.3	(7/6)
10	428.2	(14/11), 32/25
11	591.0	(7/5, 17/12)
12	753.8	(14/9, 17/11)
13	916.6	(17/10)

* In 2.3.5.7.11 CWE tuning

Notation and intervals

Each sharp or flat from MOS diatonic can be split into three distinct notes, so we use the accidental ^ to raise by a 1/3 chroma and v to lower by a 1/3 chroma (ups and downs notation). Other accidentals that are identified with this in porcupine include any accidental representing the syntonic comma (such as in Ben Johnston, sagittal, SRS, or FJS notation), any accidental representing 25/24 (also Ben Johnston), and any accidental representing 33/32 (such as the FJS or HEJI accidentals for 11).

To avoid ambiguity, systems of notation that utilize the porcupine 25/24 diesis as their chroma may use exclusively ups and downs, though it might be more natural to some to repurpose the diatonic # and b symbols, especially if diatonic notation is not used simultaneously with these other schemes.

Zarlino notation uses the ternary Zarlino scale (see #Zarlino diatonic), or Ptolemy's intense diatonic, as its basic scale. Alternatively, the MOS tempering of equable heptatonic may be used, with degrees from 1 through 7 instead of note names.

As porcupine (15 & 22 extension) is a keemic temperament, it has four evenly spaced interval qualities: subminor, nearminor, nearmajor, and supermajor. These are the qualities found in 22edo, but they may also be applied to other porcupine systems such as 15 or 37edo. They may also be used more generally in just intonation, where they are not evenly spaced.

Porcupine also therefore gives a distinction between MOS diatonic (the standard superpyth diatonic, with each chroma split in three parts) and "zarlino" diatonic, wherein zarlino may be seen as a MODMOS of onyx.

Modal harmony

Modal harmony further emphasizes the qualities of the various intervals and chords found in the different scales used in music, as opposed to things like leading tendencies. It is within modal harmony that clear "supermajor", "nearmajor", "nearminor", and "subminor" diatonic scales can be defined, rather than used as context-dependent tonal systems. These mostly follow the interval qualities suggested above, except this time it becomes applicable to an entire scale rather than just to specific chords. (And of course, additional modes of zarlino or mosdiatonic may be used.)

The modes presented here, arranged in a Tetrahedron.

Given this, it's also useful to enumerate various modal scales, as a counterpart to the various non-Ionian/Aeolian modes used throughout standard modal harmony. These will not be exclusively "real" diatonic modes, but rather combinations of qualities loosely analogous to standard modes (and sharing the quality of having notes constrained to range in certain qualities), in six "series" comprising 22 unique modes, visible on the right. This setup overall aims to generalize the idea that diatonic modes exist on a gradation of "brightness" in 12edo, where successive alterations make a mode brighter or darker. Here, bright vs. dark isn't the only axis, however - there's near vs. super/sub and stable vs. unstable as well, so series of alterations along those allow for a much more complex selection of modes to choose from. Additionally, each series of modes has one quality in common, which I've labelled here, so all the "stable" modes have only stable intervals, even if they might contain both nearmajor and subminor ones, and all the "bright" modes contain only major intervals, even if they might be both nearmajor and supermajor. Holding the fourth and fifth constant (as Lydian and Locrian are rarely used in standard diatonic modal harmony) means that there are four "vertex" modes, corresponding to the pure nearmajor, nearminor, supermajor, and subminor qualities, as well as the Ionian and Phrygian modes of Zarlino and mosdiatonic.

Choosing a mode

Much as the choice of mode in 12edo largely depends on its position on the scale from bright to dark, you might choose a mode here by selecting a series based on the common sound you want your song or section to have, and then choosing a position on that series between its two extremes. For example, for something intense and somewhat uncanny, you might start by choosing the Unstable series, and then proceed to select a mode along that series between bright/super and near/dark that embodies the feel you want, such as Unstable Dorian. Alternatively, for an excited, cheerful sound, you might choose the Bright series and a mode between the near/stable and super/unstable extremes of it, such as Didymic Major.

The "Equable" mode (which is the main porcupine MOS) serves as a somewhat 'neutral' sound - despite the lack of neutral intervals in porcupine, it still occupies that somewhat soft position in between major and minor qualities, while at the same time being more equally distributed than any other version of Dorian available. Quality-wise, it has a mix of nearmajor (bright, stable) and nearminor (dark, unstable) intervals, serving as the opposite polarity to MOS Dorian, and its equidistant nature somewhat overrides other quality-based properties from a melodic perspective. On the opposite side of things, MOS Dorian can be seen as somewhat aggressively defined by its qualities, being a mix of subminor (stable, dark) and supermajor (unstable, bright), with a subminor third on the tonic.

Other pairs of "opposing" modes include unstable Dorian vs. stable Dorian, and didymic major vs. didymic minor, both of which unlike equable vs. MOS Dorian form complementary pairs similar to Ionian and Phrygian in 12edo.

Here they have been organized into two "loops"; bolded entries represent modes that differ along the loops, and italicized entries have had their positions flipped.

All modes are given in 22edo tuning.

Class A

Loop A				Loop B
Series	Mode	Type	Name	Series	Mode	Type	Name
Super/Sub	├───┴┴───┴───┴┴───┴───┤ 4 1 4 4 1 4 4	Aeolian	Aeolian	Super/Sub	├───┴┴───┴───┴┴───┴───┤ 4 1 4 4 1 4 4	Aeolian	Aeolian
Super/Sub	├───┴┴───┴───┴───┴┴───┤ 4 1 4 4 4 1 4	Dorian	Dorian	Super/Sub	├───┴┴───┴───┴───┴┴───┤ 4 1 4 4 4 1 4	Dorian	Dorian
Super/Sub	├───┴───┴┴───┴───┴┴───┤ 4 4 1 4 4 1 4	Mixolydian	Mixolydian	Super/Sub	├───┴───┴┴───┴───┴┴───┤ 4 4 1 4 4 1 4	Mixolydian	Mixolydian
Bright, Super/Sub, Unstable	├───┴───┴┴───┴───┴───┴┤ 4 4 1 4 4 4 1	Ionian	Ionian ("Supermajor")	Bright, Super/Sub, Unstable	├───┴───┴┴───┴───┴───┴┤ 4 4 1 4 4 4 1	Ionian	Ionian ("Supermajor")
Bright	├───┴───┴┴───┴───┴──┴─┤ 4 4 1 4 4 3 2	Ionian	Harmonic major	Unstable	├───┴───┴┴───┴───┴─┴──┤ 4 4 1 4 4 2 3	Mixolydian	Unstable Mixolydian
Bright	├───┴──┴─┴───┴───┴──┴─┤ 4 3 2 4 4 3 2	Ionian	Didymic major	Unstable	├───┴─┴──┴───┴───┴─┴──┤ 4 2 3 4 4 2 3	Dorian	Unstable Dorian
Bright	├───┴──┴─┴───┴──┴───┴─┤ 4 3 2 4 3 4 2	Ionian	RH-Ionian	Unstable	├───┴─┴──┴───┴─┴───┴──┤ 4 2 3 4 2 4 3	Aeolian	LH-Aeolian
Near, Bright, Stable	├──┴───┴─┴───┴──┴───┴─┤ 3 4 2 4 3 4 2	Ionian	LH-Ionian ("Nearmajor")	Near, Dark, Unstable	├─┴───┴──┴───┴─┴───┴──┤ 2 4 3 4 2 4 3	Phrygian	RH-Phrygian ("Nearminor")
Near	├──┴───┴─┴───┴──┴──┴──┤ 3 4 2 4 3 3 3	Mixolydian	Major equable	Near	├──┴──┴──┴───┴─┴───┴──┤ 3 3 3 4 2 4 3	Aeolian	Minor equable
Near	├──┴──┴──┴───┴──┴──┴──┤ 3 3 3 4 3 3 3	Dorian	Equable	Near	├──┴──┴──┴───┴──┴──┴──┤ 3 3 3 4 3 3 3	Dorian	Equable
Near	├──┴──┴──┴───┴─┴───┴──┤ 3 3 3 4 2 4 3	Aeolian	Minor equable	Near	├──┴───┴─┴───┴──┴──┴──┤ 3 4 2 4 3 3 3	Mixolydian	Major equable
Near, Dark, Unstable	├─┴───┴──┴───┴─┴───┴──┤ 2 4 3 4 2 4 3	Phrygian	RH-Phrygian ("Nearminor")	Near, Bright, Stable	├──┴───┴─┴───┴──┴───┴─┤ 3 4 2 4 3 4 2	Ionian	LH-Ionian ("Nearmajor")
Dark	├─┴───┴──┴───┴─┴──┴───┤ 2 4 3 4 2 3 4	Phrygian	LH-Phrygian	Stable	├──┴───┴─┴───┴──┴─┴───┤ 3 4 2 4 3 2 4	Mixolydian	RH-Mixolydian
Dark	├─┴──┴───┴───┴─┴──┴───┤ 2 3 4 4 2 3 4	Phrygian	Didymic minor	Stable	├──┴─┴───┴───┴──┴─┴───┤ 3 2 4 4 3 2 4	Dorian	Stable Dorian
Dark	├─┴──┴───┴───┴┴───┴───┤ 2 3 4 4 1 4 4	Phrygian	Subharmonic minor	Stable	├──┴─┴───┴───┴┴───┴───┤ 3 2 4 4 1 4 4	Aeolian	Stable Aeolian
Dark, Super/Sub, Stable	├┴───┴───┴───┴┴───┴───┤ 1 4 4 4 1 4 4	Phrygian	Phrygian ("Subminor")	Dark, Super/Sub, Stable	├┴───┴───┴───┴┴───┴───┤ 1 4 4 4 1 4 4	Phrygian	Phrygian ("Subminor")

Note that Aeolian is not a vertex. Because of this, it might be prudent to construct a secondary, smaller tetrahedron that holds the major second constant alongside the fourth and fifth. Doing so yields six additional modes:

• A set of two additional modes between RH-Ionian and LH-Aeolian, acting as alternative near forms of Mixolydian/major equable (├───┴──┴─┴───┴──┴──┴──┤ 4 3 2 4 3 3 3) and Dorian/equable (├───┴─┴──┴───┴──┴──┴──┤ 4 2 3 4 3 3 3 )

• A set of two additional modes between LH-Aeolian and mosdiatonic Aeolian, acting as alternative dark/minor scales (├───┴─┴──┴───┴─┴──┴───┤ 4 2 3 4 2 3 4, ├───┴┴───┴───┴─┴──┴───┤ 4 1 4 4 2 3 4 ).

• Alternative stable forms of Mixolydian (├───┴──┴─┴───┴──┴─┴───┤ 4 3 2 4 3 2 4) and Dorian (├───┴┴───┴───┴──┴─┴───┤ 4 1 4 4 3 2 4 ). These differ by one note varying by two steps; between them is in fact the simplest possible 5-limit Dorian, at (├───┴─┴──┴───┴──┴─┴───┤ 4 2 3 4 3 2 4 ), which is not a mode of zarlino due to distributing the large and medium steps differently.

This appears to suggest that the sum total of all theoretically possible modes existing under this system is the complete volume of a tetrahedron with endpoints at near- Locrian and Lydian and at sub-Locrian and super-Lydian. There are 84 total modes in the scheme, which are the rotations of the following 8 base scales, including chirality. These are the set of scales that have the property that all instances of any diatonic interval between any two notes in the scale are either supermajor, nearmajor, nearminor, or subminor, which is the property that constrains the tetrahedron:

Name	Scale	Note	Symmetrical?	Exists in the set of 22 modes?
mosdiatonic	├───┴───┴┴───┴───┴───┴┤ 4 4 1 4 4 4 1		No	Yes
harmonic major	├───┴───┴┴───┴───┴──┴─┤ 4 4 1 4 4 3 2		Yes	Yes
didymic	├───┴──┴─┴───┴───┴──┴─┤ 4 3 2 4 4 3 2		Yes	Yes
zarlino	├──┴───┴─┴───┴──┴───┴─┤ 3 4 2 4 3 4 2		Yes	Yes
diatonyx-A	├───┴──┴─┴───┴──┴──┴──┤ 4 3 2 4 3 3 3	The upper tetrachord is a porcupine tetrachord.	No	No
diatonyx-B	├──┴──┴──┴───┴─┴───┴──┤ 3 3 3 4 2 4 3	The lower tetrachord is a porcupine tetrachord.	Yes	Yes
equable / onyx	├──┴──┴──┴───┴──┴──┴──┤ 3 3 3 4 3 3 3		No	Yes
symmetrical dorian	├───┴─┴──┴───┴──┴─┴───┤ 4 2 3 4 3 2 4		No	No

This reduces to a set of 35 if the fourth and fifth are held fixed, and 55 if only the fifth is.

Every mode of one of these scales has a pattern of broadly major and minor intervals corresponding to one of the standard diatonic modes. For example, the equable scale in its primary mode is a form of Dorian, as its pattern is major-minor-perfect-perfect-major-minor (in this case, nearmajor and nearminor). However, there are not in fact 12 instances of each mode! The equable scale only has Mixolydian, Dorian, and Aeolian modes, and the symmetrical Dorian scale lacks a Locrian or Lydian mode.

Mode type	84-set	35-set	22-set
Locrian	7	-	-
Phrygian	12	5	5
Aeolian	15	8	4
Dorian	16	9	4
Mixolydian	15	8	4
Ionian	12	5	5
Lydian	7	-	-

Regardless, this is simply a mathematically complete enumeration - for actual modal music, it is best to stick to the list of 22 modes provided above, as those are the ones that have clear common qualities alongside the functionally important perfect fifth and fourth.

Additional work needs to be done to determine if this can be generalized.

Tetrachords in modal analysis

Tetrachords can be used in a different way, more in accordance with their use in modern 12edo theory. In modes where the fourth and fifth are perfect, the mode can always be thought of as being comprised of two tetrachords separated by a whole tone, although the constraints on these tetrachords are entirely different from the Greek versions. In short, a modal tetrachord must comprise the unison, the fourth, a second of one of the four qualities, and a third of one of the four qualities, such that the interval between two adjacent tones is never more than four steps. This is a generalization of the constraints on modal tetrachord patterns in 12edo, which must always contain either whole tones or semitones. By this constraint there are ten distinct tetrachords in 22edo. Considering all the possible scales constructed from these, there are 10x10 = 100 distinct possibilities, compared to the 3x3 = 9 options found in 12edo. This provides an extended set, including not only the 35 modes corresponding to diatonic but 65 additional scales corresponding in some regard to melodic minor or neapolitan major. Not all intervals are necessarily within their expected quality ranges.

Loosening the constraint further to only necessitate that the two movable tones remain within the 4 qualities of their respective degrees allows for the generalization to a set of scales analogous to harmonic minor or double harmonic major, with 156 additional possibilities.

Chromatic subsets

In porcupine, multiple qualities may be combined together into a compound system. With mosdiatonic alone, there is little reason to do this, because there are only two qualities available, so the scale combining them (the chromatic scale, or some other large scale like ├─┴┴┴┴─┴┴┴─┴┤ 2 1 1 1 2 1 1 2 1 (12edo tuning)) is not particularly engaging from either a tonal or modal perspective. However, in porcupine there are four different qualities, from which two may be selected to share characteristics.

The standard chromatic scales combine nearmajor+nearminor and supermajor+subminor, which lead to somewhat of the same problem as 12edo chromatic; they are opposing pairs of qualities. However, if we make an asymmetric chromatic scale, with (for instance) supermajor and nearminor, we get a scale with the trait they have in common: being "unstable". Alternatively, you could get a generally "dark" system by combining subminor and nearminor qualities.

The following are a few examples of these kinds of scales, including diatonic and chromatic variations. (Note that in tonal music, these become less distinct from standard counterparts, as degrees are already expected to be altered between different qualities depending on context.)

Aberrismic scales may also be leveraged for this purpose.

Functional harmony

See Archy#Full 7-limit harmony.

Scales

Zarlino

The 5-limit zarlino scale, in Porcupine temperament, has its two chromas 25/24 and 81/80 equated, making it a MODMOS of onyx. This characteristic defines 5-limit Porcupine. It allows zarlino to be notated with a single pair of accidentals, which may be written as ups and downs if MOSdiatonic sharps are used, consistently with the general usage of ups and downs in porcupine.

Including degrees of both zarlino and Pythagorean diatonic results in blackdye. In fact, blackdye or several characteristics of it are likely to naturally emerge in tonal harmony in the first place, given an overall desire to avoid wolf fifths in nearmajor and nearminor tonalities, resulting in certain intervals being doubled up. Most notably, it is reasonable to consider the two forms of major second equally part of a nearmajor tonal system, so that 5-2 and 2-6 can both be perfect fifths with different versions of the 2 degree (although unless 6 is additionally sharpened, some additional harmonic movements are needed to resolve the edostep offset that results from a pumped syntonic comma if you move from fifth-bounded triads on 5 to 2 in a single motion).

Zarlino may be used as the base scale for porcupine. This encounters problems with existing familiarity with notation (for example, one of D-G and G-C must now be a flat 5th, called a "wolf fifth" and representing 16/11 as opposed to 3/2), but may be more structurally useful than considering equiheptatonic the base scale.

Equiheptatonic

The equable diatonic, 1/(18:20:22:24:27:30:33:36) (or in this case, equivalently its otonal counterpart) is represented as the MOS scale sssLsss in porcupine. It is reasonable to, for that structural reason, consider sssLsss the default mode, with a nearminor chord on the tonic - it is the unique mode which possesses both a perfect fifth and a perfect fourth. This is more generally the MOS porcupine[7]; altering several notes of this MOS yields the Zarlino diatonic, explaining 22edo Zarlino's heavy reliance on porcupine's equivalences. Porcupine also has an 8-note scale LLLLsLLL that adds an additional "blue note" to the heptatonic, and a chromatic scale sLsLsLsLsLsLsLs, a form of the Roklotian scale that may also be derived by dividing the intervals of superpyth pentatonic: sLsLs -> [sLs] [LsL] [sLs] [LsL] [sLs].

Mode	Brightness	Steps	Interval qualities
Mixolydian	3	Lssssss	AMMMMP
Mixolydian	2	sLsssss	PMMMMP
Dorian	1	ssLssss	PmMMMP
Dorian	0	sssLsss	PmmMMP
Dorian	-1	ssssLss	PmmmMP
Aeolian	-2	sssssLs	PmmmmP
Aeolian	-3	ssssssL	Pmmmmd

Solfege

A solfege provided by Vector for porcupine uses an -n coda to inflect inwards by a 1/3-chroma; -s may be used to inflect outwards by a 1/3-chroma. The standard 12-form solfege syllables (do, ra, re, me, mi...) represent Superpyth[12].

An alternate solfege proposed by Kite consistently uses the sequence -i, -u, -o, and -a for the interval qualities.

A solfege used by Andrew Heathwaite uses the following syllables:

Note	Solfege (on Do)	Note	Solfege (on Do)
P1 / P8	Do	vA4	Fi
m2	Di	vP5	Su
^m2	Ra	P5	Sol
vM2	Ru	m6	Lo
M2	Re	^m6	Le
m3	Ma	vM6	La
^m3	Me	M6	Li
vM3	Mi	m7	Ta
M3	Mo	^m7	Tu
P4	Fa	vM7	Ti
^P4	Fu	M7	Da

List of patent vals

EDO	Extension to 7	Generator tuning	25/24 tuning	Fifth tuning
7	[15 & 22]	171.4c	0c	685.7c
29	[22 & 29]	165.5c	41.4c	703.5c
51	[22 & 29]	164.7c	47.1c	705.9c
22	[15 & 22], [22 & 29]	163.6c	54.5c	709.1c
59	[15 & 22]	162.7c	61c	711.9c
37	[15 & 22]	162.2c	64.9c	713.5c
15	[15 & 22]	160c	80c	720c
8	[15 & 22]	150c	150c	750c