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 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[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
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, 48/25
8	102.5	16/15
9	265.3	(7/6)
10	428.2	(14/11), 32/25
11	591.0	(7/5)
12	753.8	(14/9)

* In 2.3.5.7.11 CWE tuning

Modal harmony

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.

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. The standard qualities of major and minor retain their "bright" and "dark" feels respectively, which can be broken down into a combination of their complexity in triads and the actual width of the intervals. This suggests that the four qualities in 22edo should have more granularity in their feels, and can be broken down into stable/unstable and bright/dark.

	Stable	Unstable
Bright	Nearmajor (warm, pleasant, comforting)	Supermajor (excited, animated, active)
Dark	Subminor (depressive, sad, bluesy)	Nearminor (angry, tense, stressful)

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.)

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

Leading tones

When considering Secor's supposed optimal leading tone at 70 cents, one may notice that porcupine in most reasonable tunings skips this category entirely. However, it instead matches with Aura's theory of functional harmony, which places the 70-cent leading tone at the intersection of two other functional categories at around 110 cents and 50 cents respectively - the collocant and gradient functions. The collocant functions as a conventional leading tone, whereas the gradient functions as a passing tone to either jump past the tonic or resolve to the collocant. Porcupine represents both of these separately (as the nearminor and subminor seconds), and in doing do presents a distinct approach to leading tones from systems like 17edo and 31edo that have Secor's leading tone instead.

(In 15edo, this becomes even more prominent, as the subminor second reduces to the unison.)

Further functional harmony

There are four distinct "keys" in porcupine (nearmajor, supermajor, nearminor, subminor), as compared to two in monocot, where a key is defined as a system of tonal hierarchy based around a certain interval quality or tonic chord (independent of absolute pitch), which will be elaborated on below. Note that relative major or minor depends on whether the key is near- or super/sub, and that, for instance, nearmajor and supermajor use different scales that are not rotations of one another. In specific, using ups and downs notation, C Nearmajor corresponds to vA Nearminor, meanwhile C Supermajor corresponds to A Subminor, and in general nearmajor-nearminor relative correspondences acquire an additional down accidental compared to standard MOSdiatonic correspondences.

The chirality of the nearmajor or nearminor scale in question is ultimately of little relevance (see blackdye; in short, the major second in nearmajor (and the fourth in nearminor) may be either note depending on context), but in general the right-handed version of nearmajor is assumed due to having a non-wolf V chord, and the left-handed version of nearminor is assumed due to having a non-wolf fourth over the tonic.

The heptatonic interval functions remain as they are in 12edo, although with the caveat that the ideal leading tone ends up at the nearminor second rather than the semitone found in MOSdiatonic, which has implications for the subminor and supermajor keys and turns the use of the diatonic scale into a balancing act between the functional utility of MOSdiatonic and the tension of the leading tones in zarlino diatonic. (In particular, it suggests the use of a "harmonic supermajor" by flattening the seventh of supermajor by an edostep.)

Nearmajor key

Natural nearmajor scale and tonic chord

In nearmajor (the key with the nearmajor tonic chord), the fourth acts as it usually does in MOS major, serving as a tendency tone towards the third. The basic tonal identity for nearmajor is 4:5:6, which extends generally to a nearmajor seventh chord, although a dominant (harmonic) seventh is also possible, and more justified in porcupine due to naturally extending the harmonic series segment corresponding to 4:5:6.

Nearminor key

Natural nearminor scale and tonic chord

Nearminor harmony functions somewhat similarly to how you expect, with the nearminor sixth functioning as a leading tone down to the fifth and the seventh being able to be raised to a nearmajor seventh in order to give a more directed dominant resolution. The whole tone also provides a lead up to the minor third, like in standard diatonic.

Melodic minor scales are somewhat interesting here as well, as there are a couple different reasonable ways to construct them, which would likely depend on the chords being used and the desired melodic contour.

Supermajor key

Natural supermajor scale and tonic chord

In supermajor, a lead to the third would be a wolf fourth (11/8), perhaps justifying its inclusion in the scale over the fourth proper, or the functional alternation between the two in different contexts.

The functionality of the seventh grows increasingly complicated in supermajor - while in 12edo, one may only see, for instance, a dominant chord replacing the I chord, in porcupine there are four different potential types of seventh, all with justifications. A fifth over the third would be a supermajor seventh (notably serving as the MOSdiatonic maj7, and distinguishing itself from the 12edo maj7 by not leading up to its own root), a tritone (neardim 5; 7/5) over the third would be a nearminor seventh, a lead up to the tonic would be a nearmajor seventh, and finally the MOS diatonic dominant chord utilizes a subminor seventh. Therefore, an alternate version of the supermajor scale usable in certain contexts makes the fourth wolf and the seventh nearmajor.

This also means that the regular perfect fourth isn't as unstable an interval or as functionally dissonant in supermajor - in fact, the third is actually somewhat of a tension compared to it (though the step between them is smaller than the size of a conventional leading tone).

Subminor key

The same kind of justification emerges for harmonic subminor, except that there is little reason to alter the seventh all the way up to a supermajor seventh if the objective is for it to function as a leading tone. In fact, the same logic can be used against a dominant chord with a nearminor seventh in nearmajor - leading inwards to a nearmajor third by equal semitones on either side requires that the initial interval be a neardiminished fifth, and that the chord to be used as a dominant is actually a harmonic 4:5:6:7 on the fifth. (Resolving to a supermajor chord actually wants a dom7 with a nearmajor third and nearminor seventh, if quartertones are not to be used).

Natural subminor scale and tonic chord

In general, porcupine's functional harmony ends up a lot more context-bound and much less scale-bound than 12edo's, due to the multiple different qualities of intervals and notes doing different things, and the ideal leading tone not matching the standard diatonic structure.

Alternative leading tones

An alternative approach to simplify things is instead to discard Aura's theory of leading in favor of treating the quartertone as the optimal leading tone (as it is the diatonic major seventh), an entirely different paradigm emerges. Supermajor and subminor become definitive, stable diatonic tonality systems, with no awkwardness around leading tones, behaving identically to any MOSdiatonic temperament (albeit with the different, somewhat inverted "moods" presented by the supermajor and subminor intervals). Meanwhile, the nearmajor and nearminor scales, which actually utilize porcupine intervals, acquire new "harmonic" variations, with the final note raised up to a quartertone below the tonic. In effect, supermajor/subminor and nearmajor/nearminor "switch" in regards to some functions. Instead of raising the fourth in supermajor, it is in this system viable to lower it in nearmajor. The best dom7 to resolve to a nearmajor triad on the tonic features a seventh lowered to one step below the subminor seventh, alongside the supermajor third, and can consequently be reanalyzed as a subminor seventh chord on the 9/7 over the tonic. The MOSdiatonic dominant seventh serves to resolve to a MOSdiatonic major triad, as in 12edo.

Scales

Onyx, pine

Onyx (1L6s) is the moment-of-symmetry scale generated by six porcupine generators. It is tuned rather soft in porcupine, resulting in a 7-form structure. Pine adds an additional "blue note" to onyx.

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

maybe move this to dedicated page for onyx

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.

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