Porcupine temperament modal harmony: Difference between revisions

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 "third", 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].

Porcupine remains the "best" such temperament even if the maximum error is relaxed somewhat, until eventually being superceded by [[mavila]] temperament, which is simpler in the 5-limit but much less accurate than porcupine temperament due to its extremely flat fifth of 670-680 cents. (using standard harmonic timbres).

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.

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.

The below mode names are given in [[Modal UDP Notation|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 chord|magical seventh]] tetrad) instead. These are sometimes just written bright/dark diminished and magical seventh modes when clear.

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.

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.  

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! William's Name

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

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in both 15edo and 22edo as G-A-B-C-D-Fv

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(melodic major)

8:9:10:11:12:14:15 on the tonic in 15edo as G-A-B-C-D-E^-F^

and suggests it as a candidate for "default major"

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<ul><li>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.</li></ul>

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(harmonic major)

Major with major V chord (most major-sounding major)

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Magical seventh

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(harmonic minor)

Minor with minor v chord (most minor-sounding minor)

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Major with major IV chord

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Symmetric diminished

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Minor with minor iv

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Magical seventh

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8:9:10:11:12:14 on the tonic in 15edo only

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(melodic minor)

Keenan and Igliashon both find this exceptionally beautiful

and suggest it as a candidate for default "minor"

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

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.

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Has both 4/3 and 3/2

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Has both 4/3 and 3/2

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

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