User:Unque/Porcupine
Porcupine is the 15 & 22 temperament generated by an interval of approximately 160 - 165 cents known as a quill; a stack of two quills represents 6/5, and a stack of three represents 4/3, thus making the quill represent 10/9 in the 5-limit.
Terminology, notation, and intervals
The most important property of the porcupine temperament is splitting 4/3 into three equal parts, and similarly splitting the apotome into three equal parts. Each third of 4/3 is the quill, representing 10/9~11/10~12/11, and each third of the apotome is the Porcupine chroma, representing 25/24~81/80.
The easiest way to notate Porcupine using a staff is to include a set of accidentals representing the Porcupine chroma. One can repurpose ♯/♭ for these purposes, but this may get confusing as these symbols traditionally represent the apotome rather than one-third of it, so the most common option is to use ^/v for the Porcupine chroma. The quill is thus a diatonic vM2 (read as "downmajor second" or "submajor second"), and 6/5 is a diatonic ^m3 (read as "upminor third" or "supraminor third").
The other possible paradigm is to redefine the ordinals to represent the equable scale (see scales) rather than the 3-limit diatonic scale, thus requiring fewer accidentals to effectively notate Porcupine structures. This requires composers and performers to unlearn familiar interval arithmetic, but it is ultimately more conducive to working with Porcupine as a system long-term. On this page, the equable labels will be used, however the conversion between equable labels and diatonic labels will be given in the interval table below.
The interval chain is shown below (for information on "Archy" and "Aberschismic," see extensions). Octave complements for diatonic and equable labels can be found using standard arithmetic (2-7, 3-6, 4-5), where major corresponds with minor, augmented corresponds with diminished, and ^ corresponds with v. Octave-reduced harmonics and subharmonics are shown in bold.
| Quills up | 5-limit ratios | 11-limit ratios | Archy ratios | Aberschismic ratios | Diatonic Label | Equable Label |
|---|---|---|---|---|---|---|
| 1 | 10/9, 27/25 | 11/10, 12/11 | vM2 | P2 | ||
| 2 | 6/5 | 11/9 | 77/64 | ^m3 | m3 | |
| 3 | 4/3 | 21/16 | P4 | m4 | ||
| 4 | 36/25, 40/27 | 16/11, 22/15 | 35/24, 63/44 | vP5 | m5 | |
| 5 | 8/5 | 44/27 | 35/22 | ^M6 | m6 | |
| 6 | 16/9 | 7/4 | 105/64 | m7 | d7 | |
| 7 | 48/25, 160/81 | 64/33, 88/45 | 21/11, 35/18 | 63/32 | vP8 / vP1 | d8 |
| 8 | 16/15 | 132/125 | 21/20 | ^m2 | d2 | |
| 9 | 32/27 | 7/6 | 77/64 | m3 | d3 | |
| 10 | 32/25 | 128/99 | 14/11 | 21/16 | vP4 | d4 |
| 11 | 64/45 | 176/125 | 7/5 | 35/24, 63/44 | ^d5 | d5 |
| 12 | 128/81 | 14/9 | 35/22 | M6 | d6 | |
| 13 | 128/75 | 56/33, 77/45 | 7/4 | vm7 | dd7 | |
| 14 | 256/135 | 28/15 | 21/11, 36/35 | ^d8 / ^d1 | dd8 / dd1 | |
| 15 | 256/243, 156/125 | 28/27 | 21/20 | m2 | dd2 | |
| 16 | 104/75 | 7/6 | vm3 | dd3 |
Supporting systems and extensions
Porcupine in the 5-limit is supported by 7, 8, 15, 22, 29, and 37edo; it also has a reasonable tuning in 36edo if we use the flat tuning for prime 5 rather than the sharp tuning borrowed from 12edo. All of these tunings additionally find prime 11 by equating the quill with 11/10~12/11, and equating the minor third with 11/9, making this extension quite ubiquitous. This also equalizes the harmonic segment 9:10:11:12 as a stack of quills, a feature which is relevant to scale construction.
Prime 7
Prime 7 is slightly more contentious than 11 and 13 in this respect, as its mapping is not so ubiquitous across these supporting tunings.
Archy Porcupine (15 & 22)
The most commonly-accepted extension to add prime 7 is via Archy tuning, where 16/9 (found at six quills up) is equated with 7/4, a range which is reasonably supported by 15, 22, and 37edo. Note that all of these tunings have a quill which is on the narrower end of the range, around 160 cents; this provides a somewhat more accurate tuning of 6/5, but a very flat tuning of 4/3.
In this tuning, the Porcupine chroma is equated with 36/35, making the 7/6 - 6/5 - 5/4 - 9/7 segment equidistant.
Aberschismic Porcupine (29 & 36c)
Another reasonable extension to prime 7 is via Aberschismic tuning, where the Porcupine chroma is equated with 64/63, making the 7/6 - 32/27 - 6/5 segment equidistant; this range is supported primarily by 29edo and 36c. Both of these tunings have a quill around 165 cents, providing near-just 4/3 but a high-error 6/5; this property makes this extension act as something of a foil to the Archy extension, which has the opposite tendency.
Because 5/4 and 11/8 are both relatively high-error in this range, but their difference 11/10 is not, these tunings may be analyzed under the fractional subgroup 2.3.7.11/5 instead; however, this analysis has the detriment of providing no clear low-complexity interpretation for the thirds of the equable scale, which are 6/5~11/9 and 5/4~27/22 in the 2.3.5.7.11 subgroup.
Prime 13
Because perfect fifth and the 5-limit thirds of porcupine tend to be tuned quite sharp, 16/13 tends to fall near the middle of 5/4 and 6/5 rather than leaning towards the sharp end; this is especially prevalent in 15edo, where 13/8 has almost precisely 50% error. This makes it rather difficult to convincingly map prime 13 to the porcupine generator chain.
Wilsormatic Porcupine (22 & 29)
Because 6/5 is equated with 11/9, making a third which is rather sharp of the former ratio and flat of the latter, we may map prime 13 by equating 16/13 with 5/4. Despite its higher error, this mapping for prime 13 is supported by 22, 29, and 36edo, but notably not 37edo.
This range has the property of making 13/8 and 5/4 into octave complements, equating the otonal chord 8:10:13:16 with its own utonal retroversion and making it symmetrical across the octave.
Harmoneutral Porcupine (30 & 37)
37edo does not support the above mapping, instead splitting the Porcupine chroma into two 65/64 intervals, which we can call a hemichroma; this makes the sequence 6/5 - 16/13 - 5/4 equidistant. This is considered a weak extension, because the 13-limit intervals cannot be reached on the original chain of quills; specifically, the generator for this temperament is the rather complex 160/117, which splits the Porcupine perfect seventh (the octave complement of the quill) in half.
15 and 22edo also come to support this mapping when doubled to 30 and 44edo, adding 16/13 in the newly-acquired gap between the 5-limit thirds. Because of this, we may consider this mapping to define a canonical "hemi-Porcupine."
Scales
MOS Scales
Equable Scale
Because the quill simultaneously represents each constituent element of 9:10:11:12, the equable tetrachord is equalized in Porcupine tunings. Using this tetrachord to form a full scale gives us a 7-note MOS with the signature sssLsss.
The mode names for this scale are given by William Lynch.
| Brightness | Name | Pattern | Interval Qualities |
|---|---|---|---|
| 3 | Chinchillian | Lssssss | A2 - M3 - M4 - M5 - M6 - P7 |
| 2 | Badgerian | sLsssss | P2 - M3 - M4 - M5 - M6 - P7 |
| 1 | Zebrian | ssLssss | P2 - m3 - M4 - M5 - M6 - P7 |
| 0 | Dingoian | sssLsss | P2 - m3 - m4 - M5 - M6 - P7 |
| -1 | Gazellian | ssssLss | P2 - m3 - m4 - m5 - M6 - P7 |
| -2 | Lemurian | sssssLs | P2 - m3 - m4 - m5 - m6 - P7 |
| -3 | Pandian | ssssssL | P2 - m3 - m4 - m5 - m6 - d7 |
Octatonic Scale
The Porcupine Octatonic scale can be formed most simply by splitting the augmented quill step of the equable scale into a quill plus a Porcupine chroma, ensuring that the scale remains MV2. This one additional note actually creates a rather expressive addition to the harmony, creating double thirds over one mode and double fifths over another.
The modes of this scale are also given by William Lynch.
| Brightness | Name | Pattern | Interval Qualities |
|---|---|---|---|
| 4 | Octopus | LLLLLLLs | P2 - m3 - m4 - m5 - m6 - d7 - d8 |
| 3 | Mantis | LLLLLLsL | P2 - m3 - m4 - m5 - m6 - d7 - P7 |
| 2 | Dolphin | LLLLLsLL | P2 - m3 - m4 - m5 - m6 - M6 - P7 |
| 1 | Crab | LLLLsLLL | P2 - m3 - m4 - m5 - M5 - M6 - P7 |
| -1 | Tuna | LLLsLLLL | P2 - m3 - m4 - M4 - M5 - M6 - P7 |
| -2 | Salmon | LLsLLLLL | P2 - m3 - M3 - M4 - M5 - M6 - P7 |
| -3 | Starfish | LsLLLLLL | P2 - A2 - M3 - M4 - M5 - M6 - P7 |
| -4 | Whale | sLLLLLLL | A1 - A2 - M3 - M4 - M5 - M6 - P7 |
Non-MOS Scales
Zarlino
The Zarlino scale can be defined by an alternating stack of 5-limit thirds, 5/4 and 6/5. Porcupine temperament has several features which make it a notable system to support this scale: firstly, the intervals 81/80 and 25/24, which are the differences between steps of the Zarlino scale, are both represented by the Porcupine chroma; this means that there exists only one type of alteration which occurs on all intervals, rather than two types as in 5-limit JI. The steps are thus all consistently qualities of the quill, namely diminished, perfect, and augmented.
Secondly, the "wolf" fourth 27/20 is equated with 11/8 in Porcupine tuning, whose concordance makes it act less like a wolf interval and more like simply a different quality of fourth.
Because the Zarlino scale is chiral -- that is, its retroversion is not a rotation of the original -- we must distinguish left-hand and right-hand modes; in the generator sequence, left-hand modes have a chain which begins with 5/4 and ends with 6/5, whereas right-hand modes have the opposite.
| Brightness | Name | Pattern | Interval Qualities |
|---|---|---|---|
| 3 | Lydian | LMLsLMs | A2 - M3 - A4 - M5 - A6 - A7 |
| 2 | Aeolian | LsLMsLM | A2 - m3 - M4 - M5 - m6 - P7 |
| 1 | Ionian | LMsLMLs | A2 - M3 - m4 - M5 - M6 - A7 |
| 0 | Phrygian | sLMLsLM | d2 - m3 - m4 - M5 - m6 - P7 |
| -1 | Mixolydian | MLsLMsL | P2 - M3 - m4 - M5 - M6 - d7 |
| -2 | Locrian | sLMsLML | d2 - m3 - m4 - d5 - m6 - d7 |
| -3 | Dorian | MsLMLsL | P2 - d3 - m4 - m5 - M6 - d7 |
| Brightness | Name | Pattern | Interval Qualities |
|---|---|---|---|
| 3 | Dorian | LsLMLsM | A2 - m3 - M4 - M5 - A6 - P7 |
| 2 | Lydian | LMLsMLs | A2 - M3 - A4 - M5 - M6 - A7 |
| 1 | Aeolian | LsMLsLM | A2 - m3 - m4 - M5 - m6 - P7 |
| 0 | Ionian | MLsLMLs | P2 - M3 - m4 - M5 - M6 - A7 |
| -1 | Phrygian | sLMLsML | d2 - m3 - m4 - M5 - m6 - d7 |
| -2 | Mixolydian | MLsMLsL | P2 - M3 - m4 - m5 - M6 - d7 |
| -3 | Locrian | sMLsLML | d2 - d3 - m4 - d5 - m6 - d7 |
Blackdye
Just as we extended the equable scale into an octatonic form by turning the augmented quill into a perfect quill plus a Porcupine chroma, we can do the same to the augmented quills in the Zarlino scale to yield a 10-form called Blackdye. The scale remains MV3; the largest step represents a perfect quill, the middle step a diminished quill, and the smallest step a Porcupine chroma.
The Blackdye scale has several benefits over Zarlino, most notably uniting the two chiralities and creating longer unbroken fifth chains. More specifically, Blackdye is made up of two separate fifth chains, with their roots offset from one another by a quill; we can disambiguate similar modes based on which of the two chains contains the root. This gives us five "acute" modes and five "grave" modes, which can be organized further based on the brightness of their pentatonic chains.
| Brightness | Name | Pattern | Interval Qualities |
|---|---|---|---|
| 2 | Aeolian | LmLsLmLsLs | P2 - d3 - d4 - m4 - m5 - d6 - dd7 - d7 - d8 |
| 1 | Dorian | LmLsLsLmLs | P2 - d3 - d4 - m4 - m5 - M5 - M6 - d7 - d8 |
| 0 | Mixolydian | LsLmLsLmLs | P2 - A2 - M3 - m4 - m5 - M5 - m6 - d7 - d8 |
| -1 | Ionian | LsLmLsLsLm | P2 - A2 - M3 - m4 - m5 - M5 - M6 - A6 - A7 |
| -2 | Lydian | LsLsLmLsLm | P2 - A2 - M3 - A3 - A4 - M5 - M6 - A6 - A7 |
| Brightness | Name | Pattern | Interval Qualities |
|---|---|---|---|
| 2 | Locrian | mLsLmLsLsL | d2 - d3 - m3 - m4 - d5 - d6 - m6 - d7 - P7 |
| 1 | Phrygian | mLsLsLmLsL | d2 - d3 - m3 - m4 - M4 - M5 - m6 - d7 - P7 |
| 0 | Aeolian | sLmLsLmLsL | A1 - A2 - m3 - m4 - M4 - M5 - m6 - d7 - P7 |
| -1 | Dorian | sLmLsLsLmL | A1 - A2 - m3 - m4 - M4 - M5 - A5 - A6 - P7 |
| -2 | Mixolydian | sLsLmLsLmL | A1 - A2 - AA2 - A3 - M4 - M5 - A5 - A6 - P7 |
15edo is notable for collapsing the Blackdye scale into a MV2 by merging the diminished quill with the Porcupine chroma, equivalent to tempering out (16/15) / (25/24) = 128/125; this means that its tuning of the scale only has two modes, a unified Acute Mode and a unified Grave Mode.
Chords and harmony
Triad Qualities
Because equable and Zarlino, the basal scales of Porcupine, are both 7-form and are easy to take as alterations of one another, they can be used to define a system of triadic harmony not dissimilar to the kind seen in Western music.
Tertiary Triads
| Quality | Intervals | Formula |
|---|---|---|
| Super Augmented | A3, A5 | A3 + M3 |
| Augmented | M3, A5 | M3 + M3 |
| Major | M3, M5 | M3 + m3 |
| Minor | m3, M5 | m3 + M3 |
| Diminished | m3, m5 | m3 + m3 |
| Sub Diminished | d3, d5 | d3 + m3 |
Further qualities as appear in Zarlino will not be enumerated, and can simply be described as alterations of these triads.
Quartal Triads
In the octatonic scale, the two qualities of thirds do not occur on the same degree; however, the major third and the minor fourth do, which creates an inversion of the major or minor triad in the form of a quartal triad.
| Quality | Intervals | Formula |
|---|---|---|
| Diminished Quartal | m4, d7 | m4 + m4 |
| Minor 6/4 | m4, M6 | m4 + M3 |
| Major 6/3 | M3, M6 | M3 + m4 |
| Augmented | M3, A5 | M3 + M3 |
Blackwood Triads
The Blackwood scale offers a different generalization of the triads, as the major and minor thirds occur alongside diminished fourths and double-augmented quills.
| Quality | Intervals | Formula |
|---|---|---|
| SusAA2 | AA2, M5 | AA2 + d4 |
| Diminished | m3, d6 | m3 + AA2 |
| Minor | m3, M5 | m3 + M3 |
| Major | M3, M5 | M3 + m3 |
| Augmented | M3, d6 | M3 + d4 |
| Susd4 | d4, M5 | d4 + AA2 |
Functional Harmony
Due to the structure of its scales, chord functions in Porcupine temperament are roughly midway between those of Meantone and Neutral temperaments. Because the equable and Zarlino scales both prioritize 5-limit major and minor triads, one can get a lot of functional leeway from simply translating familiar diatonic chord progressions into the temperament.
The quill is a useful leading interval, but its repeated occurrence throughout the equable and octatonic scales make its usage somewhat ambiguous and destabilize it as a directed step; comparatively, the diminished quill seen in the Zarlino scale is much narrower, and occurs much less frequently, allowing it to be used as a directed tension. Because of this property, major and minor triads on degree 5 both act equally as dominant functions, with the minor triad creating a soft and sweet pull while the major triad creates a powerful and confident pull back to the tonic triad.
