Porcupine: Difference between revisions
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| Mapping = 1; -3 -5 6 -4 | | Mapping = 1; -3 -5 6 -4 | ||
| Edo join 1 = 15 | Edo join 2 = 22 | | Edo join 1 = 15 | Edo join 2 = 22 | ||
| Generators = | | Generators = 10/9 | ||
| Generators tuning = 163 | | Generators tuning = 163 | ||
| Optimization method = CWE | | Optimization method = CWE | ||
| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]] | | MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]] | ||
| Pergen = (P8, P4/3) | | Pergen = (P8, P4/3) | ||
| Color name = Triyoti | | Color name = Triyoti | ||
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[[File:porcupinesymmetricminor22edo.mp3|thumb|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]] | [[File:porcupinesymmetricminor22edo.mp3|thumb|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]] | ||
'''Porcupine''' is a [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone which is tuned flat to around 160–165 [[cent]]s | '''Porcupine''' is a [[regular temperament|temperament]] that is [[generator|generated]] by a [[10/9|minor whole tone]] which is tuned flat to around 160–165 [[cent]]s. Two generators (stacked) represent [[6/5]], and three represent [[4/3]], so that the [[250/243|porcupine comma (250/243)]] is [[tempering out|tempered out]]; from this, the generator itself represents a very flat 10/9. This is in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales. | ||
One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]). | One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]). | ||
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Porcupine can be thought of as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the 6/5 minor third, due to the extreme flatness of 10/9. This also means that [[27/20]], the 5-limit "acute fourth", is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at −4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy. | Porcupine can be thought of as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the 6/5 minor third, due to the extreme flatness of 10/9. This also means that [[27/20]], the 5-limit "acute fourth", is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at −4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy. | ||
It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990 | It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990{{c}}), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720{{c}}, with the best tunings around 711–712{{c}}, which roughly splits the damage on 7/4 and 9/7. This extension sets [[7/6]], 6/5, 5/4, and 9/7 equidistant, thus tempering out [[875/864]], making porcupine a [[keemic temperaments|keemic temperament]]. | ||
See [[Porcupine family #Porcupine]] for technical data and alternative 7-limit extensions. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s. | See [[Porcupine family #Porcupine]] for technical data and alternative 7-limit extensions. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s. | ||
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| M3 | | M3 | ||
|} | |} | ||
<nowiki>* | <nowiki/>* In 11-limit [[CWE tuning]], octave reduced | ||
In the ups and downs notation, the [[enharmonic unison]] is the trudsharp, the triple-down augmented unison. The porcupine notation does not have an enharmonic unison. | In the ups and downs notation, the [[enharmonic unison]] is the trudsharp, the triple-down augmented unison. The porcupine notation does not have an enharmonic unison. | ||
Besides the specific tuning shown here, there is a range of acceptable porcupine tunings that includes generators as small as 160 | Besides the specific tuning shown here, there is a range of acceptable porcupine tunings that includes generators as small as 160{{c}} ([[15edo]]) and as large as 165.5{{c}} ([[29edo]]). However, the 29edo patent val does not support full 11-limit porcupine proper, since it does not temper out [[64/63]]. | ||
== Chords and harmony == | == Chords and harmony == | ||
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| [[12/11]] | | [[12/11]] | ||
| 150.637 | | 150.637 | ||
| Lower bound of 11- and 15-odd-limit diamond tradeoff | | Lower bound of 11-odd-limit and 11-limit 15-odd-limit diamond tradeoff | ||
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| '''160.000''' | | '''160.000''' | ||
| '''Lower bound of 7- to | | '''Lower bound of 7-odd-limit to 11-limit 15-odd-limit diamond monotone''' | ||
|- | |- | ||
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| [[8/5]] | | [[8/5]] | ||
| 162.737 | | 162.737 | ||
| 2/5-comma, 5- | | 2/5-comma, 5- and 7-odd-limit minimax | ||
|- | |- | ||
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| '''163.636''' | | '''163.636''' | ||
| '''Upper bound of 7- to | | '''Upper bound of 7-odd-limit to 11-limit 15-odd-limit diamond monotone''' | ||
|- | |- | ||
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| [[14/9]] | | [[14/9]] | ||
| 163.743 | | 163.743 | ||
| 9-, 11-, and | | 9-, 11-, and 11-limit 15-odd-limit minimax | ||
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| Untempered generator; upper bound of 9- to 15-odd-limit diamond tradeoff | | Untempered generator; upper bound of 9- to 15-odd-limit diamond tradeoff | ||
|} | |} | ||
<nowiki>* | <nowiki/>* Besides the octave | ||
== History == | == History == | ||
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; [[Jake Freivald]] | ; [[Jake Freivald]] | ||
* [ | * ''[https://soundcloud.com/jdfreivald/porcupine-comma-pump Porcupine Comma Pump]'' | ||
; [[Cody Hallenbeck]] | ; [[Cody Hallenbeck]] | ||