Porcupine: Difference between revisions
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'''Porcupine''' is a [[linear temperament]] in the [[porcupine family]] that tempers out [[250/243]], the porcupine [[comma]], and whose generator is somewhere around 160–165 cents. It can be thought of as a 5-[[Harmonic Limit|limit]], 7-limit, or 11-limit temperament, or a 2.3.5.11 [[subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy. | '''Porcupine''' is a [[linear temperament]] in the [[porcupine family]] that tempers out [[250/243]], the porcupine [[comma]], and whose generator is somewhere around 160–165 cents. It can be thought of as a 5-[[Harmonic Limit|limit]], 7-limit, or 11-limit temperament, or a 2.3.5.11 [[subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy. | ||
The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)<sup>2</sup> equivalent to (6/5)<sup>3</sup>. In perhaps more familiar musical terms, this means two "perfect fourths" equals three "minor thirds". As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. This is obviously in stark contrast to [[12edo | The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)<sup>2</sup> equivalent to (6/5)<sup>3</sup>. In perhaps more familiar musical terms, this means two "perfect fourths" equals three "minor thirds". As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. This is obviously in stark contrast to [[12edo]], and to meantone, in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales. | ||
[[File:porcupinesymmetricminor22edo.mp3]] | [[File:porcupinesymmetricminor22edo.mp3]] | ||
Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them | Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them ([[22edo]] tuning). | ||
[[File:porcupine.png]] | [[File:porcupine.png]] | ||
== Interval chain == | == Interval chain == | ||
{{Main| Porcupine intervals }} | |||
{{Main|Porcupine intervals}} | |||
{| class="wikitable center-all right-2 left-3 right-6 left-7" | {| class="wikitable center-all right-2 left-3 right-6 left-7" | ||
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| M3 | | M3 | ||
|} | |} | ||
The specific tuning shown is the full 11-limit [[POTE tuning]], but of course there is a range of acceptible porcupine tunings that includes generators as small as 160 cents ([[15edo | The specific tuning shown is the full 11-limit [[POTE tuning]], but of course there is a range of acceptible porcupine tunings that includes generators as small as 160 cents ([[15edo]]) and as large as 165.5 cents ([[29edo]]). (However, the 29edo patent val does not support 11-limit porcupine proper, since it does not temper out [[64/63]].) | ||
[[12/11]], [[11/10]], and [[10/9]] are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find. | [[12/11]], [[11/10]], and [[10/9]] are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find. | ||
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|- | |- | ||
| 8:9:10:11:12 chord, in just intonation. <br> All intervals are slightly different. | | 8:9:10:11:12 chord, in just intonation. <br> All intervals are slightly different. | ||
| Porcupine-tempered 8:9:10:11:12 chord, in [[22edo | | Porcupine-tempered 8:9:10:11:12 chord, in [[22edo]]. <br> Except the first, the intervals are the same. | ||
| Porcupine-tempered 8:9:10:11:12 chord, in [[29edo | | Porcupine-tempered 8:9:10:11:12 chord, in [[29edo]]. <br> Except the first, the intervals are the same. | ||
|} | |} | ||
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== Chords == | == Chords == | ||
{{ | {{Main| Chords of porcupine }} | ||
== Scales == | == Scales == | ||
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{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! | ! Edo<br>Generator | ||
! [[ | ! [[Eigenmonzo|Eigenmonzo<br>(Unchanged Interval]]) | ||
! | ! Generator (¢) | ||
! | ! Comments | ||
|- | |- | ||
| | | | ||
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{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! | ! Edo Generator | ||
! | ! Eigenmonzo<br>(Unchanged Interval) | ||
! | ! Generator (¢) | ||
! | ! Comments | ||
|- | |- | ||
| | | | ||
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== History == | == History == | ||
Porcupine temperament/scales were discovered by [[Dave Keenan]], but | Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his Mizarian Porcupine Overture in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[DE]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself. | ||
== See also == | == See also == | ||
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* [[Porcupine Album Project]] | * [[Porcupine Album Project]] | ||
== | == Music == | ||
* ''[https://sites.google.com/site/teamouse/home#TOC-Mizarian-music Mizarian Porcupine Overture]'', [[Herman Miller]], 1999. (15EDO, namesake of the temperament) | * ''[https://sites.google.com/site/teamouse/home#TOC-Mizarian-music Mizarian Porcupine Overture]'', [[Herman Miller]], 1999. (15EDO, namesake of the temperament) | ||
* ''[https://web.archive.org/web/20070928093239/http://66.98.148.43/~xenharmo/mp3/erlich/glassic.mp3 Glassic]'', [[Paul Erlich]], [[22edo|22EDO]] (at least the beginning part is in porcupine). | * ''[https://web.archive.org/web/20070928093239/http://66.98.148.43/~xenharmo/mp3/erlich/glassic.mp3 Glassic]'', [[Paul Erlich]], [[22edo|22EDO]] (at least the beginning part is in porcupine). | ||
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* [https://cityoftheasleep.bandcamp.com/track/second-breakfast-15edo Second Breakfast (15EDO)] by [[Igliashon Jones]] | * [https://cityoftheasleep.bandcamp.com/track/second-breakfast-15edo Second Breakfast (15EDO)] by [[Igliashon Jones]] | ||
== | == Diagrams == | ||
[[File:porcupine8.jpg]] | [[File:porcupine8.jpg]] | ||