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 usually around 160–165 [[cent]]s. It can be thought of as a [[5-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 usually around 160–165 [[cent]]s. It can be thought of as a [[5-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]], 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. | 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. | ||
== Interval chain == | == Interval chain == | ||
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! Cents | ! Cents | ||
! Ratios | ! Ratios | ||
! Ups and Downs <br> | ! Ups and Downs <br> Notation | ||
! # | ! # | ||
! 2/1 inverse | ! 2/1 inverse | ||
! Ratios | ! Ratios | ||
! Ups and Downs <br> | ! Ups and Downs <br> Notation | ||
|- | |- | ||
| 0 | | 0 | ||
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{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|+ Tuning spectrum of 13-limit porcupine | |+ Tuning spectrum of 13-limit porcupine | ||
! | ! Edo<br>generator | ||
! [[ | ! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-Interval)]] | ||
! | ! Generator (¢) | ||
! | ! Comments | ||
|- | |- | ||
| | | | ||
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{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|+ Tuning spectrum of porcupinefish | |+ Tuning spectrum of porcupinefish | ||
! | ! Edo<br>generator | ||
! | ! Eigenmonzo<br>(Unchanged-Interval) | ||
! | ! Generator (¢) | ||
! | ! Comments | ||
|- | |- | ||
| | | | ||
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== Music == | == Music == | ||
; [[Herman Miller]] | ; [[Herman Miller]] | ||
* | * [https://sites.google.com/site/teamouse/home#TOC-Mizarian-music ''Mizarian Porcupine Overture''] (1999) – in [[15edo]], namesake of the temperament | ||
; [[Paul Erlich]] | ; [[Paul Erlich]] | ||