2L 8s: Difference between revisions
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'''2L 8s''' or '''pajaroid''' is the MOS pattern of the [[decatonic|decatonic]] scale of [[Paul_Erlich|Paul Erlich]] and others.<!-- | '''2L 8s''' or '''pajaroid''' (named after the abstract temperament [[pajara]]) is the MOS pattern of the [[decatonic|decatonic]] scale of [[Paul_Erlich|Paul Erlich]] and others.<!-- | ||
The only significant harmonic entropy minimum that is [[Rothenberg_propriety|proper]] is the decatonic scale itself ([[Diaschismic_family|pajara]][10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes [[4/3|4/3]], and another generator down makes [[5/4|5/4]]. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links: | The only significant harmonic entropy minimum that is [[Rothenberg_propriety|proper]] is the decatonic scale itself ([[Diaschismic_family|pajara]][10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes [[4/3|4/3]], and another generator down makes [[5/4|5/4]]. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links: | ||
Revision as of 07:56, 28 March 2021
User:IlL/Template:RTT restriction
| ↖ 1L 7s | ↑ 2L 7s | 3L 7s ↗ |
| ← 1L 8s | 2L 8s | 3L 8s → |
| ↙ 1L 9s | ↓ 2L 9s | 3L 9s ↘ |
ssssLssssL
2L 8s or pajaroid (named after the abstract temperament pajara) is the MOS pattern of the decatonic scale of Paul Erlich and others. In addition to the true MOS form, LssssLssss, these scales also exist in a near-MOS form, LsssssLsss, in which the period is the only interval class with more than two flavors. In the case of the decatonic scale, LssssLssss is called the "symmetric" scale and LsssssLsss is called the "pentachordal" scale (because it has two identical "pentachords" in the same way that the diatonic scale has two identical tetrachords).
Notation
The notation used in this article is ssLssssLss = JKLMNOPQRSJ unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)
Thus the 12edo gamut is as follows:
J K L L&/M@ M N O P Q Q&/R@ R S J
Scale tree
| Generator | Cents | Comments | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 0\2 | 0 | |||||||||
| 1\26 | 46.15 | |||||||||
| 1\24 | 50 | |||||||||
| 2\46 | 52.17 | |||||||||
| 1\22 | 54.55 | |||||||||
| 1\20 | 60 | |||||||||
| 1\18 | 66.67 | |||||||||
| 1\16 | 75 | L/s = 4 | ||||||||
| 600/(4+pi) | ||||||||||
| 1\14 | 85.71 | L/s = 3 | ||||||||
| 600/(4+e) | ||||||||||
| 2\26 | 92.31 | |||||||||
| 5\64 | 93.75 | |||||||||
| 13\166 | 93.98 | |||||||||
| 21\268 | 94.03 | Golden pajaroid | ||||||||
| 8\102 | 94.12 | |||||||||
| 3\38 | 94.74 | |||||||||
| 4\50 | 96 | |||||||||
| 5\62 | 96.77 | |||||||||
| 1\12 | 100 | Boundary of propriety (generators
larger than this are proper) | ||||||||
| 4\46 | 104.35 | |||||||||
| 600/(4+sqrt(3)) | ||||||||||
| 3\34 | 105.88 | |||||||||
| 8\90 | 106.67 | around here 8g=18/11 | ||||||||
| 21\236 | 106.78 | |||||||||
| 34\382 | 106.81 | Golden pajaroid | ||||||||
| 13\146 | 106.85 | |||||||||
| 5\56 | 107.14 | |||||||||
| 600/(4+pi/2) | ||||||||||
| 2\22 | 109.09 | Optimum rank range (L/s=3/2) pajaroid | ||||||||
| 3\32 | 112.5 | |||||||||
| 4\42 | 114.29 | |||||||||
| 5\52 | 115.385 | |||||||||
| 6\62 | 116.13 | |||||||||
| 7\72 | 116.67 | |||||||||
| 8\82 | 117.07 | |||||||||
| 1\10 | 120 | |||||||||