2L 8s: Difference between revisions
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'''2L 8s''' | '''2L 8s''' is the MOS pattern of the 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: | ||
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Improper harmonic entropy minima include [[Meantone_family#Injera|injera]] (which is similar to pajara except that 5/4 is now four generators '''up''' and no periods) and [[Diaschismic_family#Shrutar|shrutar]] (which is basically pajara with the generator divided in two). | Improper harmonic entropy minima include [[Meantone_family#Injera|injera]] (which is similar to pajara except that 5/4 is now four generators '''up''' and no periods) and [[Diaschismic_family#Shrutar|shrutar]] (which is basically pajara with the generator divided in two). | ||
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). | 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). | ||
== Names == | |||
The [[TAMNAMS]] system calls this pattern '''pajaroid''' (named after the abstract temperament [[pajara]]). | |||
== Notation == | == Notation == | ||
Revision as of 21:11, 14 April 2021
| ↖ 1L 7s | ↑ 2L 7s | 3L 7s ↗ |
| ← 1L 8s | 2L 8s | 3L 8s → |
| ↙ 1L 9s | ↓ 2L 9s | 3L 9s ↘ |
ssssLssssL
2L 8s is the MOS pattern of the decatonic scale of Paul Erlich and others.
The only significant harmonic entropy minimum that is proper is the decatonic scale itself (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, and another generator down makes 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:
http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf
http://www.lumma.org/tuning/erlich/
Improper harmonic entropy minima include injera (which is similar to pajara except that 5/4 is now four generators up and no periods) and shrutar (which is basically pajara with the generator divided in two).
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).
Names
The TAMNAMS system calls this pattern pajaroid (named after the abstract temperament pajara).
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 | ||||||||
| 1\14 | 85.71 | L/s = 3 | ||||||||
| 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 | |||||||||
| 7\80 | 105.00 | |||||||||
| 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 | |||||||||
| 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 | |||||||||