2L 8s: Difference between revisions

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2L 8s is the ~~MOS~~ pattern of the decatonic scale of [[~~Paul_Erlich|~~Paul Erlich]] and others.

2L 8s is the mos pattern of the decatonic scale of [[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]], 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~~:

The only significant harmonic entropy minimum that is [[Rothenberg propriety|Rothenberg 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 is the next step up from the [[5L 2s|diatonic]] scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found here: [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf ''Tuning, Tonality, and Twenty-Two-Tone Temperament''] ([http://www.lumma.org/tuning/erlich/ alt link]).

~~http:~~/~~/sethares.engr.wisc.edu/paperspdf/Erlich-22~~.~~pdf~~

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).

~~http://www~~.~~lumma~~.~~org/tuning/erlich/~~

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 [[pentachord]]s in the same way that the diatonic scale has two identical [[tetrachord]]s).

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).

== Scale properties ==

~~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).

=== Intervals ===

== ~~Notation~~ ==

=== Generator chain ===

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:~~

=== Modes ===

~~'''J''' '''K''' '''L''' L& '''M''' '''N''' '''O''' '''P''' '''Q''' Q& '''R''' '''S''' '''J'''~~

== Modes ==

== Scale tree ==

{{~~Scale tree~~|~~Comments=~~6/5: [[Semimiracle]] ↑;

{{MOS tuning spectrum

3/2: [[Pajara]];

| 6/5 = [[Semimiracle]] ↑

8/5: [[Keen]];

| 3/2 = [[Pajara]]

12/7: [[Srutal]];

| 8/5 = [[Keen]]

9/5: [[Diaschismic]];

| 12/7 = [[Srutal]]

9/4: [[Bimeantone]];

| 9/5 = [[Diaschismic]]

5/2: [[Injera]];

| 9/4 = [[Bimeantone]]

9/2: [[Vishnu]] (incomplete);

| 5/2 = [[Injera]]

6/1: [[Shrutar]], [[teff]]/[[pombe]] (incomplete)↓}}

| 9/2 = [[Vishnu]] (incomplete)

| 6/1 = [[Shrutar]], [[teff]]/[[pombe]] (incomplete) ↓

}}

[[Category:jaric]]

[[Category:10-tone scales]]

@@ Line 10: / Line 10: @@
 {{MOS intro}}
-L 8s is the MOS pattern of the decatonic scale of [[Paul_Erlich|Paul Erlich]] and others.
+L 8s is the mos pattern of the decatonic scale of [[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]], 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:
+The only significant harmonic entropy minimum that is [[Rothenberg propriety|Rothenberg 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 is the next step up from the [[5L 2s|diatonic]] scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found here: [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf ''Tuning, Tonality, and Twenty-Two-Tone Temperament''] ([http://www.lumma.org/tuning/erlich/ alt link]).
-http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf
+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).
-http://www.lumma.org/tuning/erlich/
+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 [[pentachord]]s in the same way that the diatonic scale has two identical [[tetrachord]]s).
-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).
+== Scale properties ==
+{{TAMNAMS use}}
-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).
+=== Intervals ===
+{{MOS intervals}}
-== Notation ==
+=== Generator chain ===
-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".)
+{{MOS genchain}}
-Thus the 12edo gamut is as follows:
+=== Modes ===
-'''J''' '''K''' '''L''' L& '''M''' '''N''' '''O''' '''P''' '''Q''' Q& '''R''' '''S''' '''J'''
-== Modes ==
 {{MOS mode degrees}}
 == Scale tree ==
-{{Scale tree|Comments=6/5: [[Semimiracle]] ↑;
+{{MOS tuning spectrum
-/2: [[Pajara]];
+| 6/5 = [[Semimiracle]]&nbsp;↑
-/5: [[Keen]];
+| 3/2 = [[Pajara]]
-/7: [[Srutal]];
+| 8/5 = [[Keen]]
-/5: [[Diaschismic]];
+| 12/7 = [[Srutal]]
-/4: [[Bimeantone]];
+| 9/5 = [[Diaschismic]]
-/2: [[Injera]];
+| 9/4 = [[Bimeantone]]
-/2: [[Vishnu]] (incomplete);
+| 5/2 = [[Injera]]
-/1: [[Shrutar]], [[teff]]/[[pombe]] (incomplete)↓}}
+| 9/2 = [[Vishnu]] (incomplete)
+| 6/1 = [[Shrutar]], [[teff]]/[[pombe]] (incomplete)&nbsp;↓
+}}
 [[Category:jaric]]
 [[Category:10-tone scales]]