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.

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]], 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

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

== Modes ==

* 8|0(2) LssssLssss, J-K&-L&-M-N-O-P&-Q&-R-S-J

* 6|2(2) sLssssLsss, J-K-L&-M-N-O-P-Q&-R-S-J

* 4|4(2) ssLssssLss, J-K-L-M-N-O-P-Q-R-S-J

* 2|6(2) sssLssssLs, J-K-L-M@-N-O-P-Q-R@-S-J

* 0|8(2) ssssLssssL, J-K-L-M@-N@-O-P-Q-R@-S@-J

== Scale tree ==

Generator ranges:

* Chroma-positive generator: 480 cents (4\10) to 600 cents (1\2)

* Chroma-negative generator: 0 cents (0\2) to 120 cents (1\10)

{| class="wikitable center-all"

! colspan="6" rowspan="2" | Generator

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| 4\10 || || || || || || 480.000 || 120.000 || 1 || 1 || 1.000 ||

|-

| || || || || || 21\52 || 484.615 || 115.385 || 6 || 5 || 1.200 || ~~Semimiracle↑~~

| || || || || || 21\52 || 484.615 || 115.385 || 6 || 5 || 1.200 || [[Semimiracle]]↑

|-

| || || || || 17\42 || || 485.714 || 114.286 || 5 || 4 || 1.250 ||

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| || || || || 22\54 || || 488.889 || 111.111 || 7 || 5 || 1.400 ||

|-

| || || || || || 31\76 || 489.474 || 110.526 || 10 || 7 || 1.~~428~~ ||

| || || || || || 31\76 || 489.474 || 110.526 || 10 || 7 || 1.429 ||

|-

| || || 9\22 || || || || 490.909 || 107.692 || 3 || 2 || 1.500 || ~~L/s = 3/2, pajara~~

| || || 9\22 || || || || 490.909 || 107.692 || 3 || 2 || 1.500 || [[Pajara]]

|-

| || || || || || 32\78 || 492.308 || 131.250 || 11 || 7 || 1.571 ||

|-

| || || || || 23\56 || || 492.857 || 107.143 || 8 || 5 || 1.600 || Keen

| || || || || 23\56 || || 492.857 || 107.143 || 8 || 5 || 1.600 || [[Keen]]

|-

| || || || || || 37\90 || 493.333 || 106.667 || 13 || 8 || 1.625 || Golden pajaroid

| || || || || || 37\90 || 493.333 || 106.667 || 13 || 8 || 1.625 || Golden pajaroid (493.2011¢)

|-

| || || || 14\34 || || || 494.118 || 105.882 || 5 || 3 || 1.667 ||

|-

| || || || || || 33\80 || 495.000 || 105.000 || 12 || 7 || 1.714 || Srutal

| || || || || || 33\80 || 495.000 || 105.000 || 12 || 7 || 1.714 || [[Srutal]]

|-

| || || || || 19\46 || || 495.652 || 104.348 || 7 || 4 || 1.750 ||

|-

| || || || || || 24\58 || 496.552 || 103.448 || 9 || 5 || 1.800 || Diaschismic

| || || || || || 24\58 || 496.552 || 103.448 || 9 || 5 || 1.800 || [[Diaschismic]]

|-

| || 5\12 || || || || || 500.000 || 100.000 || 2 || 1 || 2.000 || Basic 2L 8s<br>(Generators smaller than this are proper)

| || 5\12 || || || || || 500.000 || 100.000 || 2 || 1 || 2.000 || Basic 2L 8s <br>(Generators smaller than this are proper)

|-

| || || || || || 21\50 || 504.000 || 96.000 || 9 || 4 || 2.250 || Bimeantone

| || || || || || 21\50 || 504.000 || 96.000 || 9 || 4 || 2.250 || [[Bimeantone]]

|-

| || || || || 16\38 || || 505.263 || 94.737 || 7 || 3 || 2.333 ||

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| || || || || || 27\64 || 506.250 || 93.750 || 12 || 5 || 2.400 ||

|-

| || || || 11\26 || || || 507.692 || 92.308 || 5 || 2 || 2.500 || Injera

| || || || 11\26 || || || 507.692 || 92.308 || 5 || 2 || 2.500 || [[Injera]]

|-

| || || || || || 28\66 || 509.091 || 90.909 || 13 || 5 || 2.600 || Golden injeroid

| || || || || || 28\66 || 509.091 || 90.909 || 13 || 5 || 2.600 || Golden injeroid (509.3386¢)

|-

| || || || || 17\40 || || 510.000 || 90.000 || 8 || 3 || 2.667 ||

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| || || || || || 23\54 || 511.111 || 88.889 || 11 || 4 || 2.750 ||

|-

| || || 6\14 || || || || 514.286 || 85.714 || 3 || 1 || 3.000 || ~~L/s = 3/1~~

| || || 6\14 || || || || 514.286 || 85.714 || 3 || 1 || 3.000 ||

|-

| || || || || || 19\44 || 518.182 || 81.818 || 10 || 3 || 3.333 ||

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| || || || 7\16 || || || 525.000 || 75.000 || 4 || 1 || 4.000 ||

|-

| || || || || || 15\34 || 529.412 || 70.588 || 9 || 2 || 4.500 || Vishnuzmic (incomplete)

| || || || || || 15\34 || 529.412 || 70.588 || 9 || 2 || 4.500 || [[Vishnuzmic]] (incomplete)

|-

| || || || || 8\18 || || 533.333 || 66.667 || 5 || 1 || 5.000 ||

|-

| || || || || || 9\20 || 540.000 || 60.000 || 6 || 1 || 6.000 || Shrutar, teff/pombe (incomplete)↓

| || || || || || 9\20 || 540.000 || 60.000 || 6 || 1 || 6.000 || [[Shrutar]], [[teff]]/[[pombe]] (incomplete)↓

|-

| 1\2 || || || || || || 600.000 || 0.000 || 1 || 0 || → inf ||

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[[Category:Abstract MOS patterns]]

[[Category:Antidimanic]]

[[Category:10-tone scales]]

@@ Line 11: / Line 11: @@
 '''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]], 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
@@ Line 27: / Line 27: @@
 '''J''' '''K''' '''L''' L&/M@ '''M''' '''N''' '''O''' '''P''' '''Q''' Q&/R@ '''R''' '''S''' '''J'''
+== Modes ==
+* 8|0(2) LssssLssss, J-K&-L&-M-N-O-P&-Q&-R-S-J
+* 6|2(2) sLssssLsss, J-K-L&-M-N-O-P-Q&-R-S-J
+* 4|4(2) ssLssssLss, J-K-L-M-N-O-P-Q-R-S-J
+* 2|6(2) sssLssssLs, J-K-L-M@-N-O-P-Q-R@-S-J
+* 0|8(2) ssssLssssL, J-K-L-M@-N@-O-P-Q-R@-S@-J
 == Scale tree ==
+Generator ranges:
+* Chroma-positive generator: 480 cents (4\10) to 600 cents (1\2)
+* Chroma-negative generator: 0 cents (0\2) to 120 cents (1\10)
 {| class="wikitable center-all"
 ! colspan="6" rowspan="2" | Generator
@@ Line 42: / Line 53: @@
 | 4\10 || || || || || || 480.000 || 120.000 || 1 || 1 || 1.000 ||
 |-
-| || || || || || 21\52 || 484.615 || 115.385 || 6 || 5 || 1.200 || Semimiracle↑
+| || || || || || 21\52 || 484.615 || 115.385 || 6 || 5 || 1.200 || [[Semimiracle]]↑
 |-
 | || || || || 17\42 || || 485.714 || 114.286 || 5 || 4 || 1.250 ||
@@ Line 54: / Line 65: @@
 | || || || || 22\54 || || 488.889 || 111.111 || 7 || 5 || 1.400 ||
 |-
-| || || || || || 31\76 || 489.474 || 110.526 || 10 || 7 || 1.428 ||
+| || || || || || 31\76 || 489.474 || 110.526 || 10 || 7 || 1.429 ||
 |-
-| || || 9\22 || || || || 490.909 || 107.692 || 3 || 2 || 1.500 || L/s = 3/2, pajara
+| || || 9\22 || || || || 490.909 || 107.692 || 3 || 2 || 1.500 || [[Pajara]]
 |-
 | || || || || || 32\78 || 492.308 || 131.250 || 11 || 7 || 1.571 ||
 |-
-| || || || || 23\56 || || 492.857 || 107.143 || 8 || 5 || 1.600 || Keen
+| || || || || 23\56 || || 492.857 || 107.143 || 8 || 5 || 1.600 || [[Keen]]
 |-
-| || || || || || 37\90 || 493.333 || 106.667 || 13 || 8 || 1.625 || Golden pajaroid
+| || || || || || 37\90 || 493.333 || 106.667 || 13 || 8 || 1.625 || Golden pajaroid (493.2011¢)
 |-
 | || || || 14\34 || || || 494.118 || 105.882 || 5 || 3 || 1.667 ||
 |-
-| || || || || || 33\80 || 495.000 || 105.000 || 12 || 7 || 1.714 || Srutal
+| || || || || || 33\80 || 495.000 || 105.000 || 12 || 7 || 1.714 || [[Srutal]]
 |-
 | || || || || 19\46 || || 495.652 || 104.348 || 7 || 4 || 1.750 ||
 |-
-| || || || || || 24\58 || 496.552 || 103.448 || 9 || 5 || 1.800 || Diaschismic
+| || || || || || 24\58 || 496.552 || 103.448 || 9 || 5 || 1.800 || [[Diaschismic]]
 |-
-| || 5\12 || || || || || 500.000 || 100.000 || 2 || 1 || 2.000 || Basic 2L 8s<br>(Generators smaller than this are proper)
+| || 5\12 || || || || || 500.000 || 100.000 || 2 || 1 || 2.000 || Basic 2L 8s <br>(Generators smaller than this are proper)
 |-
-| || || || || || 21\50 || 504.000 || 96.000 || 9 || 4 || 2.250 || Bimeantone
+| || || || || || 21\50 || 504.000 || 96.000 || 9 || 4 || 2.250 || [[Bimeantone]]
 |-
 | || || || || 16\38 || || 505.263 || 94.737 || 7 || 3 || 2.333 ||
@@ Line 80: / Line 91: @@
 | || || || || || 27\64 || 506.250 || 93.750 || 12 || 5 || 2.400 ||
 |-
-| || || || 11\26 || || || 507.692 || 92.308 || 5 || 2 || 2.500 || Injera
+| || || || 11\26 || || || 507.692 || 92.308 || 5 || 2 || 2.500 || [[Injera]]
 |-
-| || || || || || 28\66 || 509.091 || 90.909 || 13 || 5 || 2.600 || Golden injeroid
+| || || || || || 28\66 || 509.091 || 90.909 || 13 || 5 || 2.600 || Golden injeroid (509.3386¢)
 |-
 | || || || || 17\40 || || 510.000 || 90.000 || 8 || 3 || 2.667 ||
@@ Line 88: / Line 99: @@
 | || || || || || 23\54 || 511.111 || 88.889 || 11 || 4 || 2.750 ||
 |-
-| || || 6\14 || || || || 514.286 || 85.714 || 3 || 1 || 3.000 || L/s = 3/1
+| || || 6\14 || || || || 514.286 || 85.714 || 3 || 1 || 3.000 ||
 |-
 | || || || || || 19\44 || 518.182 || 81.818 || 10 || 3 || 3.333 ||
@@ Line 98: / Line 109: @@
 | || || || 7\16 || || || 525.000 || 75.000 || 4 || 1 || 4.000 ||
 |-
-| || || || || || 15\34 || 529.412 || 70.588 || 9 || 2 || 4.500 || Vishnuzmic (incomplete)
+| || || || || || 15\34 || 529.412 || 70.588 || 9 || 2 || 4.500 || [[Vishnuzmic]] (incomplete)
 |-
 | || || || || 8\18 || || 533.333 || 66.667 || 5 || 1 || 5.000 ||
 |-
-| || || || || || 9\20 || 540.000 || 60.000 || 6 || 1 || 6.000 || Shrutar, teff/pombe (incomplete)↓
+| || || || || || 9\20 || 540.000 || 60.000 || 6 || 1 || 6.000 || [[Shrutar]], [[teff]]/[[pombe]] (incomplete)↓
 |-
 | 1\2 || || || || || || 600.000 || 0.000 || 1 || 0 || → inf ||
@@ Line 108: / Line 119: @@
 [[Category:Abstract MOS patterns]]
+[[Category:Antidimanic]]
+[[Category:10-tone scales]]