User:Moremajorthanmajor/Greater whole tone scale: Difference between revisions

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'''5L 1s(<major seventh>)''', also known as the greater whole tone scale, refers to [[MOS scale]]s with 5 large steps and 1 small step. When L=s we have [[6edo|6edo]], the equal-tempered "whole tone scale" of impressionistic fame. At the other end of the spectrum, we approach [[5edo]], with five equal whole tones of 240 cents. In between, we find relatively even hexatonic scales with one irregularity: a "whole tone" which is smaller than all the others — perhaps not a "whole tone" at all.
'''5L 1s(<major seventh>)''', also known as the greater whole tone scale, refers to [[MOS scale]]s with 5 large steps and 1 small step. When L=s we have [[6edo|6edo]], the equal-tempered "whole tone scale" of impressionistic fame. At the other end of the spectrum, we approach [[5edo]], with five equal whole tones of 240 cents. In between, we find relatively even hexatonic scales with one irregularity: a "whole tone" which is smaller than all the others — perhaps not a "whole tone" at all.


The only notable low-harmonic-entropy scale with this MOS pattern is [[Gamelismic clan #Slendric|slendric]], in which the large step is 8/7 and three of them make a 3/2. Other low-harmonic-entropy scales with this MOS pattern tune the seventh (septave) 1080¢ or flatter for making a quarter 3/2.
The only notable low-harmonic-entropy scale with this MOS pattern is [[Gamelismic clan #Slendric|slendric]], in which the large step is 8/7 and three of them make a 3/2. Other low-harmonic-entropy scales with this MOS pattern tune the seventh (septave) 1050¢ or flatter for making a quarter 3/2 and between 1080¢ and 1100¢ for making a 7/2-equivalent augmented thirteenth scale.  


Scales with this pattern are always [[Rothenberg propriety|proper]], because there is only one small step.
Scales with this pattern are always [[Rothenberg propriety|proper]], because there is only one small step.
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== Scale tree ==
== Scale tree ==
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="8" | Generator
! Generator
! Normalized Cents (septave)
! Normalized Cents (septave)
! Cents
! Cents
Line 15: Line 15:
! Comments
! Comments
|-
|-
| 1\6 || || || || || || || || 171.429 || 200.000 || 1 || 1 || 1.000 ||  
| 1\6 || 171.429 || 200.000 || 1 || 1 || 1.000 ||  
|-
|-
|
|
|
|
|
|
|
|9\53
|9\53
|177.049
|177.049
Line 32: Line 25:
|
|
|-
|-
|
|
|
|
|
|
|8\47
|8\47
|
|177.778
|177.778
|204.255
|204.255
Line 47: Line 33:
|
|
|-
|-
|
|
|
|
|
|7\41
|7\41
|
|
|178.723
|178.723
|204.878
|204.878
Line 62: Line 41:
|
|
|-
|-
|
|
|
|
|
|
|13\76
|13\76
|
|179.310
|179.310
|205.263
|205.263
Line 77: Line 49:
|
|
|-
|-
| || || || || 6\35 || || || || 180.000 || 205.714 || 6 || 5 || 1.200 ||  
| 6\35|| 180.000 || 205.714 || 6 || 5 || 1.200 ||  
|-
|-
| || || || 5\29 || || || || || 181.818 || 206.897 || 5 || 4 || 1.250 ||  
| 5\29|| 181.818 || 206.897 || 5 || 4 || 1.250 ||  
|-
|-
| || || || || || 14\81 || || || 182.609 || 207.407 || 14 || 11 || 1.273 ||  
| 14\81|| 182.609 || 207.407 || 14 || 11 || 1.273 ||  
|-  
|-  
| || || || || 9\52 || || || || 183.051 || 207.692 || 9 || 7 || 1.286 ||  
| 9\52|| 183.051 || 207.692 || 9 || 7 || 1.286 ||  
|-
|-
| || || 4\23 || || || || || || 184.615 || 208.696 || 4 || 3 || 1.333 ||
| 4\23|| 184.615 || 208.696 || 4 || 3 || 1.333 ||
|-
|-
| || || || || 11\63 || || || ||185.915 || 209.524||11||8|| 1.375||
| 11\63||185.915 || 209.524||11||8|| 1.375||
|-
|-
| || || ||7\40|| || || || ||186.667|| 210.000||7||5|| 1.400||
| 7\40||186.667|| 210.000||7||5|| 1.400||
|-
|-
|  || || || ||10\57|| || || ||187.500 || 210.528|| 10||7||1.428||
| 10\57 ||187.500 || 210.528|| 10||7||1.428||
|-
|-
| || || || || ||13\74 ||  || ||187.952||210.811||13||9|| 1.444||
| 13\74||187.952||210.811||13||9|| 1.444||
|-
|-
| || || || || || ||16\91|| ||188.235||210.989||16||11 ||1.4545||
| 16\91||188.235||210.989||16||11 ||1.4545||
|-
|-
| ||3\17|| || || || || || || 189.474 || 211.765 || 3||2||1.500||L/s = 3/2
| 3\17|| 189.474 || 211.765 || 3||2||1.500||L/s = 3/2
|-
|-
| ||  ||  || ||  || 14\79 || || ||190.909||212.658 ||14 || 9|| 1.556||
| 14\79||190.909||212.658 ||14 || 9|| 1.556||
|-
|-
| || || ||  || 11\62|| || || ||191.304|| 212.903||11 ||7||1.571||
| 11\62||191.304|| 212.903||11 ||7||1.571||
|-
|-
| || || ||8\45 ||  ||  ||  || ||192.000||213.333||8 || 5||1.600||
| 8\45||192.000||213.333||8 || 5||1.600||
|-
|-
| || || || ||13\73|| || || || 192.593||213.699 ||13 || 8|| 1.625||Golden Ionianic-machine
| 13\73|| 192.593||213.699 ||13 || 8|| 1.625||Golden Ionianic-machine
|-
|-
| || ||5\28|| ||  ||  || ||  || 193.548||214.286||5||3||1.667 || Ionianic-Machine
| 5\28|| 193.548||214.286||5||3||1.667 || Ionianic-Machine
|-
|-
| || || || || 12\67|| || || || 194.595||214.925 ||12||7||1.714||
| 12\67|| 194.595||214.925 ||12||7||1.714||
|-
|-
| ||  || ||7\39|| || || || ||195.349||215.385||7||4 || 1.750||
| 7\39||195.349||215.385||7||4 || 1.750||
|-
|-
|  || || || ||9\50|| || || ||196.364||216.000 ||9||5||1.800 ||
| 9\50 ||196.364||216.000 ||9||5||1.800 ||
|-
|-
| ||  || || || ||11\61|| || || 197.015||216.393||11 ||6||1.833 ||
| 11\61|| 197.015||216.393||11 ||6||1.833 ||
|-  
|-  
| || || || || ||  || 13\72|| ||197.468||216.667 ||13||7||1.857||
| 13\72||197.468||216.667 ||13||7||1.857||
|-   
|-   
| ||  ||  || ||  || ||  || 15\83||197.802|197.802||216.867||15||8||1.875
| 15\83|| 197.802 |197.802||216.867||15||8||1.875
|
|-
|17\94
|198.058
|217.021
|17
|9
|1.889
|-  
|-  
| 2\11 ||  || || || || ||  || ||200.000||218.182||2||1||2.000 ||Basic Ionianic-machinoid
| 2\11 ||200.000||218.182||2||1||2.000 ||Basic Ionianic-machinoid
|-
|17\93
|201.980
|219.355
|17
|8
|2.125
|
|-
|-
|  || ||  || || || || ||15\82 || 202.247||219.512||15|| 7||2.143||
| 15\82 || 202.247||219.512||15|| 7||2.143||
|-
|-
|  || || || || || ||13\71|| ||202.597||219.718||13||6 || 2.167||
| 13\71 ||202.597||219.718||13||6 || 2.167||
|-
|-
| || || || || ||11\60|| || ||203.077 ||220.000 ||11|| 5|| 2.200||
| 11\60||203.077 ||220.000 ||11|| 5|| 2.200||
|-
|-
|  ||  || ||  || 9\49 || || ||  ||203.774||220.408 || 9||4||2.250||
| 9\49 ||203.774||220.408 || 9||4||2.250||
|-
|-
| || || ||7\38|| || || ||  ||204.878 ||221.053||7 || 3||2.333||
| 7\38||204.878 ||221.053||7 || 3||2.333||
|-
|-
| || || || ||12\65 || || || ||205.714||221.538||12|| 5||2.400 ||
| 12\65||205.714||221.538||12|| 5||2.400 ||
|-
|-
| || ||5\27|| ||  || || || ||206.897||222.222||5||2||2.500 ||
| 5\27||206.897||222.222||5||2||2.500 ||
|-
|-
|  || ||  || || ||18\97|| || ||207.692 || 222.680 ||18||7|| 2.571||
| 18\97 ||207.692 || 222.680 ||18||7|| 2.571||
|-
|-
| || || ||  ||13\70|| || || ||208.000||222.857||13||5||2.600||Unnamed golden tuning
| 13\70||208.000||222.857||13||5||2.600||Unnamed golden tuning
|-
|-
| ||  || || 8\43|| || || || ||208.697||223.256||8||3||2.667||
| 8\43||208.697||223.256||8||3||2.667||
|-
|-
|  || || || ||11\59||  ||  ||  || 209.524|| 223.729||11||4||2.750||
| 11\59 || 209.524|| 223.729||11||4||2.750||
|-
|-
| || || ||  ||  ||14\75 || || ||210.000||224.000||14||5 ||3.000||
| 14\75||210.000||224.000||14||5 ||3.000||
|-
|-
| ||3\16|| ||  || || ||  ||  ||211.765||225.000||3||1||3.000||L/s = 3/1, Ionianic-clyndro
| 3\16||211.765||225.000||3||1||3.000||L/s = 3/1, Ionianic-clyndro
|-
|-
| || || || || || || 16\85 || ||213.333||225.882||16|| 5|| 3.200||
|22\117
|212.903
|225.641
|22
|7
|3.143
|
|-
|19\101
|213.084
|225.743
|19
|6
|3.167
|
|-
| 16\85||213.333||225.882||16|| 5|| 3.200||
|-  
|-  
| || || || || ||13\69||  ||  ||213.698||226.087||13 || 4|| 3.250||
| 13\69||213.698||226.087||13 || 4|| 3.250||
|-
|-
|  ||  || || ||10\53|| || || ||214.857 ||226.415|| 10 ||3||3.333||
| 10\53 ||214.857 ||226.415|| 10 ||3||3.333||
|-
|-
| || || ||7\37 ||  || || || ||215.385||227.027||7||2||3.500||Ionianic-Laconic
| 7\37||215.385||227.027||7||2||3.500||Ionianic-Laconic
|-
|-
|  ||  || || ||11\58 ||  ||  ||  ||216.393||227.586||11||3 || 3.667||
| 11\58 ||216.393||227.586||11||3 || 3.667||
|-
|-
| ||  ||  ||  ||  || 15\79 ||  || ||216.867||227.848||15||4||3.750||
| 15\79||216.867||227.848||15||4||3.750||
|-
|-
|  || || || || ||  || 19\100|| ||217.143 || 228.000 || 19||5||3.800||
| 19\100 ||217.143 || 228.000 || 19||5||3.800||
|-
|-
| || ||4\21|| || || || || ||218.182||228.571||4||1||4.000||Ionianic-Gorgo
| 4\21||218.182||228.571||4||1||4.000||Ionianic-Gorgo
|-
|-
|  || ||  || || ||13\68|| || ||219.718||229.412 || 13||3||4.333 ||
| 13\68 ||219.718||229.412 || 13||3||4.333 ||
|-
|-
|  ||  || || ||9\47|| || ||  ||220.408||229.787||9||2||4.500||
| 9\47 ||220.408||229.787||9||2||4.500||
|-
|-
|  || ||  || || ||14\73|| || ||221.053 || 230.137||14||3||4.667||
| 14\73 ||221.053 || 230.137||14||3||4.667||
|-
|-
| || || ||5\26 ||  || ||  ||  ||222.222||230.769||5||1 || 5.000||Ionianic-Gidorah
| 5\26||222.222||230.769||5||1 || 5.000||Ionianic-Gidorah
|-
|-
| || || || || || ||11\57 || || 223.729|| 231.579 || 11||2||5.500 ||
| 11\57|| 223.729|| 231.579 || 11||2||5.500 ||
|-  
|-  
|  || || || || ||  ||  ||17\88||224.179||231.818 || 17||3||5.667||
| 17\88 ||224.179||231.818 || 17||3||5.667||
|-   
|-   
|  || || || || 6\31|| || || ||225.000||232.258||6||1||6.000 || Ionianic-Slendric↓
| 6\31 ||225.000||232.258||6||1||6.000 || Ionianic-Slendric↓
|-
|-
|
|
|
|
|
|
|13\67
|13\67
|
| 226.087
| 226.087
|232.839
|232.839
Line 194: Line 191:
|
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|-
|-
|
|
|
|
|
|7\36
|7\36
|
|
|227.027
|227.027
| 233.333
| 233.333
Line 209: Line 199:
|
|
|-
|-
|
|
|
|
|
|
|8\41
|8\41
|
| 228.571
| 228.571
| 234.146
| 234.146
Line 224: Line 207:
|
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|-
|-
|
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|
|
|
|
|9\46
|9\46
|229.787
|229.787
Line 239: Line 215:
|
|
|-
|-
|1\5|| || || || || || || ||240.000|| 240.000||1||0||→ inf||
|1\5||240.000|| 240.000||1||0||→ inf||
|}
|}


Revision as of 23:07, 1 August 2023

5L 1s(<major seventh>), also known as the greater whole tone scale, refers to MOS scales with 5 large steps and 1 small step. When L=s we have 6edo, the equal-tempered "whole tone scale" of impressionistic fame. At the other end of the spectrum, we approach 5edo, with five equal whole tones of 240 cents. In between, we find relatively even hexatonic scales with one irregularity: a "whole tone" which is smaller than all the others — perhaps not a "whole tone" at all.

The only notable low-harmonic-entropy scale with this MOS pattern is slendric, in which the large step is 8/7 and three of them make a 3/2. Other low-harmonic-entropy scales with this MOS pattern tune the seventh (septave) 1050¢ or flatter for making a quarter 3/2 and between 1080¢ and 1100¢ for making a 7/2-equivalent augmented thirteenth scale.

Scales with this pattern are always proper, because there is only one small step.

Scale tree

Generator Normalized Cents (septave) Cents L s L/s Comments
1\6 171.429 200.000 1 1 1.000
9\53 177.049 203.774 9 8 1.125
8\47 177.778 204.255 8 7 1.143
7\41 178.723 204.878 7 6 1.167
13\76 179.310 205.263 13 11 1.273
6\35 180.000 205.714 6 5 1.200
5\29 181.818 206.897 5 4 1.250
14\81 182.609 207.407 14 11 1.273
9\52 183.051 207.692 9 7 1.286
4\23 184.615 208.696 4 3 1.333
11\63 185.915 209.524 11 8 1.375
7\40 186.667 210.000 7 5 1.400
10\57 187.500 210.528 10 7 1.428
13\74 187.952 210.811 13 9 1.444
16\91 188.235 210.989 16 11 1.4545
3\17 189.474 211.765 3 2 1.500 L/s = 3/2
14\79 190.909 212.658 14 9 1.556
11\62 191.304 212.903 11 7 1.571
8\45 192.000 213.333 8 5 1.600
13\73 192.593 213.699 13 8 1.625 Golden Ionianic-machine
5\28 193.548 214.286 5 3 1.667 Ionianic-Machine
12\67 194.595 214.925 12 7 1.714
7\39 195.349 215.385 7 4 1.750
9\50 196.364 216.000 9 5 1.800
11\61 197.015 216.393 11 6 1.833
13\72 197.468 216.667 13 7 1.857
15\83 197.802 216.867 15 8 1.875
17\94 198.058 217.021 17 9 1.889
2\11 200.000 218.182 2 1 2.000 Basic Ionianic-machinoid
17\93 201.980 219.355 17 8 2.125
15\82 202.247 219.512 15 7 2.143
13\71 202.597 219.718 13 6 2.167
11\60 203.077 220.000 11 5 2.200
9\49 203.774 220.408 9 4 2.250
7\38 204.878 221.053 7 3 2.333
12\65 205.714 221.538 12 5 2.400
5\27 206.897 222.222 5 2 2.500
18\97 207.692 222.680 18 7 2.571
13\70 208.000 222.857 13 5 2.600 Unnamed golden tuning
8\43 208.697 223.256 8 3 2.667
11\59 209.524 223.729 11 4 2.750
14\75 210.000 224.000 14 5 3.000
3\16 211.765 225.000 3 1 3.000 L/s = 3/1, Ionianic-clyndro
22\117 212.903 225.641 22 7 3.143
19\101 213.084 225.743 19 6 3.167
16\85 213.333 225.882 16 5 3.200
13\69 213.698 226.087 13 4 3.250
10\53 214.857 226.415 10 3 3.333
7\37 215.385 227.027 7 2 3.500 Ionianic-Laconic
11\58 216.393 227.586 11 3 3.667
15\79 216.867 227.848 15 4 3.750
19\100 217.143 228.000 19 5 3.800
4\21 218.182 228.571 4 1 4.000 Ionianic-Gorgo
13\68 219.718 229.412 13 3 4.333
9\47 220.408 229.787 9 2 4.500
14\73 221.053 230.137 14 3 4.667
5\26 222.222 230.769 5 1 5.000 Ionianic-Gidorah
11\57 223.729 231.579 11 2 5.500
17\88 224.179 231.818 17 3 5.667
6\31 225.000 232.258 6 1 6.000 Ionianic-Slendric↓
13\67 226.087 232.839 13 2 6.500
7\36 227.027 233.333 7 1 7.000
8\41 228.571 234.146 8 1 8.000
9\46 229.787 234.783 9 1 9.000
1\5 240.000 240.000 1 0 → inf

See also

5L 1s (11/6-equivalent) - simplest tuning

5L 1s (15/8-equivalent) - classical tuning

5L 1s (21\11-equivalent) - Neogothic tuning

5L 1s (27/14-equivalent) - septimal tuning

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