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{{Infobox MOS
| Name = Neapolitan-dicotonic
| Periods = 1
| nLargeSteps = 7
| nSmallSteps = 3
| Equalized = 3
| Paucitonic = 2
| Pattern = LLLsLLsLLs
| Equave = 15/7}}
{{Infobox MOS
{{Infobox MOS
| Name = dicotonic
| Name = dicotonic
Line 18: Line 9:
}}
}}


'''7L 3s(<15/7>)''' refers to the structure of [[MOSScales|moment of symmetry scales]] built from a 10-tone chain of (neutral) thirds (assuming a period of an octave or a minor ninth):
'''7L 3s''' refers to the structure of [[MOSScales|moment of symmetry scales]] built from a 10-tone chain of neutral thirds (assuming a period of an octave):


L s L L L s L L s L
L s L L L s L L s L
Line 25: Line 16:


t q t t t q t t q t
t q t t t q t t q t
==Names==
== Names==
This MOS is called '''dicotonic''' (named after the abstract temperaments [[dicot]] and more specifically 11-limit [[Dicot_family#Dichotic|dichotic]]) in [[TAMNAMS]].
This MOS is called '''dicotonic''' (named after the abstract temperaments [[dicot]] and more specifically 11-limit [[Dicot_family#Dichotic|dichotic]]) in [[TAMNAMS]].


==Intervals==
==Intervals==
The generator (g) will fall between 343 cents (2\7 - two degrees of [[7edo]] and 360 cents (3\10 - three degrees of [[10edo]]), hence a neutral third.
The generator (g) will fall between 343 cents (2\7 - two degrees of [[7edo|7edo]] and 360 cents (3\10 - three degrees of [[10edo|10edo]]), hence a neutral third.


2g, then, will fall between 686 cents (4\7) and 720 cents (3\5), the range of [[5L 2s|diatonic]] fifths.
2g, then, will fall between 686 cents (4\7) and 720 cents (3\5), the range of [[5L 2s|diatonic]] fifths.
Line 43: Line 34:
!# generators up
!# generators up
!Notation (1/1 = 0)
!Notation (1/1 = 0)
!name
! name
!In L's and s's
!In L's and s's
!# generators up
!# generators up
!Notation of 2/1 inverse
! Notation of 2/1 inverse
!name
! name
!In L's and s's
! In L's and s's
|-
|-
| colspan="8" style="text-align:center" |The 10-note MOS has the following intervals (from some root):
| colspan="8" style="text-align:center" |The 10-note MOS has the following intervals (from some root):
|-
|-
|0
| 0
|0
|0
|perfect unison
|perfect unison
Line 61: Line 52:
|7L+3s
|7L+3s
|-
|-
|1
| 1
|7
| 7
|perfect 7-step
|perfect 7-step
|5L+2s
|5L+2s
Line 82: Line 73:
|1
|1
|major (1-)step
|major (1-)step
|1L
| 1L
| -3
| -3
|9v
|9v
|minor 9-step
| minor 9-step
|6L+3s
|6L+3s
|-
|-
Line 91: Line 82:
|8
|8
|major 8-step
|major 8-step
|6L+2s
| 6L+2s
| -4
| -4
|2v
|2v
Line 101: Line 92:
|major 5-step
|major 5-step
|4L+1s
|4L+1s
| -5
| -5
|5v
|5v
|minor 5-step
| minor 5-step
|3L+2s
| 3L+2s
|-
|-
|6
|6
|2
|2
|major 2-step
| major 2-step
|2L
|2L
| -6
| -6
Line 118: Line 109:
|9
|9
|major 9-step
|major 9-step
|7L+2s
| 7L+2s
| -7
| -7
|1v
|1v
|minor (1-)step
| minor (1-)step
|1s
|1s
|-
|-
Line 127: Line 118:
|6^
|6^
|major 6-step
|major 6-step
|5L+1s
| 5L+1s
| -8
| -8
|4v
|4v
|minor 4-step
|minor 4-step
|2L+2s
| 2L+2s
|-
|-
|9
|9
|3^
|3^
|augmented 3-step
| augmented 3-step
|3L
|3L
| -9
| -9
|7v
|7v
|diminished 7-step
|diminished 7-step
| 4L+3s
|4L+3s
|-
|-
|10
|10
|0^
|0^
|augmented unison
| augmented unison
|1L-1s
|1L-1s
| -10
| -10
Line 153: Line 144:
|11
|11
|7^
|7^
|augmented 7-step
| augmented 7-step
|6L+1s
|6L+1s
| -11
| -11
|3v
|3v
|diminished 3-step
|diminished 3-step
| 1L+2s
|1L+2s
|-
|-
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s]], [[10L 7s]], or [[17edo]]) also has the following intervals (from some root):
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s]], [[10L 7s]], or [[17edo]]) also has the following intervals (from some root):
Line 164: Line 155:
|12
|12
|4^
|4^
|augmented 4-step
| augmented 4-step
|4L
|4L
| -12
| -12
|6v
| 6v
|diminished 6-step
|diminished 6-step
|3L+3s
|3L+3s
Line 176: Line 167:
|2L-1s
|2L-1s
| -13
| -13
|9w
| 9w
|diminished 9-step
|diminished 9-step
|5L+4s
|5L+4s
Line 184: Line 175:
|augmented 8-step
|augmented 8-step
|8L+1s
|8L+1s
| -14
| -14
|2w
|2w
|diminished 2-step
|diminished 2-step
Line 191: Line 182:
|15
|15
|5^
|5^
| augmented 5-step
|augmented 5-step
|5L
| 5L
| -15
| -15
|5w
|5w
|diminished 5-step
| diminished 5-step
|2L+3s
|2L+3s
|-
|-
|16
|16
|2^
|2^
|augmented 2-step
| augmented 2-step
| 3L-1s
|3L-1s
| -16
| -16
|8w
|8w
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|4L+4s
|4L+4s
|}
|}
==Scale tree ==
== Scale tree ==
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 – five degrees of [[17edo]]:
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 – five degrees of [[17edo]]:


{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="4" rowspan="2" |Generator
! colspan="6" rowspan="2" | Generator
! colspan="2" | Cents
! colspan="2" | Cents
! rowspan="2" |''ed17\16''
! rowspan="2" | L
! rowspan="2" |L
! rowspan="2" | s
! rowspan="2" |s
! rowspan="2" | L/s
! rowspan="2" |L/s
! rowspan="2" | Comments
! rowspan="2" |Comments
|-
|-
!edo
! Chroma-positive
!Neapolitan (normalized)
! Chroma-negative
|-
|-
|7\10|| || || ||840.000||933.333 (1171.242 śata)
| 7\10 || || || || || || 840.000 || 360.000 || 1 || 1 || 1.000 ||  
|''1190.000''||1||1||1.000||
|-
|-
| 40\57|| || || ||842.105||923.077 (1158.371 śata)
| || || || || || 40\57 || 842.105 || 357.895 || 6 || 5 || 1.200 || Restles↑
|''1192.9825''|| 6||5||1.200||(Neapolitan-)Restles↑
|-
|-
|
| || || || || 33\47 || || 842.553 || 357.447 || 5 || 4 || 1.250 ||  
|73\104
|
|
|842.308
|922.105 (1157.152 śata)
|''1193.269''
|11
|9
|1.222
|
|-
|-
|
| || || || || || 59\84 || 842.857 || 357.143 || 9 || 7 || 1.286 ||  
|
|106\151
|
|842.384
|921.739 (1156.692 śata)
|''1193.3775''
|16
|13
|1.231
|
|-
|-
| 33\47|| || || || 842.553||920.930 (1155.677 śata)
| || || || 26\37 || || || 843.243 || 356.757 || 4 || 3 || 1.333 ||  
|''1193.617''||5|| 4||1.250 ||  
|-
|-
|
| || || || || || 71\101 || 843.564 || 356.436 || 11 || 8 || 1.375 ||  
|92\131
|
|
|842.748
|920.000 (1154.510 śata)
|''1193.893''
|14
|11
|1.273
|
|-
|-
| 59\84|| || || ||842.857 ||919.4805 (1153.858 śata)
| || || || || 45\64 || || 843.750 || 356.250 || 7 || 5 || 1.400 || Beatles
|''1194.048''||9||7||1.286||
|-
|-
|
| || || || || || 64\91 || 843.956 || 356.044 || 10 || 7 || 1.428 ||  
|85\121
|
|
|842.975
|918.919 (1153.153 śata)
|''1194.215''
|13
|10
|1.300
|
|-
|-
|
| || || 19\27 || || || || 844.444 || 355.556 || 3 || 2 || 1.500 || L/s = 3/2, suhajira/ringo
|
|111\158
|
|843.038
|918.621 (1152.779 śata)
|''1194.304''
|17
|13
|1.308
|
|-
|-
| 26\37|| || || ||843.243||917.647 (1151.557 śata)
| || || || || || 69\98 || 844.698 || 355.102 || 11 || 7 || 1.571 ||  
|''1194.595''||4 || 3||1.333 ||
|-
|-
|
| || || || || 50\71 || || 845.070 || 354.930 || 8 || 5 || 1.600 ||  
|97\138
|
|
|843.478
|916.535 (1150.162 śata)
|''1194.9275''
|15
|11
|1.364
|
|-
|-
|71\101|| || || ||843.564||916.129 (1149.652 śata)
| || || || || || 81\115 || 845.217 || 354.783 || 13 || 8 || 1.625 || Golden suhajira
|''1195.0495''||11||8||1.375||
|-
|-
|45\64|| || || || 843.750||915.254 (1148.554 śata)
| || || || 31\44 || || || 845.455 || 354.545 || 5 || 3 || 1.667 ||  
|''1195.3125''||7|| 5||1.400||(Neapolitan-)Beatles
|-
|-
|64\91|| || || ||843.956||914.286 (1147.339 śata)
| || || || || || 74\105 || 845.714 || 354.286 || 12 || 7 || 1.714 ||  
|''1195.604''||10||7 || 1.428 ||  
|-
|-
|
| || || || || 43\61 || || 845.902 || 354.098 || 7 || 4 || 1.750 ||
|83\118
|
|
|844.068
|913.7615 (1146.681 śata)
|''1195.763''
|13
|9
|1.444
|
|-
|-
|
| || || || || || 55\78 || 846.154 || 353.846 || 9 || 5 || 1.800 ||
|
|102\145
|
|844.138
|913.433 (1146.269 śata)
|''1195.862''
|16
|11
|1.4545
|
|-
|
|
|
|121\172
|844.186
|913.2075 (1145.986 śata)
|''1195.930''
|19
|13
|1.4615
|
|-
|19\27|| || || || 844.444|| 912.000 (1144.471 śata)
|''1196.296''||3||2 || 1.500||L/s = 3/2, (Neapolitan-)suhajira/ringo
|-
|
|
|
|107\152
|844.736
|910.638 (1142.762 śata)
|''1196.7105''
|17
|11
|1.5455
|
|-
|
|
|88\125
|
|844.8
|910.345 (1142.3935 śata)
|''1196.800''
|14
|9
|1.556
|
|-
| || 69\98||  || ||844.898||909.890 (1141.823 śata)
|''1196.939''||11||7 || 1.571||
|-
|50\71|| || || ||845.070|| 909.091 (1140.820 śata)
|''1197.183''||8||5||1.600||
|-
|  ||81\115|| || ||845.217||908.411 (1139.967 śata)
|''1197.391''||13||8 || 1.625||Golden (Neapolitan-)suhajira
|-
| 31\44|| || || ||845.455||907.317 (1138.594 śata)
|''1197.727''||5|| 3||1.667 ||
|-
| ||74\105|| || ||845.714 || 906.122 (1137.095 śata)
|''1198.095''||12 || 7||1.714 ||
|-
|43\61|| || || ||845.902||905.263 (1136.0165 śata)
|''1198.361''||7||4||1.750||
|-
|
|
|98\139
|
|846.043
|904.615 (1135.204 śata)
|''1198.561''
|16
|9
|1.778
|
|-
| ||55\78||  ||  ||846.154||904.110 (1134.569 śata)
|''1198.718''||9||5||1.800 ||
|-
|
|
|67\95
|
|846.316
|903.371 (1133.642 śata)
|''1198.947''
|11
|6
|1.833
|
|-
|
|
|
|79\112
|846.429
|902.857 (1132.997 śata)
|''1199.107''
|13
|7
|1.857
|
|-
|-
|12\17|| || || || 847.059||900.000 (1129.412 śata)
| || 12\17 || || || || || 847.059 || 352.941 || 2 || 1 || 2.000 || Basic dicotonic<br>(Generators smaller than this are proper)
|''1200.000''||2||1||2.000 ||Basic (Neapolitan-)dicotonic<br>(Generators smaller than this are proper)
|-
|-
| || ||53\75||  ||848.000||352.000
| || || || || || 53\75 || 848.000 || 352.000 || 9 || 4 || 2.250 ||
| ||9||4||2.250||
|-
|-
| ||41\58|| || ||848.273||351.724
| || || || || 41\58 || || 848.273 || 351.724 || 7 || 3 || 2.333 ||  
| ||7||3||2.333||
|-
|-
| || ||70\99|| ||848.485||351.515
| || || || || || 70\99 || 848.485 || 351.515 || 12 || 5 || 2.400 || Hemif/hemififths
| ||12||5||2.400||(Neapolitan-)Hemif/hemififths
|-
|-
|29\41|| || || || 848.780 || 351.220
| || || || 29\41 || || || 848.780 || 351.220 || 5 || 2 || 2.500 || Mohaha/neutrominant
| ||5|| 2||2.500||(Neapolitan-)Mohaha/neutrominant
|-
|-
| || ||75\106|| ||849.057 || 350.943
| || || || || || 75\106 || 849.057 || 350.943 || 13 || 5 || 2.600 || Hemif/salsa/karadeniz
| || 13||5 || 2.600|| (Neapolitan-)Hemif/salsa/karadeniz
|-
|-
| ||46\65|| ||  ||849.231 ||350.769
| || || || || 46\65 || || 849.231 || 350.769 || 8 || 3 || 2.667 || Mohaha/mohamaq
| ||8||3 || 2.667 || (Neapolitan-)Mohaha/mohamaq
|-
|-
| || ||63\89|| ||849.438||350.562
| || || || || || 63\89 || 849.438 || 350.562 || 11 || 4 || 2.750 ||
| ||11||4||2.750||
|-
|-
|17\24|| || || ||850.000||350.000
| || || 17\24 || || || || 850.000 || 350.000 || 3 || 1 || 3.000 || L/s = 3/1
| ||3 || 1||3.000 ||L/s = 3/1
|-
|-
| || ||56\79|| ||850.633||349.367
| || || || || || 56\79 || 850.633 || 349.367 || 10 || 3 || 3.333 ||
| ||10||3 ||3.333||
|-
|-
| ||39\55|| || ||850.909||349.091
| || || || || 39\55 || || 850.909 || 349.091 || 7 || 2 || 3.500 ||  
| ||7||2 || 3.500||  
|-
|-
| || ||61\86|| || 851.613|| 358.837
| || || || || || 61\86 || 851.613 || 358.837 || 11 || 3 || 3.667 ||  
| ||11||3||3.667||
|-
|-
|22\31|| ||  || || 851.613||348.387
| || || || 22\31 || || || 851.613 || 348.387 || 4 || 1 || 4.000 || Mohaha/migration/mohajira
| ||4 || 1||4.000||(Neapolitan-)Mohaha/migration/mohajira
|-
|-
| || ||49\69|| ||852.174|| 347.826
| || || || || || 49\69 || 852.174 || 347.826 || 9 || 2 || 4.500 ||  
| ||9||2 ||4.500||
|-
|-
| ||27\38|| ||  ||852.632||347.368
| || || || || 27\38 || || 852.632|| 347.368 || 5 || 1 || 5.000 ||  
| ||5||1||5.000 ||
|-
|-
| || ||32\45|| ||853.333||346.667
| || || || || || 32\45 || 853.333 || 346.667 || 6 || 1 || 6.000 || Mohaha/ptolemy
| || 6||1 ||6.000||(Neapolitan-)Mohaha/ptolemy
|-
|-
|5\7|| || || ||857.143|| 342.867
| 5\7 || || || || || || 857.143 || 342.867 || 1 || 0 || → inf ||
| || 1||0 ||→ inf||
|}
|}


Line 496: Line 285:
You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).
You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).


==Rank-2 temperaments==
== Rank-2 temperaments ==
==7-note subsets ==
==7-note subsets==
If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L 4s]]:
If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L_4s|3L 4s]]:


L s s L s L s
L s s L s L s
Line 512: Line 301:
which is not a complete moment of symmetry scale in itself, but a subset of one.
which is not a complete moment of symmetry scale in itself, but a subset of one.


== Tetrachordal structure==
==Tetrachordal structure==
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]]. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]]. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.


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