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{{MOS intro}}
{{Mbox|type=notice|text=This page is an in-progress rewrite for a main-namespace page. For the current page, see [[7L 3s]].}}
 
{{MOS intro|Scale Signature=7L 3s}}
Graham Breed has a  page on his website dedicated to 7+3 scales. He proposes calling the large step "t" for "tone", lowercase because the large step is a narrow neutral tone, and the small step "q" for "quartertone", because the small step is often close to a quartertone. (Note that the small step is not a quartertone in every instance of 7+3, so do not take that "q" literally.) Thus we have:
Graham Breed has a  page on his website dedicated to 7+3 scales. He proposes calling the large step "t" for "tone", lowercase because the large step is a narrow neutral tone, and the small step "q" for "quartertone", because the small step is often close to a quartertone. (Note that the small step is not a quartertone in every instance of 7+3, so do not take that "q" literally.) Thus we have:


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 '''dicoid''' (from [[dicot]], an exotemperament) in [[TAMNAMS]].
This MOS is called '''dicoid''' (from [[dicot]], an exotemperament) in [[TAMNAMS]].


==Intervals==
==Intervals==
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.
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.


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.
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!# 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 39: Line 41:
|7L+3s
|7L+3s
|-
|-
| 1
|1
| 7
|7
|perfect 7-step
| perfect 7-step
|5L+2s
|5L+2s
| -1
| -1
Line 60: Line 62:
|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 69: Line 71:
|8
|8
|major 8-step
|major 8-step
| 6L+2s
|6L+2s
| -4
| -4
|2v
|2v
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|5
|5
|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 99: Line 101:
| -7
| -7
|1v
|1v
| minor (1-)step
|minor (1-)step
|1s
|1s
|-
|-
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|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
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|10
|10
|0^
|0^
| augmented unison
|augmented unison
|1L-1s
|1L-1s
| -10
| -10
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|11
|11
|7^
|7^
| augmented 7-step
|augmented 7-step
|6L+1s
|6L+1s
| -11
| -11
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|4L
|4L
| -12
| -12
| 6v
|6v
|diminished 6-step
|diminished 6-step
|3L+3s
|3L+3s
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|2L-1s
|2L-1s
| -13
| -13
| 9w
|9w
|diminished 9-step
|diminished 9-step
|5L+4s
|5L+4s
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|augmented 8-step
|augmented 8-step
|8L+1s
|8L+1s
| -14
| -14
|2w
|2w
|diminished 2-step
|diminished 2-step
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|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
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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="6" rowspan="2" | Generator
! colspan="6" rowspan="2" |Generator
! colspan="2" | Cents
! colspan="2" |Cents
! 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
|-
|-
! Chroma-positive
!Chroma-positive
! Chroma-negative
!Chroma-negative
|-
|-
| 7\10 || || || || || || 840.000 || 360.000 || 1 || 1 || 1.000 ||  
| 7\10|| || || || || ||840.000||360.000||1 || 1||1.000||
|-
|-
| || || || || || 40\57 || 842.105 || 357.895 || 6 || 5 || 1.200 || Restles↑
| || || || || ||40\57||842.105||357.895||6||5||1.200||Restles↑
|-
|-
| || || || || 33\47 || || 842.553 || 357.447 || 5 || 4 || 1.250 ||  
| || || || || 33\47|| ||842.553||357.447||5 || 4||1.250||
|-
|-
| || || || || || 59\84 || 842.857 || 357.143 || 9 || 7 || 1.286 ||  
| || || || || ||59\84||842.857||357.143||9 || 7||1.286||
|-
|-
| || || || 26\37 || || || 843.243 || 356.757 || 4 || 3 || 1.333 ||  
| || || || 26\37|| || ||843.243||356.757||4 || 3||1.333||
|-
|-
| || || || || || 71\101 || 843.564 || 356.436 || 11 || 8 || 1.375 ||  
| || || || || ||71\101||843.564||356.436 ||11 || 8||1.375||
|-
|-
| || || || || 45\64 || || 843.750 || 356.250 || 7 || 5 || 1.400 || Beatles
| || || || || 45\64|| ||843.750||356.250||7||5||1.400||Beatles
|-
|-
| || || || || || 64\91 || 843.956 || 356.044 || 10 || 7 || 1.428 ||  
| || || || || ||64\91||843.956||356.044 ||10 || 7||1.428||
|-
|-
| || || 19\27 || || || || 844.444 || 355.556 || 3 || 2 || 1.500 || L/s = 3/2, suhajira/ringo
| || ||19\27 || || || ||844.444||355.556|| 3 ||2 ||1.500||L/s = 3/2, suhajira/ringo
|-
|-
| || || || || || 69\98 || 844.698 || 355.102 || 11 || 7 || 1.571 ||  
| || || || || ||69\98||844.698||355.102 ||11 || 7||1.571||
|-
|-
| || || || || 50\71 || || 845.070 || 354.930 || 8 || 5 || 1.600 ||  
| || || || || 50\71|| ||845.070||354.930||8 || 5||1.600||
|-
|-
| || || || || || 81\115 || 845.217 || 354.783 || 13 || 8 || 1.625 || Golden suhajira
| || || || || ||81\115||845.217||354.783 ||13||8||1.625||Golden suhajira
|-
|-
| || || || 31\44 || || || 845.455 || 354.545 || 5 || 3 || 1.667 ||  
| || || || 31\44|| || ||845.455||354.545||5 || 3||1.667||
|-
|-
| || || || || || 74\105 || 845.714 || 354.286 || 12 || 7 || 1.714 ||  
| || || || || ||74\105||845.714||354.286 ||12 || 7||1.714||
|-
|-
| || || || || 43\61 || || 845.902 || 354.098 || 7 || 4 || 1.750 ||
| || || || || 43\61|| ||845.902||354.098||7||4 || 1.750 ||
|-
|-
| || || || || || 55\78 || 846.154 || 353.846 || 9 || 5 || 1.800 ||
| || || || || ||55\78||846.154||353.846||9||5|| 1.800 ||
|-
|-
| || 12\17 || || || || || 847.059 || 352.941 || 2 || 1 || 2.000 || Basic dicoid<br>(Generators smaller than this are proper)
| ||12\17 || || || || ||847.059||352.941||2 ||1||2.000||Basic dicoid<br>(Generators smaller than this are proper)
|-
|-
| || || || || || 53\75 || 848.000 || 352.000 || 9 || 4 || 2.250 ||
| || || || || ||53\75||848.000||352.000||9||4 || 2.250||
|-
|-
| || || || || 41\58 || || 848.273 || 351.724 || 7 || 3 || 2.333 ||  
| || || || || 41\58|| ||848.273||351.724||7 || 3||2.333||
|-
|-
| || || || || || 70\99 || 848.485 || 351.515 || 12 || 5 || 2.400 || Hemif/hemififths
| || || || || ||70\99||848.485||351.515 ||12||5||2.400||Hemif/hemififths
|-
|-
| || || || 29\41 || || || 848.780 || 351.220 || 5 || 2 || 2.500 || Mohaha/neutrominant
| || || || 29\41|| || ||848.780||351.220||5||2||2.500||Mohaha/neutrominant
|-
|-
| || || || || || 75\106 || 849.057 || 350.943 || 13 || 5 || 2.600 || Hemif/salsa/karadeniz
| || || || || ||75\106||849.057||350.943 ||13||5||2.600||Hemif/salsa/karadeniz
|-
|-
| || || || || 46\65 || || 849.231 || 350.769 || 8 || 3 || 2.667 || Mohaha/mohamaq
| || || || || 46\65|| ||849.231||350.769||8||3||2.667 || Mohaha/mohamaq
|-
|-
| || || || || || 63\89 || 849.438 || 350.562 || 11 || 4 || 2.750 ||
| || || || || ||63\89||849.438||350.562||11||4 ||2.750 ||
|-
|-
| || || 17\24 || || || || 850.000 || 350.000 || 3 || 1 || 3.000 || L/s = 3/1
| || ||17\24 || || || ||850.000||350.000|| 3 ||1||3.000||L/s = 3/1
|-
|-
| || || || || || 56\79 || 850.633 || 349.367 || 10 || 3 || 3.333 ||
| || || || || ||56\79||850.633||349.367||10||3 || 3.333||
|-
|-
| || || || || 39\55 || || 850.909 || 349.091 || 7 || 2 || 3.500 ||  
| || || || || 39\55|| ||850.909||349.091||7 || 2||3.500||
|-
|-
| || || || || || 61\86 || 851.163 || 348.837 || 11 || 3 || 3.667 ||  
| || || || || ||61\86||851.163||348.837 ||11 || 3||3.667||
|-
|-
| || || || 22\31 || || || 851.613 || 348.387 || 4 || 1 || 4.000 || Mohaha/migration/mohajira
| || || || 22\31|| || ||851.613||348.387||4||1||4.000||Mohaha/migration/mohajira
|-
|-
| || || || || || 49\69 || 852.174 || 347.826 || 9 || 2 || 4.500 ||  
| || || || || ||49\69||852.174||347.826||9 || 2||4.500||
|-
|-
| || || || || 27\38 || || 852.632|| 347.368 || 5 || 1 || 5.000 ||  
| || || || ||27\38|| || 852.632||347.368||5 || 1||5.000||
|-
|-
| || || || || || 32\45 || 853.333 || 346.667 || 6 || 1 || 6.000 || Mohaha/ptolemy
| || || || || ||32\45||853.333||346.667||6||1||6.000 ||Mohaha/ptolemy
|-
|-
| 5\7 || || || || || || 857.143 || 342.857 || 1 || 0 || → inf ||
|5\7|| || || || || ||857.143||342.857||1||0||→ inf||
|}
|}


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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|3L 4s]]:
If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L 4s]]:


L s s L s L s
L s s L s L s
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q t a
q t a
[[Category:10-tone scales]]
Retrieved from "https://en.xen.wiki/w/User:Ganaram_inukshuk/7L_3s"