3L 5s: Difference between revisions

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{{Infobox MOS
{{Infobox MOS
| Name = sensoid
| Name = checkertonic
| Periods = 1
| Periods = 1
| nLargeSteps = 3
| nLargeSteps = 3
Line 19: Line 19:


== Names ==
== Names ==
The [[TAMNAMS]] name for 3L 5s is '''sensoid''' (named after the regular temperament [[sensi]]).
The [[TAMNAMS]] name for 3L 5s is '''checkertonic''' (named after the regular temperament [[sensi]]).


== Intervals ==
== Intervals ==
Note: In TAMNAMS, a k-step interval class in sensoid may be called a "k-step", "k-mosstep", or "k-senstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
Note: In TAMNAMS, a k-step interval class in checkertonic may be called a "k-step", "k-mosstep", or "k-checkstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.


== Tuning ranges ==
== Tuning ranges ==
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=== Parasoft ===
=== Parasoft ===
Parasoft sensoid is the narrow region between 7\19 (442.1¢) and 10\27 (444.4¢).
Parasoft checkertonic is the narrow region between 7\19 (442.1¢) and 10\27 (444.4¢).


Sortable table of major and minor intervals in parasoft sensoid tunings:
Sortable table of major and minor intervals in parasoft checkertonic tunings:


{| class="wikitable right-2 right-3 right-4 sortable "
{| class="wikitable right-2 right-3 right-4 sortable "
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== Modes ==
== Modes ==
Sensoid modes can be named by prefixing ''anti-'' to their counterpart modes in the MOS sister [[oneirotonic]].
checkertonic modes can be named by prefixing ''anti-'' to their counterpart modes in the MOS sister [[oneirotonic]].


# Anti-Sarnathian (sar-NA(H)TH-iən): LsLssLss  
# Anti-Sarnathian (sar-NA(H)TH-iən): LsLssLss  
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== Temperaments ==
== Temperaments ==
The major temperaments in this area are:
The major temperaments in this area are:
* [[Sensi]] (Parasoft sensoid)
* [[Sensi]] (Parasoft checkertonic)
* [[Squares]] (Parahard sensoid)
* [[Squares]] (Parahard checkertonic)


== Scale tree ==
== Scale tree ==
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| || || || || 31\49 || || 759.184 || 8 || 5 || 1.600 ||  
| || || || || 31\49 || || 759.184 || 8 || 5 || 1.600 ||  
|-
|-
| || || || || || 50\79 || 759.494 || 13 || 8 || 1.625 || Golden sensoid/[[sentry]] (759.4078¢)
| || || || || || 50\79 || 759.494 || 13 || 8 || 1.625 || Golden checkertonic/[[sentry]] (759.4078¢)
|-
|-
| || || || 19\30 || || || 760.000 || 5 || 3 || 1.667 ||  
| || || || 19\30 || || || 760.000 || 5 || 3 || 1.667 ||  
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| || || || || || 33\52 || 761.538 || 9 || 5 || 1.800 ||  
| || || || || || 33\52 || 761.538 || 9 || 5 || 1.800 ||  
|-
|-
| || 7\11 || || || || || 763.636 || 2 || 1 || 2.000 || Basic sensoid <br>(Generators smaller than this are proper)
| || 7\11 || || || || || 763.636 || 2 || 1 || 2.000 || Basic checkertonic <br>(Generators smaller than this are proper)
|-
|-
| || || || || || 30\47 || 765.957 || 9 || 4 || 2.250 ||  
| || || || || || 30\47 || 765.957 || 9 || 4 || 2.250 ||  
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[[Category:8-tone scales]]
[[Category:8-tone scales]]
[[Category:Sensoid]]
[[Category:checkertonic]]

Revision as of 19:35, 14 August 2022

↖ 2L 4s ↑ 3L 4s 4L 4s ↗
← 2L 5s 3L 5s 4L 5s →
↙ 2L 6s ↓ 3L 6s 4L 6s ↘ 
┌╥┬╥┬┬╥┬┬┐
│║│║││║│││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLssLss
ssLssLsL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 5\8 to 2\3 (750.0 ¢ to 800.0 ¢)
Dark 1\3 to 3\8 (400.0 ¢ to 450.0 ¢)
TAMNAMS information
Name checkertonic
Prefix check-
Abbrev. chk
Related MOS scales
Parent 3L 2s
Sister 5L 3s
Daughters 8L 3s, 3L 8s
Neutralized 6L 2s
2-Flought 11L 5s, 3L 13s
Equal tunings
Equalized (L:s = 1:1) 5\8 (750.0 ¢)
Supersoft (L:s = 4:3) 17\27 (755.6 ¢)
Soft (L:s = 3:2) 12\19 (757.9 ¢)
Semisoft (L:s = 5:3) 19\30 (760.0 ¢)
Basic (L:s = 2:1) 7\11 (763.6 ¢)
Semihard (L:s = 5:2) 16\25 (768.0 ¢)
Hard (L:s = 3:1) 9\14 (771.4 ¢)
Superhard (L:s = 4:1) 11\17 (776.5 ¢)
Collapsed (L:s = 1:0) 2\3 (800.0 ¢)

3L 5s refers to the structure of octave-equivalent MOS scales with generators ranging from 1\3 (one degree of 3edo = 400¢) to 3\8 (three degrees of 8edo = 450¢). In the case of 8edo, L and s are the same size; in the case of 3edo, s becomes so small it disappears (and all that remains are the three equal L's). The pattern is also named antioneirotonic because it is the oneirotonic (5L 3s) MOS pattern with large and small steps switched.

There are two significant harmonic entropy minima with this MOS pattern. Sensi, in which the generator is a 9/7, two of them make a 5/3, and seven of them make a 3/2, is the proper one. Squares, in which the generator is also a 9/7, but two of them make an 18/11 and four of them make a 4/3, is improper.

Standing assumptions

The TAMNAMS system is used in this article to refer to 3L 5s step size ratios and step ratio ranges.

The notation used in this article is JKLMNOPQJ = sLssLsLs (Anti-Ultharian), &/@ = up/down by chroma.

Names

The TAMNAMS name for 3L 5s is checkertonic (named after the regular temperament sensi).

Intervals

Note: In TAMNAMS, a k-step interval class in checkertonic may be called a "k-step", "k-mosstep", or "k-checkstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.

Tuning ranges

Simple tunings

Degree Size in 11edo (basic) Size in 14edo (hard) Size in 19edo (soft) Note name on J #Gens up
min. sen2nd 1\11, 109.1 1\14, 85.7 2\19, 126.3 K +3
maj. sen2nd 2\11, 218.2 3\14, 257.1 3\19, 189.5 K& -5
min. sen3rd 2\11, 218.2 2\14, 171.4 4\19, 252.6 L@ +6
maj. sen3rd 3\11, 327.3 4\14, 342.9 5\19, 315.8 L -2
perf. sen4th 4\11, 436.4 5\14, 428.6 7\19, 442.1 M +1
aug. sen4th 5\11, 545.5 7\14, 600.0 8\19, 505.3 M& -7
min. sen5th 5\11, 545.5 6\14, 514.3 9\19, 568.4 N +4
maj. sen5th 6\11, 656.6 8\14, 685.7 10\19, 631.6 N& -4
dim. sen6th 6\11, 656.6 7\14, 600.0 11\19, 694.7 O@ +7
perf. sen6th 7\11, 763.6 8\14, 771.4 12\19, 757.9 O -1
min. sen7th 8\11, 872.7 10\14, 857.1 14\19, 884.2 P +2
maj. sen7th 9\11, 981.8 12\14, 1028.6 15\19, 947.4 P& -6
min. sen8th 9\11, 981.8 11\14, 942.9 16\19, 1010.5 Q@ +5
maj. sen8th 10\11, 1090.9 13\14, 1114.3 17\19, 1073.7 Q -3

Parasoft

Parasoft checkertonic is the narrow region between 7\19 (442.1¢) and 10\27 (444.4¢).

Sortable table of major and minor intervals in parasoft checkertonic tunings:

Degree Size in 19edo (soft) Size in 27edo (supersoft) Size in 46edo Note name on J Approximate ratios #Gens up
unison 0\19, 0.00 0\27, 0.00 0\46, 0.00 J 1/1 0
min. sen2nd 2\19, 126.3 3\27, 133.3 5\46, 130.4 K 14/13 +3
maj. sen2nd 3\19, 189.5 4\27, 177.8 7\46, 182.6 K& 10/9 -5
min. sen3rd 4\19, 252.6 6\27, 266.7 10\46, 260.9 L@ 7/6 +6
maj. sen3rd 5\19, 315.8 7\27, 311.1 12\46, 313.0 L 6/5 -2
perf. sen4th 7\19, 442.1 10\27, 444.4 17\46, 443.5 M 9/7, 13/10 +1
aug. sen4th 8\19, 505.3 11\27, 488.9 19\46, 495.7 M& 4/3 -7
min. sen5th 9\19, 568.4 13\27, 577.8 22\46, 573.9 N 7/5, 18/13 +4
maj. sen5th 10\19, 631.6 14\27, 622.2 24\46, 626.1 N& 10/7, 13/9 -4
dim. sen6th 11\19, 694.7 16\27, 711.1 27\46, 704.3 O@ 3/2 +7
perf. sen6th 12\19, 757.9 17\27, 755.6 20\46, 756.5 O 14/9, 20/13 -1
min. sen7th 14\19, 884.2 20\27, 888.9 34\46, 887.0 P 5/3 +2
maj. sen7th 15\19, 947.4 21\27, 933.3 36\46, 939.1 P& 12/7 -6
min. sen8th 16\19, 1010.5 23\27, 1022.2 39\46, 1017.4 Q@ 9/5 +5
maj. sen8th 17\19, 1073.7 24\27, 1066.7 41\46, 1069.6 Q 13/7 -3

Tunings in this region have a regular temperament interpretation called Sensi.

Modes

checkertonic modes can be named by prefixing anti- to their counterpart modes in the MOS sister oneirotonic.

  1. Anti-Sarnathian (sar-NA(H)TH-iən): LsLssLss
  2. Anti-Hlanithian (lə-NITH-iən): LssLsLss
  3. Anti-Kadathian (kə-DA(H)TH-iən): LssLssLs
  4. Anti-Mnarian (mə-NA(I)R-iən): sLsLssLs
  5. Anti-Ultharian (ul-THA(I)R-iən): sLssLsLs
  6. Anti-Celephaïsian (kel-ə-FAY-zhən): sLssLssL
  7. Anti-Illarnekian (ill-ar-NEK-iən): ssLsLssL
  8. Anti-Dylathian (də-LA(H)TH-iən): ssLssLsL

The modes on the white keys JKLMNOPQJ are:

  • J Anti-Ultharian
  • K Anti-Hlanithian
  • L Anti-Illarnekian
  • M Anti-Mnarian
  • N Anti-Sarnathian
  • O Anti-Celephaïsian
  • P Anti-Kadathian
  • Q Anti-Dylathian
Table of modes (based on J, from brightest to darkest)
Mode 1 2 3 4 5 6 7 8 (9)
Anti-Sarnathian J K& L M& N& O P& Q (J)
Anti-Hlanithian J K& L M N& O P& Q (J)
Anti-Kadathian J K& L M N& O P Q (J)
Anti-Mnarian J K L M N& O P Q (J)
Anti-Ultharian J K L M N O P Q (J)
Anti-Celephaïsian J K L M N O P Q@ (J)
Anti-Illarnekian J K L@ M N O P Q@ (J)
Anti-Dylathian J K L@ M N O@ P Q@ (J)

Temperaments

The major temperaments in this area are:

  • Sensi (Parasoft checkertonic)
  • Squares (Parahard checkertonic)

Scale tree

Generator ranges:

  • Chroma-positive generator: 750 cents (5\8) to 800 cents (2\3)
  • Chroma-negative generator: 400 cents (1\3) to 450 cents (3\8)
Generator Cents L s L/s Comments
5\8 750.000 1 1 1.000
27\43 753.488 6 5 1.200
22\35 754.286 5 4 1.250
39\62 754.839 9 7 1.286
17\27 755.556 4 3 1.333
46\73 756.164 11 8 1.375
29\46 756.522 7 5 1.400 Sensi is in this region
41\65 756.923 10 7 1.429
12\19 757.895 3 2 1.500
43\68 758.824 11 7 1.571 Clyde
31\49 759.184 8 5 1.600
50\79 759.494 13 8 1.625 Golden checkertonic/sentry (759.4078¢)
19\30 760.000 5 3 1.667
45\71 760.563 12 7 1.714
26\41 760.976 7 4 1.750
33\52 761.538 9 5 1.800
7\11 763.636 2 1 2.000 Basic checkertonic
(Generators smaller than this are proper)
30\47 765.957 9 4 2.250
23\36 766.667 7 3 2.333
39\61 767.213 12 5 2.400
16\25 768.000 5 2 2.500
41\64 768.750 13 5 2.600 Unnamed golden tuning (768.8815¢)
25\39 769.231 8 3 2.667
34\53 769.811 11 4 2.750 Hamity
9\14 771.429 3 1 3.000
29\45 773.333 10 3 3.333
20\31 774.194 7 2 3.500 Squares is in this region
31\48 775.000 11 3 3.667
11\17 776.471 4 1 4.000
24\37 778.378 9 2 4.500
13\20 780.000 5 1 5.000
15\23 782.609 6 1 6.000 Roman↓, Hocus↓
2\3 800.000 1 0 → inf
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