3L 5s: Difference between revisions

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{{Todo|cleanup|inline=1|text=Adopt zero-indexed intervals/degrees, add scale tree entries from previous page version}}{{Infobox MOS
{{Infobox MOS
| Name = checkertonic
| Name = checkertonic
| Periods = 1
| Periods = 1
Line 8: Line 8:
| Pattern = LsLssLss
| Pattern = LsLssLss
}}
}}
{{MOS intro|Other Names=anti-oneirotonic}}


{{MOS intro|Other Names=anti-oneirotonic}}
== Name ==
== Name==
[[TAMNAMS]] suggests the temperament-agnostic name '''checkertonic''' for this scale.
[[TAMNAMS]] suggests the temperament-agnostic name '''checkertonic''' for this scale.


==Intervals==
== Scale properties ==
{{TAMNAMS use}}
{{TAMNAMS use}}
=== Intervals ===
{{MOS intervals}}
{{MOS intervals}}


== Notation ==
=== Generator chain ===
The [[TAMNAMS]] system is used in this article to refer to {{PAGENAME}} step size ratios and step ratio ranges.
{{MOS genchain}}


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


== Theory ==
==== Proposed mode names ====
In contrast to oneirotonic ([[5L 3s]]), which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth ([[3/2]], usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.
The modes of checkertonic can be named after its sister mos [[5L 3s]] (oneirotonic). {{u|R-4981}} has also proposed names based on {{w|grand chess}} pieces.
 
{{MOS modes
=== Low harmonic entropy scales ===
| Table Headers=
There are two significant harmonic entropy minima with this MOS pattern:
Anti-modes of 5L 3s $
 
Grand chess names<sup>[proposed]</sup>
* [[Sensipent family|Sensi]], in which the generator is a 9/7, two of them make a 5/3, and seven of them make a 3/2, which is proper.
| Table Entries=
* [[Meantone family #Squares|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, which is improper.
Anti-Sarnathian (sar-NA(H)TH-iən) $
 
King $
==Tuning ranges==
Anti-Hlanithian (lə-NITH-iən) $
===Simple tunings===
Queen $
{{MOS tunings}}
Anti-Kadathian (kə-DA(H)TH-iən) $
 
Marshall $
===Parasoft tunings===
Anti-Mnarian (mə-NA(I)R-iən) $
Parasoft tunings (step ratios 4:3 to 3:2) are associated with [[sensi]] tempermament.
Cardinal $
{{MOS tunings|Step Ratios=Parasoft|JI Ratios=Subgroup: 2.3.5.7.13; Int Limit: 50; Tenney Height: 8; Complements Only: 1|Tolerance=10}}
Anti-Ultharian (ul-THA(I)R-iən) $
 
Rook $
==Modes==
Anti-Celephaïsian (kel-ə-FAY-zhən) $
{{MOS mode degrees}}
Bishop $
===Proposed Names===
Anti-Illarnekian (ill-ar-NEK-iən) $
The modes of checkertonic can be named after its sister MOS [[5L 3s]] (oneirotonic). [[User:R-4981|R-4981]] has also proposed names based on [[wikipedia:Grand_Chess|grand chess]] pieces.
Knight $
{{MOS modes|Table Headers=Anti-modes of 5L 3s;Grand chess names<sup>[proposed]</sup>|Table Entries=Anti-Sarnathian (sar-NA(H)TH-iən);
Anti-Dylathian (də-LA(H)TH-iən) $
King;
Pawn $
Anti-Hlanithian (lə-NITH-iən);
}}
Queen;
Anti-Kadathian (kə-DA(H)TH-iən);
Marshall;
Anti-Mnarian (mə-NA(I)R-iən);
Cardinal;
Anti-Ultharian (ul-THA(I)R-iən);
Rook;
Anti-Celephaïsian (kel-ə-FAY-zhən);
Bishop;
Anti-Illarnekian (ill-ar-NEK-iən);
Knight;
Anti-Dylathian (də-LA(H)TH-iən);
Pawn;}}
The order of modes on the white keys JKLMNOPQ are:
The order of modes on the white keys JKLMNOPQ are:


Line 71: Line 61:


{| class="wikitable"
{| class="wikitable"
|+ style="font-size: 105%;" | Scale degrees (on J, {{nowrap|sLssLsLs {{=}} JKLMNOPQ}})
|-
|-
|+Scale degrees (on J, sLssLsLs = JKLMNOPQ)
! [[UDP]]
|-
! Anti-modes of 5L 3s
![[UDP]]
! Chess-based names
!Anti-modes of 5L 3s
! Step pattern
!Chess-based names
!Step pattern
! 1
! 1
!2
! 2
!3
! 3
! 4
! 4
!5
! 5
!6
! 6
!7
! 7
!8
! 8
!(9)
! (9)
|-
|-
|<nowiki>7|0</nowiki>
| 7{{pipe}}0
|Anti-Sarnathian
| Anti-Sarnathian
|King
| King
|LsLssLss
| LsLssLss
| J
| J
|K&
| K&amp;
|L
| L
|M&
| M&amp;
|N&
| N&amp;
|O
| O
|P&
| P&amp;
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>6|1</nowiki>
| 6{{pipe}}1
|Anti-Hlanithian
| Anti-Hlanithian
|Queen
| Queen
|LssLsLss
| LssLsLss
|J
| J
|K&
| K&amp;
| L
| L
| M
| M
| N&
| N&amp;
|O
| O
|P&
| P&amp;
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>5|2</nowiki>
| 5{{pipe}}2
|Anti-Kadathian
| Anti-Kadathian
|Marshall
| Marshall
|LssLssLs
| LssLssLs
|J
| J
| K&
| K&amp;
|L
| L
|M
| M
|N&
| N&amp;
|O
| O
|P
| P
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>4|3</nowiki>
| 4{{pipe}}3
|Anti-Mnarian
| Anti-Mnarian
|Cardinal
| Cardinal
|sLsLssLs
| sLsLssLs
|J
| J
|K
| K
|L
| L
|M
| M
|N&
| N&amp;
|O
| O
|P
| P
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>3|4</nowiki>
| 3{{pipe}}4
| Anti-Ultharian
| Anti-Ultharian
|Rook
| Rook
|sLssLsLs
| sLssLsLs
|J
| J
|K
| K
|L
| L
|M
| M
| N
| N
|O
| O
|P
| P
|Q
| Q
|(J)
| (J)
|-
|-
|<nowiki>2|5</nowiki>
| 2{{pipe}}5
|Anti-Celephaïsian
| Anti-Celephaïsian
|Bishop
| Bishop
|sLssLssL
| sLssLssL
|J
| J
|K
| K
|L
| L
|M
| M
|N
| N
| O
| O
|P
| P
|Q@
| Q@
|(J)
| (J)
|-
|-
|<nowiki>1|6</nowiki>
| 1{{pipe}}6
|Anti-Illarnekian
| Anti-Illarnekian
|Knight
| Knight
|ssLsLssL
| ssLsLssL
|J
| J
|K
| K
|L@
| L@
|M
| M
|N
| N
|O
| O
|P
| P
|Q@
| Q@
|(J)
| (J)
|-
|-
|<nowiki>0|7</nowiki>
| 0{{pipe}}7
|Anti-Dylathian
| Anti-Dylathian
|Pawn
| Pawn
|ssLssLsL
| ssLssLsL
|J
| J
|K
| K
|L@
| L@
|M
| M
|N
| N
|O@
| O@
|P
| P
|Q@
| Q@
|(J)
| (J)
|}
|}


==Temperaments==
== Notation ==
The [[TAMNAMS]] system is used in this article to refer to {{PAGENAME}} step size ratios and step ratio ranges.
 
The notation used in this article is JKLMNOPQJ = sLssLsLs (Anti-Ultharian), &amp;/@ = up/down by chroma.
 
== Theory ==
In contrast to oneirotonic ([[5L&nbsp;3s]]), which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth ([[3/2]], usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.
 
=== Low harmonic entropy scales ===
There are two significant harmonic entropy minima with this MOS pattern:
 
* [[Sensipent family|Sensi]], in which the generator is a 9/7, two of them make a 5/3, and seven of them make a 3/2, which is proper.
* [[Meantone family #Squares|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, which is improper.
 
== Tuning ranges ==
=== Simple tunings ===
{{MOS tunings}}
 
=== Parasoft tunings ===
Parasoft tunings (step ratios 4:3 to 3:2) are associated with [[sensi]] tempermament.
{{MOS tunings|Step Ratios=Parasoft|JI Ratios=Subgroup: 2.3.5.7.13; Int Limit: 50; Tenney Height: 8; Complements Only: 1|Tolerance=10}}
 
== Temperaments ==
The major temperaments in this area are:
The major temperaments in this area are:
*[[Sensi]] (Parasoft checkertonic)
* [[Sensi]] (Parasoft checkertonic)
*[[Squares]] (Parahard checkertonic)
* [[Squares]] (Parahard checkertonic)


== Music ==
== Music ==
* [[Uncreative Name]], [https://www.youtube.com/watch?v=XZ3zB3EDKOM The Nachtlandian Somersault] (19edo)
; [[Uncreative Name]]
* [https://www.youtube.com/watch?v=XZ3zB3EDKOM ''The Nachtlandian Somersault''] (19edo)


== Scale tree ==
== Scale tree ==
Generator ranges:
Generator ranges:
* Chroma-positive generator: 750 cents (5\8) to 800 cents (2\3)
* Chroma-positive generator: 750{{c}} (5\8) to 800{{c}} (2\3)
* Chroma-negative generator: 400 cents (1\3) to 450 cents (3\8)
* Chroma-negative generator: 400{{c}} (1\3) to 450{{c}} (3\8)
{{Scale tree}}
{{MOS tuning spectrum
 
| 7/5 = [[Sensi]] (optimal around here)
[[Category:8-tone scales]]
| 11/7 = [[Clyde]]
[[Category:checkertonic]]
| 13/8 = Golden [[sentry]] (759.4078{{c}})
| 13/5 = Unnamed golden tuning (768.8815{{c}})
| 11/4 = [[Hamity]]
| 7/2 = [[Squares]] (optimal around here)
| 6/1 = [[Roman]]↓, [[hocus]]↓
}}

Latest revision as of 22:25, 27 April 2025

↖ 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, named checkertonic in TAMNAMS (also known as anti-oneirotonic), is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 5 small steps, repeating every octave. Generators that produce this scale range from 750 ¢ to 800 ¢, or from 400 ¢ to 450 ¢.

Name

TAMNAMS suggests the temperament-agnostic name checkertonic for this scale.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 3L 5s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-checkstep Perfect 0-checkstep P0chks 0 0.0 ¢
1-checkstep Minor 1-checkstep m1chks s 0.0 ¢ to 150.0 ¢
Major 1-checkstep M1chks L 150.0 ¢ to 400.0 ¢
2-checkstep Minor 2-checkstep m2chks 2s 0.0 ¢ to 300.0 ¢
Major 2-checkstep M2chks L + s 300.0 ¢ to 400.0 ¢
3-checkstep Perfect 3-checkstep P3chks L + 2s 400.0 ¢ to 450.0 ¢
Augmented 3-checkstep A3chks 2L + s 450.0 ¢ to 800.0 ¢
4-checkstep Minor 4-checkstep m4chks L + 3s 400.0 ¢ to 600.0 ¢
Major 4-checkstep M4chks 2L + 2s 600.0 ¢ to 800.0 ¢
5-checkstep Diminished 5-checkstep d5chks L + 4s 400.0 ¢ to 750.0 ¢
Perfect 5-checkstep P5chks 2L + 3s 750.0 ¢ to 800.0 ¢
6-checkstep Minor 6-checkstep m6chks 2L + 4s 800.0 ¢ to 900.0 ¢
Major 6-checkstep M6chks 3L + 3s 900.0 ¢ to 1200.0 ¢
7-checkstep Minor 7-checkstep m7chks 2L + 5s 800.0 ¢ to 1050.0 ¢
Major 7-checkstep M7chks 3L + 4s 1050.0 ¢ to 1200.0 ¢
8-checkstep Perfect 8-checkstep P8chks 3L + 5s 1200.0 ¢

Generator chain

Generator chain of 3L 5s
Bright gens Scale degree Abbrev.
10 Augmented 2-checkdegree A2chkd
9 Augmented 5-checkdegree A5chkd
8 Augmented 0-checkdegree A0chkd
7 Augmented 3-checkdegree A3chkd
6 Major 6-checkdegree M6chkd
5 Major 1-checkdegree M1chkd
4 Major 4-checkdegree M4chkd
3 Major 7-checkdegree M7chkd
2 Major 2-checkdegree M2chkd
1 Perfect 5-checkdegree P5chkd
0 Perfect 0-checkdegree
Perfect 8-checkdegree		 P0chkd
P8chkd
−1 Perfect 3-checkdegree P3chkd
−2 Minor 6-checkdegree m6chkd
−3 Minor 1-checkdegree m1chkd
−4 Minor 4-checkdegree m4chkd
−5 Minor 7-checkdegree m7chkd
−6 Minor 2-checkdegree m2chkd
−7 Diminished 5-checkdegree d5chkd
−8 Diminished 8-checkdegree d8chkd
−9 Diminished 3-checkdegree d3chkd
−10 Diminished 6-checkdegree d6chkd

Modes

Scale degrees of the modes of 3L 5s
UDP Cyclic
order 		Step
pattern 		Scale degree (checkdegree)
0 1 2 3 4 5 6 7 8
7|0 1 LsLssLss Perf. Maj. Maj. Aug. Maj. Perf. Maj. Maj. Perf.
6|1 6 LssLsLss Perf. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Perf.
5|2 3 LssLssLs Perf. Maj. Maj. Perf. Maj. Perf. Min. Maj. Perf.
4|3 8 sLsLssLs Perf. Min. Maj. Perf. Maj. Perf. Min. Maj. Perf.
3|4 5 sLssLsLs Perf. Min. Maj. Perf. Min. Perf. Min. Maj. Perf.
2|5 2 sLssLssL Perf. Min. Maj. Perf. Min. Perf. Min. Min. Perf.
1|6 7 ssLsLssL Perf. Min. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 4 ssLssLsL Perf. Min. Min. Perf. Min. Dim. Min. Min. Perf.

Proposed mode names

The modes of checkertonic can be named after its sister mos 5L 3s (oneirotonic). R-4981 has also proposed names based on grand chess pieces.

Modes of 3L 5s
UDP Cyclic
order		 Step
pattern		 Anti-modes of 5L 3s Grand chess names[proposed]
7|0 1 LsLssLss Anti-Sarnathian (sar-NA(H)TH-iən) King
6|1 6 LssLsLss Anti-Hlanithian (lə-NITH-iən) Queen
5|2 3 LssLssLs Anti-Kadathian (kə-DA(H)TH-iən) Marshall
4|3 8 sLsLssLs Anti-Mnarian (mə-NA(I)R-iən) Cardinal
3|4 5 sLssLsLs Anti-Ultharian (ul-THA(I)R-iən) Rook
2|5 2 sLssLssL Anti-Celephaïsian (kel-ə-FAY-zhən) Bishop
1|6 7 ssLsLssL Anti-Illarnekian (ill-ar-NEK-iən) Knight
0|7 4 ssLssLsL Anti-Dylathian (də-LA(H)TH-iən) Pawn

The order of modes on the white keys JKLMNOPQ are:

  • J Anti-Ultharian, Rook
  • K Anti-Hlanithian, Queen
  • L Anti-Illarnekian, Knight
  • M Anti-Mnarian, Cardinal
  • N Anti-Sarnathian, King
  • O Anti-Celephaïsian, Bishop
  • P Anti-Kadathian, Marshall
  • Q Anti-Dylathian, Pawn
Scale degrees (on J, sLssLsLs = JKLMNOPQ)
UDP Anti-modes of 5L 3s Chess-based names Step pattern 1 2 3 4 5 6 7 8 (9)
7|0 Anti-Sarnathian King LsLssLss J K& L M& N& O P& Q (J)
6|1 Anti-Hlanithian Queen LssLsLss J K& L M N& O P& Q (J)
5|2 Anti-Kadathian Marshall LssLssLs J K& L M N& O P Q (J)
4|3 Anti-Mnarian Cardinal sLsLssLs J K L M N& O P Q (J)
3|4 Anti-Ultharian Rook sLssLsLs J K L M N O P Q (J)
2|5 Anti-Celephaïsian Bishop sLssLssL J K L M N O P Q@ (J)
1|6 Anti-Illarnekian Knight ssLsLssL J K L@ M N O P Q@ (J)
0|7 Anti-Dylathian Pawn ssLssLsL J K L@ M N O@ P Q@ (J)

Notation

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.

Theory

In contrast to oneirotonic (5L 3s), which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth (3/2, usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.

Low harmonic entropy scales

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, which is proper.
  • 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, which is improper.

Tuning ranges

Simple tunings

Simple Tunings of 3L 5s
Scale degree Abbrev. Basic (2:1)
11edo 		Hard (3:1)
14edo 		Soft (3:2)
19edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-checkdegree P0chkd 0\11 0.0 0\14 0.0 0\19 0.0
Minor 1-checkdegree m1chkd 1\11 109.1 1\14 85.7 2\19 126.3
Major 1-checkdegree M1chkd 2\11 218.2 3\14 257.1 3\19 189.5
Minor 2-checkdegree m2chkd 2\11 218.2 2\14 171.4 4\19 252.6
Major 2-checkdegree M2chkd 3\11 327.3 4\14 342.9 5\19 315.8
Perfect 3-checkdegree P3chkd 4\11 436.4 5\14 428.6 7\19 442.1
Augmented 3-checkdegree A3chkd 5\11 545.5 7\14 600.0 8\19 505.3
Minor 4-checkdegree m4chkd 5\11 545.5 6\14 514.3 9\19 568.4
Major 4-checkdegree M4chkd 6\11 654.5 8\14 685.7 10\19 631.6
Diminished 5-checkdegree d5chkd 6\11 654.5 7\14 600.0 11\19 694.7
Perfect 5-checkdegree P5chkd 7\11 763.6 9\14 771.4 12\19 757.9
Minor 6-checkdegree m6chkd 8\11 872.7 10\14 857.1 14\19 884.2
Major 6-checkdegree M6chkd 9\11 981.8 12\14 1028.6 15\19 947.4
Minor 7-checkdegree m7chkd 9\11 981.8 11\14 942.9 16\19 1010.5
Major 7-checkdegree M7chkd 10\11 1090.9 13\14 1114.3 17\19 1073.7
Perfect 8-checkdegree P8chkd 11\11 1200.0 14\14 1200.0 19\19 1200.0

Parasoft tunings

Parasoft tunings (step ratios 4:3 to 3:2) are associated with sensi tempermament.

Parasoft Tunings of 3L 5s
Scale degree Abbrev. Supersoft (4:3)
27edo 		7:5
46edo 		Soft (3:2)
19edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-checkdegree P0chkd 0\27 0.0 0\46 0.0 0\19 0.0
Minor 1-checkdegree m1chkd 3\27 133.3 5\46 130.4 2\19 126.3
Major 1-checkdegree M1chkd 4\27 177.8 7\46 182.6 3\19 189.5
Minor 2-checkdegree m2chkd 6\27 266.7 10\46 260.9 4\19 252.6
Major 2-checkdegree M2chkd 7\27 311.1 12\46 313.0 5\19 315.8
Perfect 3-checkdegree P3chkd 10\27 444.4 17\46 443.5 7\19 442.1
Augmented 3-checkdegree A3chkd 11\27 488.9 19\46 495.7 8\19 505.3
Minor 4-checkdegree m4chkd 13\27 577.8 22\46 573.9 9\19 568.4
Major 4-checkdegree M4chkd 14\27 622.2 24\46 626.1 10\19 631.6
Diminished 5-checkdegree d5chkd 16\27 711.1 27\46 704.3 11\19 694.7
Perfect 5-checkdegree P5chkd 17\27 755.6 29\46 756.5 12\19 757.9
Minor 6-checkdegree m6chkd 20\27 888.9 34\46 887.0 14\19 884.2
Major 6-checkdegree M6chkd 21\27 933.3 36\46 939.1 15\19 947.4
Minor 7-checkdegree m7chkd 23\27 1022.2 39\46 1017.4 16\19 1010.5
Major 7-checkdegree M7chkd 24\27 1066.7 41\46 1069.6 17\19 1073.7
Perfect 8-checkdegree P8chkd 27\27 1200.0 46\46 1200.0 19\19 1200.0

Temperaments

The major temperaments in this area are:

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

Music

Uncreative Name

Scale tree

Generator ranges:

  • Chroma-positive generator: 750 ¢ (5\8) to 800 ¢ (2\3)
  • Chroma-negative generator: 400 ¢ (1\3) to 450 ¢ (3\8)
Scale tree and tuning spectrum of 3L 5s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
5\8 750.000 450.000 1:1 1.000 Equalized 3L 5s
27\43 753.488 446.512 6:5 1.200
22\35 754.286 445.714 5:4 1.250
39\62 754.839 445.161 9:7 1.286
17\27 755.556 444.444 4:3 1.333 Supersoft 3L 5s
46\73 756.164 443.836 11:8 1.375
29\46 756.522 443.478 7:5 1.400 Sensi (optimal around here)
41\65 756.923 443.077 10:7 1.429
12\19 757.895 442.105 3:2 1.500 Soft 3L 5s
43\68 758.824 441.176 11:7 1.571 Clyde
31\49 759.184 440.816 8:5 1.600
50\79 759.494 440.506 13:8 1.625 Golden sentry (759.4078 ¢)
19\30 760.000 440.000 5:3 1.667 Semisoft 3L 5s
45\71 760.563 439.437 12:7 1.714
26\41 760.976 439.024 7:4 1.750
33\52 761.538 438.462 9:5 1.800
7\11 763.636 436.364 2:1 2.000 Basic 3L 5s
Scales with tunings softer than this are proper
30\47 765.957 434.043 9:4 2.250
23\36 766.667 433.333 7:3 2.333
39\61 767.213 432.787 12:5 2.400
16\25 768.000 432.000 5:2 2.500 Semihard 3L 5s
41\64 768.750 431.250 13:5 2.600 Unnamed golden tuning (768.8815 ¢)
25\39 769.231 430.769 8:3 2.667
34\53 769.811 430.189 11:4 2.750 Hamity
9\14 771.429 428.571 3:1 3.000 Hard 3L 5s
29\45 773.333 426.667 10:3 3.333
20\31 774.194 425.806 7:2 3.500 Squares (optimal around here)
31\48 775.000 425.000 11:3 3.667
11\17 776.471 423.529 4:1 4.000 Superhard 3L 5s
24\37 778.378 421.622 9:2 4.500
13\20 780.000 420.000 5:1 5.000
15\23 782.609 417.391 6:1 6.000 Roman↓, hocus↓
2\3 800.000 400.000 1:0 → ∞ Collapsed 3L 5s
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