14L 22s (12/1-equivalent)

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This page presents a novelty topic. It may contain ideas which are less likely to find practical applications in xenharmonic music, or numbers that are impractically large, exceedingly complex, or chosen arbitrarily.

Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks.

Assume hemipyth[10] nominal names and intervals (and zero-indexing) unless otherwise stated. This article is meant to apply MMTM's theory on this scale, but it is attempted to be explained better here.

↖ 13L 21s⟨12/1⟩ ↑ 14L 21s⟨12/1⟩ 15L 21s⟨12/1⟩ ↗
← 13L 22s⟨12/1⟩ 14L 22s <12/1> 15L 22s⟨12/1⟩ →
↙ 13L 23s⟨12/1⟩ ↓ 14L 23s⟨12/1⟩ 15L 23s⟨12/1⟩ ↘ 
┌╥┬╥┬┬╥┬╥┬┬╥┬╥┬┬╥┬┬╥┬╥┬┬╥┬╥┬┬╥┬╥┬┬╥┬┬┐
│║│║││║│║││║│║││║││║│║││║│║││║│║││║│││
││││││││││││││││││││││││││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLssLsLssLsLssLssLsLssLsLssLsLssLss
ssLssLsLssLsLssLsLssLssLsLssLsLssLsL
Equave 12/1 (4302.0 ¢)
Period 1\2 (2151.0 ¢)
Generator size(ed12/1)
Bright 5\36 to 2\14 (597.5 ¢ to 614.6 ¢)
Dark 5\14 to 13\36 (1536.4 ¢ to 1553.5 ¢)
Related MOS scales
Parent 14L 8s⟨12/1⟩
Sister 22L 14s⟨12/1⟩
Daughters 36L 14s⟨12/1⟩, 14L 36s⟨12/1⟩
Neutralized 28L 8s⟨12/1⟩
2-Flought 50L 22s⟨12/1⟩, 14L 58s⟨12/1⟩
Equal tunings(ed12/1)
Equalized (L:s = 1:1) 5\36 (597.5 ¢)
Supersoft (L:s = 4:3) 17\122 (599.5 ¢)
Soft (L:s = 3:2) 12\86 (600.3 ¢)
Semisoft (L:s = 5:3) 19\136 (601.0 ¢)
Basic (L:s = 2:1) 7\50 (602.3 ¢)
Semihard (L:s = 5:2) 16\114 (603.8 ¢)
Hard (L:s = 3:1) 9\64 (605.0 ¢)
Superhard (L:s = 4:1) 11\78 (606.7 ¢)
Collapsed (L:s = 1:0) 2\14 (614.6 ¢)

14L 22s <12/1>, also pochhammeroid (see below), colianexoid, greater f-enhar electric or greater f-enhar smitonic is a MOS scale. The notation "<12/1>" means the period of the MOS is 12/1, disambiguating it from octave-repeating 14L 22s. The name of the period interval of this scale is called the oktokaidekatave, and the . It is also equivalent to 7L 11s <√12>. Its basic tuning is 50ed12 or 25ed√12. However, the √12-based form will be used for most of this article as it is far more practical and is the original form of the scale when it was discovered.

The generator range is 597 to 615 cents (5\18<√12> to 2\7<√12>, or 5\36<12/1> to 1\7<12/1>) . The dark generator is its √12-complement. Because this is a perfect eighteenth-repeating scale, each tone has an √12 perfect eighteenth above it.

The period can range from 24/7 to 7/2, including the pure-hemipyth √12 and 1/QPochhammer[1/2]. It is because of the latter constant that Cole proposes naming this scale pochhammeroid.

Standing assumptions

The notation used in this article is 0 Pacific-Hemipyth = 0123456789ABCDEFGH (see the section on modes below), unless specified otherwise. (Alternatively, one can use any octodecimal or niftimal/triacontaheximal digit set as the numbers, as long as it is clear which set one is using.) Octodecimal digitsets will be used for naming notes as it more practical. We denote raising and lowering by a chroma (L − s, about √(256/243) using the hemipyth interpretation) by # and ♭.

Scale properties

Assume 36-nominal notation for this section.

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 diatonic interval categories.

{{subst:MOS data temporary}}

Modes

Cole proposes naming the modes this way: Each period is split up into three pentachords and a trichord. There are three modes which defy the classification of three 2L 3s pentachords and a 1L 2s trichord, having pentachords with one large step and four small steps. However, they are still included for completeness. An ambiguous mode is named after the first two pentachords. Pacific-Hemipyth is regarded as mode 0 or 18.

Mode UDP Cyclic order Name Name origin
LsLss LsLss LsLss Lss LsLss LsLss LsLss Lss 34|0(2) 1 Atlantic-QFind The canonically first 'core' of the Colian Nexus, qfind.
LsLss LsLss LssLs Lss LsLss LsLss LssLs Lss 32|2(2) 6 Atlantic-Planet Named after PlanetN9ne.
LsLss LssLs LssLs Lss LsLss LssLs LssLs Lss 30|4(2) 11 Atlantic-Lumian This mode's first dekatave has an Atlantic pentachord and a Lumian pentachord.
LssLs LssLs LssLs Lss LssLs LssLs LssLs Lss 28|6(2) 16 Lumian-Q-Series The canonically second 'core' of the Colian Nexus, 2-analogue-related functions. Named after the Q-series.
LssLs LssLs LssLs sLs LssLs LssLs LssLs sLs 26|8(2) 3 Lumian-Moosey Named after Moosey.
LssLs LssLs sLsLs sLs LssLs LssLs sLsLs sLs 24|10(2) 8 Lumian-Drone Named after DroneBetter.
LssLs sLsLs sLsLs sLs LssLs sLsLs sLsLs sLs 22|12(2) 13 Lumian-Pacific This mode's first dekatave has a Lumian pentachord and a Pacific pentachord.
sLsLs sLsLs sLsLs sLs sLsLs sLsLs sLsLs sLs 20|14(2) 18 or 0 Pacific-Hemipyth The canonically third 'core' of the Colian Nexus, hemipyth.
sLsLs sLsLs sLssL sLs sLsLs sLsLs sLssL sLs 18|16(2) 5 Pacific-NimbleRogue Named after NimbleRogue.
sLsLs sLssL sLssL sLs sLsLs sLssL sLssL sLs 16|18(2) 10 Pacific-Taliesin This mode's first dekatave has a Pacific pentachord and a Taliesin pentachord.
sLssL sLssL sLssL sLs sLssL sLssL sLssL sLs 14|20(2) 15 Taliesin-Riemannic The canonically fourth 'core' of the Colian Nexus, Riemannic, a conlang created by Cole.
sLssL sLssL sLssL ssL sLssL sLssL sLssL ssL 12|22(2) 2 Taliesin-Wwei Named after wwei47.
sLssL sLssL ssLsL ssL sLssL sLssL ssLsL ssL 10|24(2) 7 Taliesin-LaundryPizza Named after LaundryPizza03.
sLssL ssLsL ssLsL ssL sLssL ssLsL ssLsL ssL 8|26(2) 12 Taliesin-Dresden This mode's first dekatave has a Taliesin pentachord and a Dresden pentachord.
ssLsL ssLsL ssLsL ssL ssLsL ssLsL ssLsL ssL 6|28(2) 17 Dresden-Heav Named after Heav.
ssLsL ssLsL ssLss LsL ssLsL ssLsL ssLss LsL 4|30(2) 4 Subdresden-Nimrgod Named after the canonically first of three tuppers created by User:2^67-1, Nimrgod.
ssLsL ssLss LsLss LsL ssLsL ssLss LsLss LsL 2|32(2) 9 Subdresden-Boris Named after the canonically second of three tuppers created by User:2^67-1, Boris Grothendieck.
ssLss LsLss LsLss LsL ssLss LsLss LsLss LsL 0|34(2) 14 Subdresden-Pergele Named after the canonically third of three tuppers created by User:2^67-1, Pergele (originator of this idea is Frostburn in a meme post).

Intervals on 0

Here are some intervals on the note 0.

5♭♭ A♭♭ F♭♭ 2♭♭ 7♭♭ C♭♭ H♭♭ 4♭♭ 9♭♭ E♭♭ →
dd5 d10 d15 d2 d7 d12 d17 d4 d9 d14
1♭ 6♭ B♭ G♭ 3♭ 8♭ D♭ 0♭ 5♭ A♭ F♭ 2♭ 7♭ C♭ H♭ 4♭ 9♭ E♭ →
d1 d6 d11 d16 d3 d8 d13 d0 d5 m10 m15 m2 m7 m12 m17 m4 m9 m14
1 6 B G 3 8 D 0 5 A F 2 7 C H 4 9 E →
m1 m6 m11 m16 m3 m8 P13 P0 P5 M10 M15 M2 M7 M12 M17 M4 M9 M14
1# 6# B# G# 3# 8# D# 0# 5# A# F# 2# 7# C# H# 4# 9# E# →
M1 M6 M11 M16 M3 M8 A13 A0 A5 A10 A15 A2 A7 A12 A17 A4 A9 A14
1## 6## B## G## 3## 8## D## →
A1 A6 A11 A16 A3 A8 AA13

Simple tunings

Simple Tunings of 14L 22s⟨12/1⟩
Scale degree Abbrev. Basic (2:1)
50ed12 		Hard (3:1)
64ed12 		Soft (3:2)
86ed12
Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\50 0.0 0\64 0.0 0\86 0.0
Minor 1-mosdegree m1md 1\50 86.0 1\64 67.2 2\86 100.0
Major 1-mosdegree M1md 2\50 172.1 3\64 201.7 3\86 150.1
Minor 2-mosdegree m2md 2\50 172.1 2\64 134.4 4\86 200.1
Major 2-mosdegree M2md 3\50 258.1 4\64 268.9 5\86 250.1
Minor 3-mosdegree m3md 4\50 344.2 5\64 336.1 7\86 350.2
Major 3-mosdegree M3md 5\50 430.2 7\64 470.5 8\86 400.2
Minor 4-mosdegree m4md 5\50 430.2 6\64 403.3 9\86 450.2
Major 4-mosdegree M4md 6\50 516.2 8\64 537.7 10\86 500.2
Diminished 5-mosdegree d5md 6\50 516.2 7\64 470.5 11\86 550.3
Perfect 5-mosdegree P5md 7\50 602.3 9\64 605.0 12\86 600.3
Minor 6-mosdegree m6md 8\50 688.3 10\64 672.2 14\86 700.3
Major 6-mosdegree M6md 9\50 774.4 12\64 806.6 15\86 750.3
Minor 7-mosdegree m7md 9\50 774.4 11\64 739.4 16\86 800.4
Major 7-mosdegree M7md 10\50 860.4 13\64 873.8 17\86 850.4
Minor 8-mosdegree m8md 11\50 946.4 14\64 941.1 19\86 950.4
Major 8-mosdegree M8md 12\50 1032.5 16\64 1075.5 20\86 1000.5
Minor 9-mosdegree m9md 12\50 1032.5 15\64 1008.3 21\86 1050.5
Major 9-mosdegree M9md 13\50 1118.5 17\64 1142.7 22\86 1100.5
Minor 10-mosdegree m10md 13\50 1118.5 16\64 1075.5 23\86 1150.5
Major 10-mosdegree M10md 14\50 1204.5 18\64 1209.9 24\86 1200.5
Minor 11-mosdegree m11md 15\50 1290.6 19\64 1277.1 26\86 1300.6
Major 11-mosdegree M11md 16\50 1376.6 21\64 1411.6 27\86 1350.6
Minor 12-mosdegree m12md 16\50 1376.6 20\64 1344.4 28\86 1400.6
Major 12-mosdegree M12md 17\50 1462.7 22\64 1478.8 29\86 1450.7
Perfect 13-mosdegree P13md 18\50 1548.7 23\64 1546.0 31\86 1550.7
Augmented 13-mosdegree A13md 19\50 1634.7 25\64 1680.5 32\86 1600.7
Minor 14-mosdegree m14md 19\50 1634.7 24\64 1613.2 33\86 1650.8
Major 14-mosdegree M14md 20\50 1720.8 26\64 1747.7 34\86 1700.8
Minor 15-mosdegree m15md 20\50 1720.8 25\64 1680.5 35\86 1750.8
Major 15-mosdegree M15md 21\50 1806.8 27\64 1814.9 36\86 1800.8
Minor 16-mosdegree m16md 22\50 1892.9 28\64 1882.1 38\86 1900.9
Major 16-mosdegree M16md 23\50 1978.9 30\64 2016.5 39\86 1950.9
Minor 17-mosdegree m17md 23\50 1978.9 29\64 1949.3 40\86 2000.9
Major 17-mosdegree M17md 24\50 2064.9 31\64 2083.8 41\86 2050.9
Perfect 18-mosdegree P18md 25\50 2151.0 32\64 2151.0 43\86 2151.0
Minor 19-mosdegree m19md 26\50 2237.0 33\64 2218.2 45\86 2251.0
Major 19-mosdegree M19md 27\50 2323.1 35\64 2352.6 46\86 2301.0
Minor 20-mosdegree m20md 27\50 2323.1 34\64 2285.4 47\86 2351.1
Major 20-mosdegree M20md 28\50 2409.1 36\64 2419.8 48\86 2401.1
Minor 21-mosdegree m21md 29\50 2495.1 37\64 2487.1 50\86 2501.1
Major 21-mosdegree M21md 30\50 2581.2 39\64 2621.5 51\86 2551.2
Minor 22-mosdegree m22md 30\50 2581.2 38\64 2554.3 52\86 2601.2
Major 22-mosdegree M22md 31\50 2667.2 40\64 2688.7 53\86 2651.2
Diminished 23-mosdegree d23md 31\50 2667.2 39\64 2621.5 54\86 2701.2
Perfect 23-mosdegree P23md 32\50 2753.3 41\64 2755.9 55\86 2751.3
Minor 24-mosdegree m24md 33\50 2839.3 42\64 2823.2 57\86 2851.3
Major 24-mosdegree M24md 34\50 2925.3 44\64 2957.6 58\86 2901.3
Minor 25-mosdegree m25md 34\50 2925.3 43\64 2890.4 59\86 2951.3
Major 25-mosdegree M25md 35\50 3011.4 45\64 3024.8 60\86 3001.4
Minor 26-mosdegree m26md 36\50 3097.4 46\64 3092.0 62\86 3101.4
Major 26-mosdegree M26md 37\50 3183.4 48\64 3226.5 63\86 3151.4
Minor 27-mosdegree m27md 37\50 3183.4 47\64 3159.2 64\86 3201.5
Major 27-mosdegree M27md 38\50 3269.5 49\64 3293.7 65\86 3251.5
Minor 28-mosdegree m28md 38\50 3269.5 48\64 3226.5 66\86 3301.5
Major 28-mosdegree M28md 39\50 3355.5 50\64 3360.9 67\86 3351.5
Minor 29-mosdegree m29md 40\50 3441.6 51\64 3428.1 69\86 3451.6
Major 29-mosdegree M29md 41\50 3527.6 53\64 3562.6 70\86 3501.6
Minor 30-mosdegree m30md 41\50 3527.6 52\64 3495.3 71\86 3551.6
Major 30-mosdegree M30md 42\50 3613.6 54\64 3629.8 72\86 3601.6
Perfect 31-mosdegree P31md 43\50 3699.7 55\64 3697.0 74\86 3701.7
Augmented 31-mosdegree A31md 44\50 3785.7 57\64 3831.4 75\86 3751.7
Minor 32-mosdegree m32md 44\50 3785.7 56\64 3764.2 76\86 3801.7
Major 32-mosdegree M32md 45\50 3871.8 58\64 3898.6 77\86 3851.8
Minor 33-mosdegree m33md 45\50 3871.8 57\64 3831.4 78\86 3901.8
Major 33-mosdegree M33md 46\50 3957.8 59\64 3965.9 79\86 3951.8
Minor 34-mosdegree m34md 47\50 4043.8 60\64 4033.1 81\86 4051.8
Major 34-mosdegree M34md 48\50 4129.9 62\64 4167.5 82\86 4101.9
Minor 35-mosdegree m35md 48\50 4129.9 61\64 4100.3 83\86 4151.9
Major 35-mosdegree M35md 49\50 4215.9 63\64 4234.7 84\86 4201.9
Perfect 36-mosdegree P36md 50\50 4302.0 64\64 4302.0 86\86 4302.0

Scale tree

Scale tree and tuning spectrum of 14L 22s⟨12/1⟩
Generator(ed12/1) Cents Step ratio Comments
Bright Dark L:s Hardness
5\36 597.494 1553.484 1:1 1.000 Equalized 14L 22s⟨12/1⟩
27\194 598.726 1552.252 6:5 1.200
22\158 599.006 1551.971 5:4 1.250
39\280 599.201 1551.777 9:7 1.286
17\122 599.453 1551.525 4:3 1.333 Supersoft 14L 22s⟨12/1⟩
46\330 599.666 1551.311 11:8 1.375
29\208 599.792 1551.186 7:5 1.400
41\294 599.933 1551.045 10:7 1.429 Hemipyth tuning for this scale is around here;
12\86 600.273 1550.705 3:2 1.500 Soft 14L 22s⟨12/1⟩
Approximately the step ratio for the scale with 1/QPochhammer[1/2] as period.
43\308 600.598 1550.380 11:7 1.571 The sum of the sizes of all the small steps is equal to the sum of the sizes of all the large steps.
31\222 600.723 1550.254 8:5 1.600
50\358 600.832 1550.146 13:8 1.625
19\136 601.008 1549.969 5:3 1.667 Semisoft 14L 22s⟨12/1⟩
45\322 601.205 1549.773 12:7 1.714
26\186 601.349 1549.629 7:4 1.750
33\236 601.545 1549.433 9:5 1.800
7\50 602.274 1548.704 2:1 2.000 Basic 14L 22s⟨12/1⟩
Scales with tunings softer than this are proper
30\214 603.078 1547.900 9:4 2.250
23\164 603.323 1547.655 7:3 2.333
39\278 603.512 1547.466 12:5 2.400
16\114 603.783 1547.194 5:2 2.500 Semihard 14L 22s⟨12/1⟩
41\292 604.042 1546.936 13:5 2.600
25\178 604.207 1546.770 8:3 2.667
34\242 604.407 1546.571 11:4 2.750
9\64 604.962 1546.015 3:1 3.000 Hard 14L 22s⟨12/1⟩
29\206 605.615 1545.362 10:3 3.333
20\142 605.909 1545.068 7:2 3.500
31\220 606.185 1544.793 11:3 3.667
11\78 606.686 1544.292 4:1 4.000 Superhard 14L 22s⟨12/1⟩
24\170 607.335 1543.643 9:2 4.500
13\92 607.885 1543.093 5:1 5.000
15\106 608.767 1542.210 6:1 6.000
2\14 614.565 1536.413 1:0 → ∞ Collapsed 14L 22s⟨12/1⟩
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