User:2^67-1/14L 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⟩ ↘ 
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Scale structure
Step pattern LsLssLsLssLsLssLssLsLssLsLssLsLssLss
ssLssLsLssLsLssLsLssLssLsLssLsLssLsL
Equave 12/1 (4302.0¢)
Period 1\2ed12 (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 TAMNAMS system is used in this article to name 14L 22s <12/1> intervals, step size ratios and step ratio ranges. However, the equave is taken to be the period of the scale, √12, as it is more practical.

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.) We denote raising and lowering by a chroma (L − s, about √(256/243) using the hemipyth interpretation) by # and ♭ "flat (F molle)".

Modes

For clarity, each scale, which is taken to have 18 notes, 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.

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

Genchain

The genchain for this scale is as follows:

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

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⟩
43\308 600.598 1550.380 11:7 1.571
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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