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Scale 363: "Soptimic"

Scale 363: Soptimic, Ian Ring Music Theory

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Common Names

Zeitler
Soptimic
Dozenal
Cehian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

6 (hexatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,3,5,6,8}

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

6-Z25

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.

none

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

yes
enantiomorph: 2769

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

2 (dihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

Imperfections

An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.

2

Modes

Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

5

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

yes

Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[1, 2, 2, 1, 2, 4]

Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<2, 3, 3, 2, 4, 1>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p4m2n3s3d2t

Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,2,4}
<2> = {3,4,5,6}
<3> = {5,7}
<4> = {6,7,8,9}
<5> = {8,10,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

2.333

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.

no

Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.

2.232

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.767

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

no

Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.

none

Propriety

Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".

Improper

Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(12, 9, 55)

Common Triads

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsC♯{1,5,8}131.5
G♯{8,0,3}221
Minor Triadsfm{5,8,0}221
Diminished Triads{0,3,6}131.5
Parsimonious Voice Leading Between Common Triads of Scale 363. Created by Ian Ring ©2019 G# G# c°->G# C# C# fm fm C#->fm fm->G#

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter3
Radius2
Self-Centeredno
Central Verticesfm, G♯
Peripheral Verticesc°, C♯

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Diminished: {0, 3, 6}
Major: {1, 5, 8}

Modes

Modes are the rotational transformation of this scale. Scale 363 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2229
Scale 2229: Raga Nalinakanti, Ian Ring Music TheoryRaga Nalinakanti
3rd mode:
Scale 1581
Scale 1581: Raga Bagesri, Ian Ring Music TheoryRaga Bagesri
4th mode:
Scale 1419
Scale 1419: Raga Kashyapi, Ian Ring Music TheoryRaga Kashyapi
5th mode:
Scale 2757
Scale 2757: Raga Nishadi, Ian Ring Music TheoryRaga Nishadi
6th mode:
Scale 1713
Scale 1713: Raga Khamas, Ian Ring Music TheoryRaga Khamas

Prime

This is the prime form of this scale.

Complement

The hexatonic modal family [363, 2229, 1581, 1419, 2757, 1713] (Forte: 6-Z25) is the complement of the hexatonic modal family [663, 741, 1209, 1833, 2379, 3237] (Forte: 6-Z47)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 363 is 2769

Scale 2769Scale 2769: Dyrimic, Ian Ring Music TheoryDyrimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 363 is chiral, and its enantiomorph is scale 2769

Scale 2769Scale 2769: Dyrimic, Ian Ring Music TheoryDyrimic

Transformations:

In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 363       T0I <11,0> 2769
T1 <1,1> 726      T1I <11,1> 1443
T2 <1,2> 1452      T2I <11,2> 2886
T3 <1,3> 2904      T3I <11,3> 1677
T4 <1,4> 1713      T4I <11,4> 3354
T5 <1,5> 3426      T5I <11,5> 2613
T6 <1,6> 2757      T6I <11,6> 1131
T7 <1,7> 1419      T7I <11,7> 2262
T8 <1,8> 2838      T8I <11,8> 429
T9 <1,9> 1581      T9I <11,9> 858
T10 <1,10> 3162      T10I <11,10> 1716
T11 <1,11> 2229      T11I <11,11> 3432
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 123      T0MI <7,0> 3009
T1M <5,1> 246      T1MI <7,1> 1923
T2M <5,2> 492      T2MI <7,2> 3846
T3M <5,3> 984      T3MI <7,3> 3597
T4M <5,4> 1968      T4MI <7,4> 3099
T5M <5,5> 3936      T5MI <7,5> 2103
T6M <5,6> 3777      T6MI <7,6> 111
T7M <5,7> 3459      T7MI <7,7> 222
T8M <5,8> 2823      T8MI <7,8> 444
T9M <5,9> 1551      T9MI <7,9> 888
T10M <5,10> 3102      T10MI <7,10> 1776
T11M <5,11> 2109      T11MI <7,11> 3552

The transformations that map this set to itself are: T0

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 361Scale 361: Bocritonic, Ian Ring Music TheoryBocritonic
Scale 365Scale 365: Marimic, Ian Ring Music TheoryMarimic
Scale 367Scale 367: Aerodian, Ian Ring Music TheoryAerodian
Scale 355Scale 355: Aeoloritonic, Ian Ring Music TheoryAeoloritonic
Scale 359Scale 359: Bothimic, Ian Ring Music TheoryBothimic
Scale 371Scale 371: Rythimic, Ian Ring Music TheoryRythimic
Scale 379Scale 379: Aeragian, Ian Ring Music TheoryAeragian
Scale 331Scale 331: Raga Chhaya Todi, Ian Ring Music TheoryRaga Chhaya Todi
Scale 347Scale 347: Barimic, Ian Ring Music TheoryBarimic
Scale 299Scale 299: Raga Chitthakarshini, Ian Ring Music TheoryRaga Chitthakarshini
Scale 427Scale 427: Raga Suddha Simantini, Ian Ring Music TheoryRaga Suddha Simantini
Scale 491Scale 491: Aeolyrian, Ian Ring Music TheoryAeolyrian
Scale 107Scale 107: Ansian, Ian Ring Music TheoryAnsian
Scale 235Scale 235: Bihian, Ian Ring Music TheoryBihian
Scale 619Scale 619: Double-Phrygian Hexatonic, Ian Ring Music TheoryDouble-Phrygian Hexatonic
Scale 875Scale 875: Locrian Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 7
Scale 1387Scale 1387: Locrian, Ian Ring Music TheoryLocrian
Scale 2411Scale 2411: Aeolorian, Ian Ring Music TheoryAeolorian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler (http://allthescales.org) used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with the Carnatic ragas.