The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 423: "Sogimic"

Scale 423: Sogimic, 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

41161837294116105072918310504116183
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
Sogimic

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

{0,1,2,5,7,8}

Forte Number

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

6-18

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: 3249

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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

Deep Scale

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

no

Interval Formula

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

[1, 1, 3, 2, 1, 4] 9

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.

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

Interval Spectrum

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

p4m2n2s2d3t2

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

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.

2.116

Polygon Perimeter

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

5.699

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

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}210.67
Minor Triadsfm{5,8,0}121
Diminished Triads{2,5,8}121

The following pitch classes are not present in any of the common triads: {7}

Parsimonious Voice Leading Between Common Triads of Scale 423. Created by Ian Ring ©2019 C# C# C#->d° fm fm C#->fm

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.

Diameter2
Radius1
Self-Centeredno
Central VerticesC♯
Peripheral Verticesd°, fm

Modes

Modes are the rotational transformation of this scale. Scale 423 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2259
Scale 2259: Raga Mandari, Ian Ring Music TheoryRaga Mandari
3rd mode:
Scale 3177
Scale 3177: Rothimic, Ian Ring Music TheoryRothimic
4th mode:
Scale 909
Scale 909: Katarimic, Ian Ring Music TheoryKatarimic
5th mode:
Scale 1251
Scale 1251: Sylimic, Ian Ring Music TheorySylimic
6th mode:
Scale 2673
Scale 2673: Mythimic, Ian Ring Music TheoryMythimic

Prime

This is the prime form of this scale.

Complement

The hexatonic modal family [423, 2259, 3177, 909, 1251, 2673] (Forte: 6-18) is the complement of the hexatonic modal family [423, 909, 1251, 2259, 2673, 3177] (Forte: 6-18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 423 is 3249

Scale 3249Scale 3249: Raga Tilang, Ian Ring Music TheoryRaga Tilang

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 423 is chiral, and its enantiomorph is scale 3249

Scale 3249Scale 3249: Raga Tilang, Ian Ring Music TheoryRaga Tilang

Transformations:

T0 423  T0I 3249
T1 846  T1I 2403
T2 1692  T2I 711
T3 3384  T3I 1422
T4 2673  T4I 2844
T5 1251  T5I 1593
T6 2502  T6I 3186
T7 909  T7I 2277
T8 1818  T8I 459
T9 3636  T9I 918
T10 3177  T10I 1836
T11 2259  T11I 3672

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 421Scale 421: Han-kumoi, Ian Ring Music TheoryHan-kumoi
Scale 419Scale 419: Hon-kumoi-joshi, Ian Ring Music TheoryHon-kumoi-joshi
Scale 427Scale 427: Raga Suddha Simantini, Ian Ring Music TheoryRaga Suddha Simantini
Scale 431Scale 431: Epyrian, Ian Ring Music TheoryEpyrian
Scale 439Scale 439: Bythian, Ian Ring Music TheoryBythian
Scale 391Scale 391, Ian Ring Music Theory
Scale 407Scale 407: Zylimic, Ian Ring Music TheoryZylimic
Scale 455Scale 455: Messiaen Mode 5, Ian Ring Music TheoryMessiaen Mode 5
Scale 487Scale 487: Dynian, Ian Ring Music TheoryDynian
Scale 295Scale 295: Gyritonic, Ian Ring Music TheoryGyritonic
Scale 359Scale 359: Bothimic, Ian Ring Music TheoryBothimic
Scale 167Scale 167, Ian Ring Music Theory
Scale 679Scale 679: Lanimic, Ian Ring Music TheoryLanimic
Scale 935Scale 935: Chromatic Dorian, Ian Ring Music TheoryChromatic Dorian
Scale 1447Scale 1447: Mela Ratnangi, Ian Ring Music TheoryMela Ratnangi
Scale 2471Scale 2471: Mela Ganamurti, Ian Ring Music TheoryMela Ganamurti

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. 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.