The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2393: "Zathimic"

Scale 2393: Zathimic, 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
Zathimic
Dozenal
Opuian

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

{0,3,4,6,8,11}

Forte Number

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

6-31

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: 851

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.

3

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.

no
prime: 691

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.

[3, 1, 2, 2, 3, 1]

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, 2, 3, 4, 3, 1>

Interval Spectrum

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

p3m4n3s2d2t

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

Spectra Variation

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

1.667

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.

2.366

Polygon Perimeter

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

5.864

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".

Proper

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.

(0, 8, 54)

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 TriadsE{4,8,11}231.5
G♯{8,0,3}321.17
B{11,3,6}231.5
Minor Triadsg♯m{8,11,3}321.17
Augmented TriadsC+{0,4,8}231.5
Diminished Triads{0,3,6}231.5
Parsimonious Voice Leading Between Common Triads of Scale 2393. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B C+ C+ E E C+->E C+->G# g#m g#m E->g#m g#m->G# g#m->B

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 Verticesg♯m, G♯
Peripheral Verticesc°, C+, E, B

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 are 2 ways that this hexatonic scale can be split into two common triads.


Diminished: {0, 3, 6}
Major: {4, 8, 11}

Augmented: {0, 4, 8}
Major: {11, 3, 6}

Modes

Modes are the rotational transformation of this scale. Scale 2393 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 811
Scale 811: Radimic, Ian Ring Music TheoryRadimic
3rd mode:
Scale 2453
Scale 2453: Raga Latika, Ian Ring Music TheoryRaga Latika
4th mode:
Scale 1637
Scale 1637: Syptimic, Ian Ring Music TheorySyptimic
5th mode:
Scale 1433
Scale 1433: Dynimic, Ian Ring Music TheoryDynimic
6th mode:
Scale 691
Scale 691: Raga Kalavati, Ian Ring Music TheoryRaga KalavatiThis is the prime mode

Prime

The prime form of this scale is Scale 691

Scale 691Scale 691: Raga Kalavati, Ian Ring Music TheoryRaga Kalavati

Complement

The hexatonic modal family [2393, 811, 2453, 1637, 1433, 691] (Forte: 6-31) is the complement of the hexatonic modal family [691, 811, 1433, 1637, 2393, 2453] (Forte: 6-31)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2393 is 851

Scale 851Scale 851: Raga Hejjajji, Ian Ring Music TheoryRaga Hejjajji

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2393 is chiral, and its enantiomorph is scale 851

Scale 851Scale 851: Raga Hejjajji, Ian Ring Music TheoryRaga Hejjajji

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> 2393       T0I <11,0> 851
T1 <1,1> 691      T1I <11,1> 1702
T2 <1,2> 1382      T2I <11,2> 3404
T3 <1,3> 2764      T3I <11,3> 2713
T4 <1,4> 1433      T4I <11,4> 1331
T5 <1,5> 2866      T5I <11,5> 2662
T6 <1,6> 1637      T6I <11,6> 1229
T7 <1,7> 3274      T7I <11,7> 2458
T8 <1,8> 2453      T8I <11,8> 821
T9 <1,9> 811      T9I <11,9> 1642
T10 <1,10> 1622      T10I <11,10> 3284
T11 <1,11> 3244      T11I <11,11> 2473
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 473      T0MI <7,0> 881
T1M <5,1> 946      T1MI <7,1> 1762
T2M <5,2> 1892      T2MI <7,2> 3524
T3M <5,3> 3784      T3MI <7,3> 2953
T4M <5,4> 3473      T4MI <7,4> 1811
T5M <5,5> 2851      T5MI <7,5> 3622
T6M <5,6> 1607      T6MI <7,6> 3149
T7M <5,7> 3214      T7MI <7,7> 2203
T8M <5,8> 2333      T8MI <7,8> 311
T9M <5,9> 571      T9MI <7,9> 622
T10M <5,10> 1142      T10MI <7,10> 1244
T11M <5,11> 2284      T11MI <7,11> 2488

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 2395Scale 2395: Zoptian, Ian Ring Music TheoryZoptian
Scale 2397Scale 2397: Stagian, Ian Ring Music TheoryStagian
Scale 2385Scale 2385: Aeolanitonic, Ian Ring Music TheoryAeolanitonic
Scale 2389Scale 2389: Eskimo Hexatonic 2, Ian Ring Music TheoryEskimo Hexatonic 2
Scale 2377Scale 2377: Bartók Gamma Chord, Ian Ring Music TheoryBartók Gamma Chord
Scale 2409Scale 2409: Zacrimic, Ian Ring Music TheoryZacrimic
Scale 2425Scale 2425: Rorian, Ian Ring Music TheoryRorian
Scale 2329Scale 2329: Styditonic, Ian Ring Music TheoryStyditonic
Scale 2361Scale 2361: Docrimic, Ian Ring Music TheoryDocrimic
Scale 2457Scale 2457: Augmented, Ian Ring Music TheoryAugmented
Scale 2521Scale 2521: Mela Dhatuvardhani, Ian Ring Music TheoryMela Dhatuvardhani
Scale 2137Scale 2137: Nalian, Ian Ring Music TheoryNalian
Scale 2265Scale 2265: Raga Rasamanjari, Ian Ring Music TheoryRaga Rasamanjari
Scale 2649Scale 2649: Aeolythimic, Ian Ring Music TheoryAeolythimic
Scale 2905Scale 2905: Aeolian Flat 1, Ian Ring Music TheoryAeolian Flat 1
Scale 3417Scale 3417: Golian, Ian Ring Music TheoryGolian
Scale 345Scale 345: Gylitonic, Ian Ring Music TheoryGylitonic
Scale 1369Scale 1369: Boptimic, Ian Ring Music TheoryBoptimic

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.