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Scale 3655: "Mathian"

Scale 3655: Mathian, 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
Mathian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,1,2,6,9,10,11}

Forte Number

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

7-3

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: 3151

Hemitonia

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

5 (multihemitonic)

Cohemitonia

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

4 (multicohemitonic)

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.

4

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.

6

Prime Form

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

no
prime: 319

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, 4, 3, 1, 1, 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.

<5, 4, 4, 4, 3, 1>

Interval Spectrum

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

p3m4n4s4d5t

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

Spectra Variation

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

3.714

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.

2.183

Polygon Perimeter

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

5.734

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

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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 TriadsD{2,6,9}221.33
F♯{6,10,1}221.33
Minor Triadsf♯m{6,9,1}331.33
bm{11,2,6}142
Augmented TriadsD+{2,6,10}331.33
Diminished Triadsf♯°{6,9,0}142
Parsimonious Voice Leading Between Common Triads of Scale 3655. Created by Ian Ring ©2019 D D D+ D+ D->D+ f#m f#m D->f#m F# F# D+->F# bm bm D+->bm f#° f#° f#°->f#m f#m->F#

view full size

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.

Diameter4
Radius2
Self-Centeredno
Central VerticesD, F♯
Peripheral Verticesf♯°, bm

Modes

Modes are the rotational transformation of this scale. Scale 3655 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3875
Scale 3875: Aeryptian, Ian Ring Music TheoryAeryptian
3rd mode:
Scale 3985
Scale 3985: Thadian, Ian Ring Music TheoryThadian
4th mode:
Scale 505
Scale 505: Sanian, Ian Ring Music TheorySanian
5th mode:
Scale 575
Scale 575: Ionydian, Ian Ring Music TheoryIonydian
6th mode:
Scale 2335
Scale 2335: Epydian, Ian Ring Music TheoryEpydian
7th mode:
Scale 3215
Scale 3215: Katydian, Ian Ring Music TheoryKatydian

Prime

The prime form of this scale is Scale 319

Scale 319Scale 319: Epodian, Ian Ring Music TheoryEpodian

Complement

The heptatonic modal family [3655, 3875, 3985, 505, 575, 2335, 3215] (Forte: 7-3) is the complement of the pentatonic modal family [55, 1795, 2075, 2945, 3085] (Forte: 5-3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3655 is 3151

Scale 3151Scale 3151: Pacrian, Ian Ring Music TheoryPacrian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3655 is chiral, and its enantiomorph is scale 3151

Scale 3151Scale 3151: Pacrian, Ian Ring Music TheoryPacrian

Transformations:

T0 3655  T0I 3151
T1 3215  T1I 2207
T2 2335  T2I 319
T3 575  T3I 638
T4 1150  T4I 1276
T5 2300  T5I 2552
T6 505  T6I 1009
T7 1010  T7I 2018
T8 2020  T8I 4036
T9 4040  T9I 3977
T10 3985  T10I 3859
T11 3875  T11I 3623

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 3653Scale 3653: Sathimic, Ian Ring Music TheorySathimic
Scale 3651Scale 3651, Ian Ring Music Theory
Scale 3659Scale 3659: Polian, Ian Ring Music TheoryPolian
Scale 3663Scale 3663: Sonyllic, Ian Ring Music TheorySonyllic
Scale 3671Scale 3671: Aeonyllic, Ian Ring Music TheoryAeonyllic
Scale 3687Scale 3687: Zonyllic, Ian Ring Music TheoryZonyllic
Scale 3591Scale 3591, Ian Ring Music Theory
Scale 3623Scale 3623: Aerocrian, Ian Ring Music TheoryAerocrian
Scale 3719Scale 3719, Ian Ring Music Theory
Scale 3783Scale 3783: Phrygyllic, Ian Ring Music TheoryPhrygyllic
Scale 3911Scale 3911: Katyryllic, Ian Ring Music TheoryKatyryllic
Scale 3143Scale 3143: Polimic, Ian Ring Music TheoryPolimic
Scale 3399Scale 3399: Zonian, Ian Ring Music TheoryZonian
Scale 2631Scale 2631: Macrimic, Ian Ring Music TheoryMacrimic
Scale 1607Scale 1607: Epytimic, Ian Ring Music TheoryEpytimic

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.