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Scale 837: "Epaditonic"

Scale 837: Epaditonic, 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
Epaditonic

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,2,6,8,9}

Forte Number

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

5-28

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: 1113

Hemitonia

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

1 (unhemitonic)

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.

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.

4

Prime Form

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

no
prime: 333

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.

[2, 4, 2, 1, 3]

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.

<1, 2, 2, 2, 1, 2>

Interval Spectrum

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

pm2n2s2dt2

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

Spectra Variation

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

2.4

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.

2.049

Polygon Perimeter

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

5.664

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 TriadsD{2,6,9}110.5
Diminished Triadsf♯°{6,9,0}110.5

The following pitch classes are not present in any of the common triads: {8}

Parsimonious Voice Leading Between Common Triads of Scale 837. Created by Ian Ring ©2019 D D f#° f#° D->f#°

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 837 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1233
Scale 1233: Ionoditonic, Ian Ring Music TheoryIonoditonic
3rd mode:
Scale 333
Scale 333: Bogitonic, Ian Ring Music TheoryBogitonicThis is the prime mode
4th mode:
Scale 1107
Scale 1107: Mogitonic, Ian Ring Music TheoryMogitonic
5th mode:
Scale 2601
Scale 2601: Raga Chandrakauns, Ian Ring Music TheoryRaga Chandrakauns

Prime

The prime form of this scale is Scale 333

Scale 333Scale 333: Bogitonic, Ian Ring Music TheoryBogitonic

Complement

The pentatonic modal family [837, 1233, 333, 1107, 2601] (Forte: 5-28) is the complement of the heptatonic modal family [747, 1431, 1629, 1881, 2421, 2763, 3429] (Forte: 7-28)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 837 is 1113

Scale 1113Scale 1113: Locrian Pentatonic 2, Ian Ring Music TheoryLocrian Pentatonic 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 837 is chiral, and its enantiomorph is scale 1113

Scale 1113Scale 1113: Locrian Pentatonic 2, Ian Ring Music TheoryLocrian Pentatonic 2

Transformations:

T0 837  T0I 1113
T1 1674  T1I 2226
T2 3348  T2I 357
T3 2601  T3I 714
T4 1107  T4I 1428
T5 2214  T5I 2856
T6 333  T6I 1617
T7 666  T7I 3234
T8 1332  T8I 2373
T9 2664  T9I 651
T10 1233  T10I 1302
T11 2466  T11I 2604

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 839Scale 839: Ionathimic, Ian Ring Music TheoryIonathimic
Scale 833Scale 833, Ian Ring Music Theory
Scale 835Scale 835, Ian Ring Music Theory
Scale 841Scale 841: Phrothitonic, Ian Ring Music TheoryPhrothitonic
Scale 845Scale 845: Raga Neelangi, Ian Ring Music TheoryRaga Neelangi
Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
Scale 869Scale 869: Kothimic, Ian Ring Music TheoryKothimic
Scale 773Scale 773, Ian Ring Music Theory
Scale 805Scale 805: Rothitonic, Ian Ring Music TheoryRothitonic
Scale 901Scale 901, Ian Ring Music Theory
Scale 965Scale 965: Ionothimic, Ian Ring Music TheoryIonothimic
Scale 581Scale 581: Eporic 2, Ian Ring Music TheoryEporic 2
Scale 709Scale 709: Raga Shri Kalyan, Ian Ring Music TheoryRaga Shri Kalyan
Scale 325Scale 325: Messiaen Truncated Mode 6, Ian Ring Music TheoryMessiaen Truncated Mode 6
Scale 1349Scale 1349: Tholitonic, Ian Ring Music TheoryTholitonic
Scale 1861Scale 1861: Phrygimic, Ian Ring Music TheoryPhrygimic
Scale 2885Scale 2885: Byrimic, Ian Ring Music TheoryByrimic

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.