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Scale 1365: "Whole Tone"

Scale 1365: Whole Tone, 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

Western
Whole Tone
Dozenal
Holian
Messiaen
Messiaen Mode 1
Carnatic
Raga Gopriya
Western Modern
Anhemitonic Hexatonic
Anhemic Hexatonic
Auxiliary Augmented
Zeitler
Kylimic

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

{0,2,4,6,8,10}

Forte Number

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

6-35

Rotational Symmetry

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

[2, 4, 6, 8, 10]

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.

[0, 1, 2, 3, 4, 5]

Palindromicity

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

yes

Chirality

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

no

Hemitonia

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

0 (anhemitonic)

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.

6

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.

0

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.

generator: 2
origin: 0

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.

[2, 2, 2, 2, 2, 2]

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.

<0, 6, 0, 6, 0, 3>

Interval Spectrum

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

m6s6t3

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> = {2}
<2> = {4}
<3> = {6}
<4> = {8}
<5> = {10}

Spectra Variation

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

0

Maximally Even

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

yes

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.

yes

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.

2.598

Polygon Perimeter

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

6

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.

yes

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.

[0,2,4,6,8,10]

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".

Strictly 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, 0, 0)

Generator

This scale has a generator of 2, originating on 0.

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
Augmented TriadsC+{0,4,8}0n/an/a
D+{2,6,10}0n/an/a
Parsimonious Voice Leading Between Common Triads of Scale 1365. Created by Ian Ring ©2019 C+ C+ D+ D+

Above is a graph showing opportunities for parsimonious voice leading between triads*. There are no lines connecting nodes in the graph, because the chords are not adjacent with common tones.

Diametern/a
Radiusn/a
Self-Centeredyes

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.


Augmented: {0, 4, 8}
Augmented: {2, 6, 10}

Modes

Modes are the rotational transformation of this scale. This scale has no modes, becaue any rotation of this scale will produce another copy of itself.

Prime

This is the prime form of this scale.

Complement

The hexatonic modal family [1365] (Forte: 6-35) is the complement of the hexatonic modal family [1365] (Forte: 6-35)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1365 is itself, because it is a palindromic scale!

Scale 1365Scale 1365: Whole Tone, Ian Ring Music TheoryWhole Tone

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> 1365       T0I <11,0> 1365
T1 <1,1> 2730      T1I <11,1> 2730
T2 <1,2> 1365       T2I <11,2> 1365
T3 <1,3> 2730      T3I <11,3> 2730
T4 <1,4> 1365       T4I <11,4> 1365
T5 <1,5> 2730      T5I <11,5> 2730
T6 <1,6> 1365       T6I <11,6> 1365
T7 <1,7> 2730      T7I <11,7> 2730
T8 <1,8> 1365       T8I <11,8> 1365
T9 <1,9> 2730      T9I <11,9> 2730
T10 <1,10> 1365       T10I <11,10> 1365
T11 <1,11> 2730      T11I <11,11> 2730
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1365       T0MI <7,0> 1365
T1M <5,1> 2730      T1MI <7,1> 2730
T2M <5,2> 1365       T2MI <7,2> 1365
T3M <5,3> 2730      T3MI <7,3> 2730
T4M <5,4> 1365       T4MI <7,4> 1365
T5M <5,5> 2730      T5MI <7,5> 2730
T6M <5,6> 1365       T6MI <7,6> 1365
T7M <5,7> 2730      T7MI <7,7> 2730
T8M <5,8> 1365       T8MI <7,8> 1365
T9M <5,9> 2730      T9MI <7,9> 2730
T10M <5,10> 1365       T10MI <7,10> 1365
T11M <5,11> 2730      T11MI <7,11> 2730

The transformations that map this set to itself are: T0, T2, T4, T6, T8, T10, T0I, T2I, T4I, T6I, T8I, T10I, T0M, T2M, T4M, T6M, T8M, T10M, T0MI, T2MI, T4MI, T6MI, T8MI, T10MI

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 1367Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone Inverse
Scale 1361Scale 1361: Bolitonic, Ian Ring Music TheoryBolitonic
Scale 1363Scale 1363: Gygimic, Ian Ring Music TheoryGygimic
Scale 1369Scale 1369: Boptimic, Ian Ring Music TheoryBoptimic
Scale 1373Scale 1373: Storian, Ian Ring Music TheoryStorian
Scale 1349Scale 1349: Tholitonic, Ian Ring Music TheoryTholitonic
Scale 1357Scale 1357: Takemitsu Linea Mode 2, Ian Ring Music TheoryTakemitsu Linea Mode 2
Scale 1381Scale 1381: Padimic, Ian Ring Music TheoryPadimic
Scale 1397Scale 1397: Major Locrian, Ian Ring Music TheoryMajor Locrian
Scale 1301Scale 1301: Koditonic, Ian Ring Music TheoryKoditonic
Scale 1333Scale 1333: Lyptimic, Ian Ring Music TheoryLyptimic
Scale 1429Scale 1429: Bythimic, Ian Ring Music TheoryBythimic
Scale 1493Scale 1493: Lydian Minor, Ian Ring Music TheoryLydian Minor
Scale 1109Scale 1109: Kataditonic, Ian Ring Music TheoryKataditonic
Scale 1237Scale 1237: Salimic, Ian Ring Music TheorySalimic
Scale 1621Scale 1621: Scriabin's Prometheus, Ian Ring Music TheoryScriabin's Prometheus
Scale 1877Scale 1877: Aeroptian, Ian Ring Music TheoryAeroptian
Scale 341Scale 341: Bothitonic, Ian Ring Music TheoryBothitonic
Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
Scale 2389Scale 2389: Eskimo Hexatonic 2, Ian Ring Music TheoryEskimo Hexatonic 2
Scale 3413Scale 3413: Leading Whole-tone, Ian Ring Music TheoryLeading Whole-tone

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.