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Scale 585: "Diminished Seventh"

Scale 585: Diminished Seventh, 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

Chord Names
Diminished Seventh



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

4 (tetratonic)

Pitch Class Set

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


Rotational Symmetry

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

[3, 6, 9]

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, 1.5, 3, 4.5]


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



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



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

0 (anhemitonic)


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

0 (ancohemitonic)


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.



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.


Prime Form

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



Indicates if the scale can be constructed using a generator, and an origin.

generator: 3
origin: 0

Deep Scale

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


Interval Structure

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

[3, 3, 3, 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.

<0, 0, 4, 0, 0, 2>

Interval Spectrum

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


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> = {3}
<2> = {6}
<3> = {9}

Spectra Variation

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


Maximally Even

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


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.


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.


Polygon Perimeter

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


Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.



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.


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.



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)


This scale has a generator of 3, 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
Diminished Triads{0,3,6}221
Parsimonious Voice Leading Between Common Triads of Scale 585. Created by Ian Ring ©2019 d#° d#° c°->d#° c°->a° f#° f#° d#°->f#° f#°->a°

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.



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.


This is the prime form of this scale.


The tetratonic modal family [585] (Forte: 4-28) is the complement of the octatonic modal family [1755, 2925] (Forte: 8-28)


The inverse of a scale is a reflection using the root as its axis. The inverse of 585 is itself, because it is a palindromic scale!

Scale 585Scale 585: Diminished Seventh, Ian Ring Music TheoryDiminished Seventh


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> 585       T0I <11,0> 585
T1 <1,1> 1170      T1I <11,1> 1170
T2 <1,2> 2340      T2I <11,2> 2340
T3 <1,3> 585       T3I <11,3> 585
T4 <1,4> 1170      T4I <11,4> 1170
T5 <1,5> 2340      T5I <11,5> 2340
T6 <1,6> 585       T6I <11,6> 585
T7 <1,7> 1170      T7I <11,7> 1170
T8 <1,8> 2340      T8I <11,8> 2340
T9 <1,9> 585       T9I <11,9> 585
T10 <1,10> 1170      T10I <11,10> 1170
T11 <1,11> 2340      T11I <11,11> 2340
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 585       T0MI <7,0> 585
T1M <5,1> 1170      T1MI <7,1> 1170
T2M <5,2> 2340      T2MI <7,2> 2340
T3M <5,3> 585       T3MI <7,3> 585
T4M <5,4> 1170      T4MI <7,4> 1170
T5M <5,5> 2340      T5MI <7,5> 2340
T6M <5,6> 585       T6MI <7,6> 585
T7M <5,7> 1170      T7MI <7,7> 1170
T8M <5,8> 2340      T8MI <7,8> 2340
T9M <5,9> 585       T9MI <7,9> 585
T10M <5,10> 1170      T10MI <7,10> 1170
T11M <5,11> 2340      T11MI <7,11> 2340

The transformations that map this set to itself are: T0, T3, T6, T9, T0I, T3I, T6I, T9I, T0M, T3M, T6M, T9M, T0MI, T3MI, T6MI, T9MI

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 587Scale 587: Pathitonic, Ian Ring Music TheoryPathitonic
Scale 589Scale 589: Ionalitonic, Ian Ring Music TheoryIonalitonic
Scale 577Scale 577: Illian, Ian Ring Music TheoryIllian
Scale 581Scale 581: Eporic 2, Ian Ring Music TheoryEporic 2
Scale 593Scale 593: Saric, Ian Ring Music TheorySaric
Scale 601Scale 601: Bycritonic, Ian Ring Music TheoryBycritonic
Scale 617Scale 617: Katycritonic, Ian Ring Music TheoryKatycritonic
Scale 521Scale 521: Astian, Ian Ring Music TheoryAstian
Scale 553Scale 553: Rothic 2, Ian Ring Music TheoryRothic 2
Scale 649Scale 649: Byptic, Ian Ring Music TheoryByptic
Scale 713Scale 713: Thoptitonic, Ian Ring Music TheoryThoptitonic
Scale 841Scale 841: Phrothitonic, Ian Ring Music TheoryPhrothitonic
Scale 73Scale 73: Diminished Triad, Ian Ring Music TheoryDiminished Triad
Scale 329Scale 329: Mynic 2, Ian Ring Music TheoryMynic 2
Scale 1097Scale 1097: Aeraphic, Ian Ring Music TheoryAeraphic
Scale 1609Scale 1609: Thyritonic, Ian Ring Music TheoryThyritonic
Scale 2633Scale 2633: Bartók Beta Chord, Ian Ring Music TheoryBartók Beta Chord

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 ( 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.