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Scale 585: "Diminished Seventh"

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

Zeitler: Phrynic

Analysis

Cardinality 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	{0,3,6,9}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	4-28
Rotational Symmetry Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.	[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]
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.	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.	0
Prime Form Describes if this scale is in prime form, using the Rahn/Ring formula.	yes
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.	[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.	n⁴t²
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.	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
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	5.657
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.	[0,3,6,9]
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

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 Type	Triad^*	Pitch Classes	Degree	Eccentricity	Closeness Centrality
Diminished Triads	c°	{0,3,6}	2	2	1
	d♯°	{3,6,9}	2	2	1
	f♯°	{6,9,0}	2	2	1
	a°	{9,0,3}	2	2	1

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.

Diameter	2
Radius	2
Self-Centered	yes

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

Inverse

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 585

Diminished Seventh

Transformations:

T₀	585	T₀I	585
T₁	1170	T₁I	1170
T₂	2340	T₂I	2340
T₃	585	T₃I	585
T₄	1170	T₄I	1170
T₅	2340	T₅I	2340
T₆	585	T₆I	585
T₇	1170	T₇I	1170
T₈	2340	T₈I	2340
T₉	585	T₉I	585
T₁₀	1170	T₁₀I	1170
T₁₁	2340	T₁₁I	2340

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 587				Pathitonic
Scale 589				Ionalitonic
Scale 577
Scale 581				Eporic 2
Scale 593				Saric
Scale 601				Bycritonic
Scale 617				Katycritonic
Scale 521
Scale 553				Rothic 2
Scale 649				Byptic
Scale 713				Thoptitonic
Scale 841				Phrothitonic
Scale 73				Diminished Triad
Scale 329				Mynic 2
Scale 1097				Aeraphic
Scale 1609				Thyritonic
Scale 2633				Bartók 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. 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.