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CardinalityCardinality is the count of how many pitches are in the scale. |
4 (tetratonic) |
Pitch Class SetThe tones in this scale, expressed as numbers from 0 to 11 |
{0,3,6,9} |
Forte NumberA code assigned by theorist Alan Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations. |
4-28 |
Rotational SymmetrySome 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 AxesIf 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] |
PalindromicityA palindromic scale has the same pattern of intervals both ascending and descending. |
yes |
ChiralityA chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph. |
no |
HemitoniaA hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist. |
0 (anhemitonic) |
CohemitoniaA cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist. |
0 (ancohemitonic) |
ImperfectionsAn 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 |
ModesModes 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 FormDescribes if this scale is in prime form, using the Rahn/Ring formula. |
yes |
Deep ScaleA deep scale is one where the interval vector has 6 different digits. |
no |
Interval VectorDescribes 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 SpectrumThe same as the Interval Vector, but expressed in a syntax used by Howard Hansen. |
n4t2 |
Distribution SpectraDescribes 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 VariationDetermined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality. |
0 |
Maximally EvenA scale is maximally even if the tones are optimally spaced apart from each other. |
yes |
Maximal Area SetA 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 AreaArea 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 PerimeterPerimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle. |
5.657 |
Myhill PropertyA scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval. |
no |
BalancedA 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 TonesRidge 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] |
ProprietyAlso 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 ProfileDefined 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. |
(0,0,0) |
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 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 585 | ![]() | Diminished Seventh |
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 |
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 | ![]() | ILLIAN | ||
Scale 581 | ![]() | Bolic | ||
Scale 593 | ![]() | Saric | ||
Scale 601 | ![]() | Bycritonic | ||
Scale 617 | ![]() | Katycritonic | ||
Scale 521 | ![]() | ASTIAN | ||
Scale 553 | ![]() | Phradic | ||
Scale 649 | ![]() | Byptic | ||
Scale 713 | ![]() | Thoptitonic | ||
Scale 841 | ![]() | Phrothitonic | ||
Scale 73 | ![]() | Diminished Triad | ||
Scale 329 | ![]() | Lonic | ||
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. All the calculations on this page are done in real time by proprietary software created by Ian Ring. Scale notation is generated by VexFlow and Lilypond, graph visualization by Graphviz, and MIDI playback and audio manipulation by MIDI.js, TiMIDIty, and SOX. 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.