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CardinalityCardinality is the count of how many pitches are in the scale. |
7 (heptatonic) |
Pitch Class SetThe tones in this scale, expressed as numbers from 0 to 11 |
{0,1,2,3,9,10,11} |
Forte NumberA code assigned by theorist Alan Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations. |
7-1 |
Rotational SymmetrySome scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity. |
none |
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] |
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. |
6 (multihemitonic) |
CohemitoniaA cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist. |
5 (multicohemitonic) |
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. |
5 |
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. |
6 |
Prime FormDescribes if this scale is in prime form, using the Rahn/Ring formula. |
no prime: 127 |
Deep ScaleA deep scale is one where the interval vector has 6 different digits. |
yes |
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. |
[6, 5, 4, 3, 2, 1] |
Interval SpectrumThe same as the Interval Vector, but expressed in a syntax used by Howard Hansen. |
p2m3n4s5d6t |
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> = {1,6} <2> = {2,7} <3> = {3,8} <4> = {4,9} <5> = {5,10} <6> = {6,11} |
Spectra VariationDetermined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality. |
4.286 |
Maximally EvenA scale is maximally even if the tones are optimally spaced apart from each other. |
no |
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. |
no |
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. |
1.5 |
Polygon PerimeterPerimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle. |
5.106 |
Myhill PropertyA scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval. |
yes |
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. |
no |
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] |
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". | Improper |
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 | a° | {9,0,3} | 0 | 0 | 0 |
The following pitch classes are not present in any of the common triads: {1,2,10,11}
Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.
Modes are the rotational transformation of this scale. Scale 3599 can be rotated to make 6 other scales. The 1st mode is itself.
2nd mode: Scale 3847 | ![]() | Heptatonic Chromatic 5 | |||
3rd mode: Scale 3971 | ![]() | Heptatonic Chromatic 6 | |||
4th mode: Scale 4033 | ![]() | Heptatonic Chromatic Descending | |||
5th mode: Scale 127 | ![]() | Heptatonic Chromatic | This is the prime mode | ||
6th mode: Scale 2111 | ![]() | Heptatonic Chromatic 2 | |||
7th mode: Scale 3103 | ![]() | Heptatonic Chromatic 3 |
The prime form of this scale is Scale 127
Scale 127 | ![]() | Heptatonic Chromatic |
The heptatonic modal family [3599, 3847, 3971, 4033, 127, 2111, 3103] (Forte: 7-1) is the complement of the pentatonic modal family [31, 2063, 3079, 3587, 3841] (Forte: 5-1)
The inverse of a scale is a reflection using the root as its axis. The inverse of 3599 is itself, because it is a palindromic scale!
Scale 3599 | ![]() | Heptatonic Chromatic 4 |
T0 | 3599 | T0I | 3599 | |||||
T1 | 3103 | T1I | 3103 | |||||
T2 | 2111 | T2I | 2111 | |||||
T3 | 127 | T3I | 127 | |||||
T4 | 254 | T4I | 254 | |||||
T5 | 508 | T5I | 508 | |||||
T6 | 1016 | T6I | 1016 | |||||
T7 | 2032 | T7I | 2032 | |||||
T8 | 4064 | T8I | 4064 | |||||
T9 | 4033 | T9I | 4033 | |||||
T10 | 3971 | T10I | 3971 | |||||
T11 | 3847 | T11I | 3847 |
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 3597 | ![]() | WIJIAN | ||
Scale 3595 | ![]() | WIHIAN | ||
Scale 3591 | ![]() | WIFIAN | ||
Scale 3607 | ![]() | WOPIAN | ||
Scale 3615 | ![]() | Octatonic Chromatic 4 | ||
Scale 3631 | ![]() | Gydyllic | ||
Scale 3663 | ![]() | Sonyllic | ||
Scale 3727 | ![]() | Tholyllic | ||
Scale 3855 | ![]() | Octatonic Chromatic 5 | ||
Scale 3087 | ![]() | Hexatonic Chromatic 3 | ||
Scale 3343 | ![]() | VAJIAN | ||
Scale 2575 | ![]() | PUMIAN | ||
Scale 1551 | ![]() | JORIAN |
This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. 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.