The Exciting Universe Of Music Theory

presents

more than you ever wanted to know about...

Scale 3599: "Heptatonic Chromatic 4"

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 Modern: Heptatonic Chromatic 4

Dozenal: Wikian

Analysis

Cardinality Cardinality is the count of how many pitches are in the scale.	7 (heptatonic)
Pitch Class Set The tones in this scale, expressed as numbers from 0 to 11	{0,1,2,3,9,10,11}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	7-1
Rotational Symmetry Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.	none
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]
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.	6 (multihemitonic)
Cohemitonia A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.	5 (multicohemitonic)
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.	5
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.	6
Prime Form Describes if this scale is in prime form, using the Starr/Rahn algorithm.	no prime: 127
Generator Indicates if the scale can be constructed using a generator, and an origin.	generator: 1 origin: 9
Deep Scale A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.	yes
Interval Structure Defines the scale as the sequence of intervals between one tone and the next.	[1, 1, 1, 6, 1, 1, 1]
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.	<6, 5, 4, 3, 2, 1>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	p²m³n⁴s⁵d⁶t
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> = {1,6} <2> = {2,7} <3> = {3,8} <4> = {4,9} <5> = {5,10} <6> = {6,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	4.286
Maximally Even A scale is maximally even if the tones are optimally spaced apart from each other.	no
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.	no
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.	1.5
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	5.106
Myhill Property A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.	yes
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.	no
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]
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".	Improper
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.	(69, 1, 56)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

Generator

This scale has a generator of 1, originating on 9.

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	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

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

Prime

The prime form of this scale is Scale 127

Scale 127

Heptatonic Chromatic

Complement

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)

Inverse

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

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>	3599	T₀I	<11,0>	3599
T₁	<1,1>	3103	T₁I	<11,1>	3103
T₂	<1,2>	2111	T₂I	<11,2>	2111
T₃	<1,3>	127	T₃I	<11,3>	127
T₄	<1,4>	254	T₄I	<11,4>	254
T₅	<1,5>	508	T₅I	<11,5>	508
T₆	<1,6>	1016	T₆I	<11,6>	1016
T₇	<1,7>	2032	T₇I	<11,7>	2032
T₈	<1,8>	4064	T₈I	<11,8>	4064
T₉	<1,9>	4033	T₉I	<11,9>	4033
T₁₀	<1,10>	3971	T₁₀I	<11,10>	3971
T₁₁	<1,11>	3847	T₁₁I	<11,11>	3847
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	1709	T₀MI	<7,0>	1709
T₁M	<5,1>	3418	T₁MI	<7,1>	3418
T₂M	<5,2>	2741	T₂MI	<7,2>	2741
T₃M	<5,3>	1387	T₃MI	<7,3>	1387
T₄M	<5,4>	2774	T₄MI	<7,4>	2774
T₅M	<5,5>	1453	T₅MI	<7,5>	1453
T₆M	<5,6>	2906	T₆MI	<7,6>	2906
T₇M	<5,7>	1717	T₇MI	<7,7>	1717
T₈M	<5,8>	3434	T₈MI	<7,8>	3434
T₉M	<5,9>	2773	T₉MI	<7,9>	2773
T₁₀M	<5,10>	1451	T₁₀MI	<7,10>	1451
T₁₁M	<5,11>	2902	T₁₁MI	<7,11>	2902

The transformations that map this set to itself are: T₀, T₀I

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 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. 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.