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Scale 3037: "Nine Tone Scale"

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: Nine Tone Scale

Western: Diminishing Nonamode 5th Rotation

Zeitler: Staptygic

Dozenal: TALian

Analysis

Cardinality Cardinality is the count of how many pitches are in the scale.	9 (enneatonic)
Pitch Class Set The tones in this scale, expressed as numbers from 0 to 11	{0,2,3,4,6,7,8,9,11}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	9-11
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.	none
Palindromicity A palindromic scale has the same pattern of intervals both ascending and descending.	no
Chirality A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.	yes enantiomorph: 1915
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.	3 (tricohemitonic)
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.	2
Modes Modes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.	9
Prime Form Describes if this scale is in prime form, using the Starr/Rahn algorithm.	no prime: 1775
Generator Indicates if the scale can be constructed using a generator, and an origin.	none
Deep Scale A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.	no
Interval Structure Defines the scale as the sequence of intervals between one tone and the next.	[2, 1, 1, 2, 1, 1, 1, 2, 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, 6, 7, 7, 7, 3>
Proportional Saturation Vector First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.	<0, 0, 0.5, 0.333, 0.5, 0>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	p⁷m⁷n⁷s⁶d⁶t³
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> = {1,2} <2> = {2,3} <3> = {3,4,5} <4> = {5,6} <5> = {6,7} <6> = {7,8,9} <7> = {9,10} <8> = {10,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	1.111
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.	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.	2.799
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	6.106
Myhill Property A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.	no
Centre of Gravity Distance When tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero.	0.057516
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.	none
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".	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, 51, 138)
Coherence Quotient The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.	0.89
Sameness Quotient The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.	0.521

Generator

This scale has no generator.

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
Major Triads	C	{0,4,7}	3	4	2.44
	D	{2,6,9}	3	4	2.67
	E	{4,8,11}	3	4	2.44
	G	{7,11,2}	3	4	2.39
	G♯	{8,0,3}	4	4	2.22
	B	{11,3,6}	4	4	2.28
Minor Triads	cm	{0,3,7}	4	4	2.17
	em	{4,7,11}	3	4	2.39
	g♯m	{8,11,3}	4	4	2.17
	am	{9,0,4}	3	4	2.56
	bm	{11,2,6}	3	4	2.5
Augmented Triads	C+	{0,4,8}	4	4	2.33
Augmented Triads	D♯+	{3,7,11}	5	4	2
Diminished Triads	c°	{0,3,6}	2	4	2.56
	d♯°	{3,6,9}	2	4	2.72
	f♯°	{6,9,0}	2	4	2.72
	g♯°	{8,11,2}	2	4	2.67
	a°	{9,0,3}	2	4	2.67

view full size

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	4
Radius	4
Self-Centered	yes

Modes

Modes are the rotational transformation of this scale. Scale 3037 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode: Scale 1783				Youlan
3rd mode: Scale 2939				Diminishing Nonamode 2nd Rotation
4th mode: Scale 3517				Diminishing Nonamode 1st Rotation
5th mode: Scale 1903				Diminishing Nonamode Basic
6th mode: Scale 2999				Diminishing Nonamode
7th mode: Scale 3547				Diminishing Nonamode 9th Rotation
8th mode: Scale 3821				Diminishing Nonamode 8th Rotation
9th mode: Scale 1979				Diminishing Nonamode 6th Rotation

Prime

The prime form of this scale is Scale 1775

Scale 1775

Lyrygic

Complement

The enneatonic modal family [3037, 1783, 2939, 3517, 1903, 2999, 3547, 3821, 1979] (Forte: 9-11) is the complement of the tritonic modal family [137, 289, 529] (Forte: 3-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3037 is 1915

Scale 1915

Thydygic

Interval Matrix

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

Contradictions (0)

Ambiguities(51)

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

k	Hierarchizability	Breakdown Pattern
1	1	`101110111101`
2	3	`10([11]0[11])([11]0[11])`
3	3	`10([11]0[11])([11]0[11])`
4	3	`10([11]0[11])([11]0[11])`
5	3	`10([11]0[11])([11]0[11])`

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3037 is chiral, and its enantiomorph is scale 1915

Scale 1915

Thydygic

Center of Gravity

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position with origin in the center	(-0.014886, 0.055556)
Distance from Center	0.057516
Angle in degrees measured clockwise starting from the root.	195
Angle in cents 100 cents = 1 semitone.	650

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev	Operation	Result	Abbrev	Operation	Result
T₀	<1,0>	3037	T₀I	<11,0>	1915
T₁	<1,1>	1979	T₁I	<11,1>	3830
T₂	<1,2>	3958	T₂I	<11,2>	3565
T₃	<1,3>	3821	T₃I	<11,3>	3035
T₄	<1,4>	3547	T₄I	<11,4>	1975
T₅	<1,5>	2999	T₅I	<11,5>	3950
T₆	<1,6>	1903	T₆I	<11,6>	3805
T₇	<1,7>	3806	T₇I	<11,7>	3515
T₈	<1,8>	3517	T₈I	<11,8>	2935
T₉	<1,9>	2939	T₉I	<11,9>	1775
T₁₀	<1,10>	1783	T₁₀I	<11,10>	3550
T₁₁	<1,11>	3566	T₁₁I	<11,11>	3005
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	4057	T₀MI	<7,0>	895
T₁M	<5,1>	4019	T₁MI	<7,1>	1790
T₂M	<5,2>	3943	T₂MI	<7,2>	3580
T₃M	<5,3>	3791	T₃MI	<7,3>	3065
T₄M	<5,4>	3487	T₄MI	<7,4>	2035
T₅M	<5,5>	2879	T₅MI	<7,5>	4070
T₆M	<5,6>	1663	T₆MI	<7,6>	4045
T₇M	<5,7>	3326	T₇MI	<7,7>	3995
T₈M	<5,8>	2557	T₈MI	<7,8>	3895
T₉M	<5,9>	1019	T₉MI	<7,9>	3695
T₁₀M	<5,10>	2038	T₁₀MI	<7,10>	3295
T₁₁M	<5,11>	4076	T₁₁MI	<7,11>	2495

The transformations that map this set to itself are: T₀

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 3039				Godyllian
Scale 3033				Doptyllic
Scale 3035				Gocrygic
Scale 3029				Ionocryllic
Scale 3021				Gyptyllic
Scale 3053				Zycrygic
Scale 3069				Bacryllian
Scale 2973				Panyllic
Scale 3005				Gycrygic
Scale 2909				Mocryllic
Scale 2781				Gycryllic
Scale 2525				Aeolaryllic
Scale 3549				Messiaen Mode 3
Scale 4061				Staptyllian
Scale 989				Phrolyllic
Scale 2013				Mocrygic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

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 naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Thanks to u/howaboot for inventing the Center of Gravity metrics.