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Scale 3037: "Nine Tone Scale"

Scale 3037: Nine Tone Scale, Ian Ring Music Theory

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

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.

 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 TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsC{0,4,7}342.44
D{2,6,9}342.67
E{4,8,11}342.44
G{7,11,2}342.39
G♯{8,0,3}442.22
B{11,3,6}442.28
Minor Triadscm{0,3,7}442.17
em{4,7,11}342.39
g♯m{8,11,3}442.17
am{9,0,4}342.56
bm{11,2,6}342.5
Augmented TriadsC+{0,4,8}442.33
D♯+{3,7,11}542
Diminished Triadsc°{0,3,6}242.56
d♯°{3,6,9}242.72
f♯°{6,9,0}242.72
g♯°{8,11,2}242.67
a°{9,0,3}242.67
Parsimonious Voice Leading Between Common Triads of Scale 3037. Created by Ian Ring ©2019 c° c° cm cm c°->cm B B c°->B C C cm->C D#+ D#+ cm->D#+ G# G# cm->G# C+ C+ C->C+ em em C->em E E C+->E C+->G# am am C+->am D D d#° d#° D->d#° f#° f#° D->f#° bm bm D->bm d#°->B D#+->em Parsimonious Voice Leading Between Common Triads of Scale 3037. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m D#+->B em->E E->g#m f#°->am g#° g#° G->g#° G->bm g#°->g#m g#m->G# a° a° G#->a° a°->am bm->B

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.

Diameter4
Radius4
Self-Centeredyes

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

Prime

The prime form of this scale is Scale 1775

Scale 1775Scale 1775: Lyrygic, Ian Ring Music TheoryLyrygic

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 1915Scale 1915: Thydygic, Ian Ring Music TheoryThydygic

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.

kHierarchizabilityBreakdown PatternDiagram
111011101111013037k = 1h = 1
2310([11]0[11])([11]0[11])3037k = 2h = 3
3310([11]0[11])([11]0[11])3037k = 3h = 3
4310([11]0[11])([11]0[11])3037k = 4h = 3
5310([11]0[11])([11]0[11])3037k = 5h = 3

Enantiomorph

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

Scale 1915Scale 1915: Thydygic, Ian Ring Music TheoryThydygic

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 3039Scale 3039: Godyllian, Ian Ring Music TheoryGodyllian
Scale 3033Scale 3033: Doptyllic, Ian Ring Music TheoryDoptyllic
Scale 3035Scale 3035: Gocrygic, Ian Ring Music TheoryGocrygic
Scale 3029Scale 3029: Ionocryllic, Ian Ring Music TheoryIonocryllic
Scale 3021Scale 3021: Gyptyllic, Ian Ring Music TheoryGyptyllic
Scale 3053Scale 3053: Zycrygic, Ian Ring Music TheoryZycrygic
Scale 3069Scale 3069: Bacryllian, Ian Ring Music TheoryBacryllian
Scale 2973Scale 2973: Panyllic, Ian Ring Music TheoryPanyllic
Scale 3005Scale 3005: Gycrygic, Ian Ring Music TheoryGycrygic
Scale 2909Scale 2909: Mocryllic, Ian Ring Music TheoryMocryllic
Scale 2781Scale 2781: Gycryllic, Ian Ring Music TheoryGycryllic
Scale 2525Scale 2525: Aeolaryllic, Ian Ring Music TheoryAeolaryllic
Scale 3549Scale 3549: Messiaen Mode 3, Ian Ring Music TheoryMessiaen Mode 3
Scale 4061Scale 4061: Staptyllian, Ian Ring Music TheoryStaptyllian
Scale 989Scale 989: Phrolyllic, Ian Ring Music TheoryPhrolyllic
Scale 2013Scale 2013: Mocrygic, Ian Ring Music TheoryMocrygic

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. (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.