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Scale 2969: "Tholian"

Scale 2969: Tholian, 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

Zeitler
Tholian
Dozenal
Sowian

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

{0,3,4,7,8,9,11}

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

7-21

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: 827

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

4 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

1 (uncohemitonic)

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.

3

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 Rahn/Ring formula.

no
prime: 823

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.

no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[3, 1, 3, 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.

<4, 2, 4, 6, 4, 1>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p4m6n4s2d4t

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> = {1,2,3}
<2> = {2,3,4}
<3> = {4,5,6,7}
<4> = {5,6,7,8}
<5> = {8,9,10}
<6> = {9,10,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

2

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.

2.433

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.899

Myhill Property

A scale has Myhill Property if the Interval Spectra has 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".

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.

(7, 33, 90)

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.

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}331.7
E{4,8,11}331.7
G♯{8,0,3}431.5
Minor Triadscm{0,3,7}331.7
em{4,7,11}341.9
g♯m{8,11,3}331.7
am{9,0,4}242.1
Augmented TriadsC+{0,4,8}431.5
D♯+{3,7,11}341.9
Diminished Triads{9,0,3}242.1
Parsimonious Voice Leading Between Common Triads of Scale 2969. Created by Ian Ring ©2019 cm cm 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#+->em g#m g#m D#+->g#m em->E E->g#m g#m->G# G#->a° a°->am

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
Radius3
Self-Centeredno
Central Verticescm, C, C+, E, g♯m, G♯
Peripheral VerticesD♯+, em, a°, am

Modes

Modes are the rotational transformation of this scale. Scale 2969 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 883
Scale 883: Ralian, Ian Ring Music TheoryRalian
3rd mode:
Scale 2489
Scale 2489: Mela Gangeyabhusani, Ian Ring Music TheoryMela Gangeyabhusani
4th mode:
Scale 823
Scale 823: Stodian, Ian Ring Music TheoryStodianThis is the prime mode
5th mode:
Scale 2459
Scale 2459: Ionocrian, Ian Ring Music TheoryIonocrian
6th mode:
Scale 3277
Scale 3277: Mela Nitimati, Ian Ring Music TheoryMela Nitimati
7th mode:
Scale 1843
Scale 1843: Ionygian, Ian Ring Music TheoryIonygian

Prime

The prime form of this scale is Scale 823

Scale 823Scale 823: Stodian, Ian Ring Music TheoryStodian

Complement

The heptatonic modal family [2969, 883, 2489, 823, 2459, 3277, 1843] (Forte: 7-21) is the complement of the pentatonic modal family [307, 787, 817, 2201, 2441] (Forte: 5-21)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2969 is 827

Scale 827Scale 827: Mixolocrian, Ian Ring Music TheoryMixolocrian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2969 is chiral, and its enantiomorph is scale 827

Scale 827Scale 827: Mixolocrian, Ian Ring Music TheoryMixolocrian

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> 2969       T0I <11,0> 827
T1 <1,1> 1843      T1I <11,1> 1654
T2 <1,2> 3686      T2I <11,2> 3308
T3 <1,3> 3277      T3I <11,3> 2521
T4 <1,4> 2459      T4I <11,4> 947
T5 <1,5> 823      T5I <11,5> 1894
T6 <1,6> 1646      T6I <11,6> 3788
T7 <1,7> 3292      T7I <11,7> 3481
T8 <1,8> 2489      T8I <11,8> 2867
T9 <1,9> 883      T9I <11,9> 1639
T10 <1,10> 1766      T10I <11,10> 3278
T11 <1,11> 3532      T11I <11,11> 2461
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2969       T0MI <7,0> 827
T1M <5,1> 1843      T1MI <7,1> 1654
T2M <5,2> 3686      T2MI <7,2> 3308
T3M <5,3> 3277      T3MI <7,3> 2521
T4M <5,4> 2459      T4MI <7,4> 947
T5M <5,5> 823      T5MI <7,5> 1894
T6M <5,6> 1646      T6MI <7,6> 3788
T7M <5,7> 3292      T7MI <7,7> 3481
T8M <5,8> 2489      T8MI <7,8> 2867
T9M <5,9> 883      T9MI <7,9> 1639
T10M <5,10> 1766      T10MI <7,10> 3278
T11M <5,11> 3532      T11MI <7,11> 2461

The transformations that map this set to itself are: T0, T0M

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 2971Scale 2971: Aeolynyllic, Ian Ring Music TheoryAeolynyllic
Scale 2973Scale 2973: Panyllic, Ian Ring Music TheoryPanyllic
Scale 2961Scale 2961: Bygimic, Ian Ring Music TheoryBygimic
Scale 2965Scale 2965: Darian, Ian Ring Music TheoryDarian
Scale 2953Scale 2953: Ionylimic, Ian Ring Music TheoryIonylimic
Scale 2985Scale 2985: Epacrian, Ian Ring Music TheoryEpacrian
Scale 3001Scale 3001: Lonyllic, Ian Ring Music TheoryLonyllic
Scale 3033Scale 3033: Doptyllic, Ian Ring Music TheoryDoptyllic
Scale 2841Scale 2841: Sothimic, Ian Ring Music TheorySothimic
Scale 2905Scale 2905: Aeolian Flat 1, Ian Ring Music TheoryAeolian Flat 1
Scale 2713Scale 2713: Porimic, Ian Ring Music TheoryPorimic
Scale 2457Scale 2457: Augmented, Ian Ring Music TheoryAugmented
Scale 3481Scale 3481: Katathian, Ian Ring Music TheoryKatathian
Scale 3993Scale 3993: Ioniptyllic, Ian Ring Music TheoryIoniptyllic
Scale 921Scale 921: Bogimic, Ian Ring Music TheoryBogimic
Scale 1945Scale 1945: Zarian, Ian Ring Music TheoryZarian

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.