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Scale 313: "Goritonic"

Scale 313: Goritonic, 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
Goritonic
Dozenal
Cacian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

5 (pentatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,3,4,5,8}

Forte Number

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

5-Z37

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.

[4]

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.

no

Hemitonia

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

2 (dihemitonic)

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.

4

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

yes

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, 1, 3, 4]

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.

<2, 1, 2, 3, 2, 0>

Interval Spectrum

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

p2m3n2sd2

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

Spectra Variation

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

3.2

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.

1.933

Polygon Perimeter

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

5.596

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.

[8]

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.

(10, 4, 32)

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 TriadsG♯{8,0,3}121
Minor Triadsfm{5,8,0}121
Augmented TriadsC+{0,4,8}210.67
Parsimonious Voice Leading Between Common Triads of Scale 313. Created by Ian Ring ©2019 C+ C+ fm fm C+->fm G# G# C+->G#

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.

Diameter2
Radius1
Self-Centeredno
Central VerticesC+
Peripheral Verticesfm, G♯

Modes

Modes are the rotational transformation of this scale. Scale 313 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 551
Scale 551: Aeoloditonic, Ian Ring Music TheoryAeoloditonic
3rd mode:
Scale 2323
Scale 2323: Doptitonic, Ian Ring Music TheoryDoptitonic
4th mode:
Scale 3209
Scale 3209: Aeraphitonic, Ian Ring Music TheoryAeraphitonic
5th mode:
Scale 913
Scale 913: Aeolyritonic, Ian Ring Music TheoryAeolyritonic

Prime

This is the prime form of this scale.

Complement

The pentatonic modal family [313, 551, 2323, 3209, 913] (Forte: 5-Z37) is the complement of the heptatonic modal family [443, 1591, 1891, 2269, 2843, 2993, 3469] (Forte: 7-Z37)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 313 is 913

Scale 913Scale 913: Aeolyritonic, Ian Ring Music TheoryAeolyritonic

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> 313       T0I <11,0> 913
T1 <1,1> 626      T1I <11,1> 1826
T2 <1,2> 1252      T2I <11,2> 3652
T3 <1,3> 2504      T3I <11,3> 3209
T4 <1,4> 913      T4I <11,4> 2323
T5 <1,5> 1826      T5I <11,5> 551
T6 <1,6> 3652      T6I <11,6> 1102
T7 <1,7> 3209      T7I <11,7> 2204
T8 <1,8> 2323      T8I <11,8> 313
T9 <1,9> 551      T9I <11,9> 626
T10 <1,10> 1102      T10I <11,10> 1252
T11 <1,11> 2204      T11I <11,11> 2504
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 283      T0MI <7,0> 2833
T1M <5,1> 566      T1MI <7,1> 1571
T2M <5,2> 1132      T2MI <7,2> 3142
T3M <5,3> 2264      T3MI <7,3> 2189
T4M <5,4> 433      T4MI <7,4> 283
T5M <5,5> 866      T5MI <7,5> 566
T6M <5,6> 1732      T6MI <7,6> 1132
T7M <5,7> 3464      T7MI <7,7> 2264
T8M <5,8> 2833      T8MI <7,8> 433
T9M <5,9> 1571      T9MI <7,9> 866
T10M <5,10> 3142      T10MI <7,10> 1732
T11M <5,11> 2189      T11MI <7,11> 3464

The transformations that map this set to itself are: T0, T8I

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 315Scale 315: Stodimic, Ian Ring Music TheoryStodimic
Scale 317Scale 317: Korimic, Ian Ring Music TheoryKorimic
Scale 305Scale 305: Gonic, Ian Ring Music TheoryGonic
Scale 309Scale 309: Palitonic, Ian Ring Music TheoryPalitonic
Scale 297Scale 297: Mynic, Ian Ring Music TheoryMynic
Scale 281Scale 281: Lanic, Ian Ring Music TheoryLanic
Scale 345Scale 345: Gylitonic, Ian Ring Music TheoryGylitonic
Scale 377Scale 377: Kathimic, Ian Ring Music TheoryKathimic
Scale 441Scale 441: Thycrimic, Ian Ring Music TheoryThycrimic
Scale 57Scale 57: Ahoian, Ian Ring Music TheoryAhoian
Scale 185Scale 185: Becian, Ian Ring Music TheoryBecian
Scale 569Scale 569: Mothitonic, Ian Ring Music TheoryMothitonic
Scale 825Scale 825: Thyptimic, Ian Ring Music TheoryThyptimic
Scale 1337Scale 1337: Epogimic, Ian Ring Music TheoryEpogimic
Scale 2361Scale 2361: Docrimic, Ian Ring Music TheoryDocrimic

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.