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Scale 303: "Golimic"

Scale 303: Golimic, 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
Golimic
Dozenal
Buxian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,1,2,3,5,8}

Forte Number

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

6-Z40

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: 3729

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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.

5

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.

[1, 1, 1, 2, 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.

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

Interval Spectrum

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

p3m2n3s3d3t

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

Spectra Variation

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

3.667

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.

2.116

Polygon Perimeter

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

5.699

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.

(29, 16, 64)

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♯{1,5,8}221
G♯{8,0,3}131.5
Minor Triadsfm{5,8,0}221
Diminished Triads{2,5,8}131.5
Parsimonious Voice Leading Between Common Triads of Scale 303. Created by Ian Ring ©2019 C# C# C#->d° fm fm C#->fm G# G# fm->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.

Diameter3
Radius2
Self-Centeredno
Central VerticesC♯, fm
Peripheral Verticesd°, G♯

Modes

Modes are the rotational transformation of this scale. Scale 303 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2199
Scale 2199: Dyptimic, Ian Ring Music TheoryDyptimic
3rd mode:
Scale 3147
Scale 3147: Ryrimic, Ian Ring Music TheoryRyrimic
4th mode:
Scale 3621
Scale 3621: Gylimic, Ian Ring Music TheoryGylimic
5th mode:
Scale 1929
Scale 1929: Aeolycrimic, Ian Ring Music TheoryAeolycrimic
6th mode:
Scale 753
Scale 753: Aeronimic, Ian Ring Music TheoryAeronimic

Prime

This is the prime form of this scale.

Complement

The hexatonic modal family [303, 2199, 3147, 3621, 1929, 753] (Forte: 6-Z40) is the complement of the hexatonic modal family [183, 1761, 1803, 2139, 2949, 3117] (Forte: 6-Z11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 303 is 3729

Scale 3729Scale 3729: Starimic, Ian Ring Music TheoryStarimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 303 is chiral, and its enantiomorph is scale 3729

Scale 3729Scale 3729: Starimic, Ian Ring Music TheoryStarimic

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> 303       T0I <11,0> 3729
T1 <1,1> 606      T1I <11,1> 3363
T2 <1,2> 1212      T2I <11,2> 2631
T3 <1,3> 2424      T3I <11,3> 1167
T4 <1,4> 753      T4I <11,4> 2334
T5 <1,5> 1506      T5I <11,5> 573
T6 <1,6> 3012      T6I <11,6> 1146
T7 <1,7> 1929      T7I <11,7> 2292
T8 <1,8> 3858      T8I <11,8> 489
T9 <1,9> 3621      T9I <11,9> 978
T10 <1,10> 3147      T10I <11,10> 1956
T11 <1,11> 2199      T11I <11,11> 3912
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1083      T0MI <7,0> 2949
T1M <5,1> 2166      T1MI <7,1> 1803
T2M <5,2> 237      T2MI <7,2> 3606
T3M <5,3> 474      T3MI <7,3> 3117
T4M <5,4> 948      T4MI <7,4> 2139
T5M <5,5> 1896      T5MI <7,5> 183
T6M <5,6> 3792      T6MI <7,6> 366
T7M <5,7> 3489      T7MI <7,7> 732
T8M <5,8> 2883      T8MI <7,8> 1464
T9M <5,9> 1671      T9MI <7,9> 2928
T10M <5,10> 3342      T10MI <7,10> 1761
T11M <5,11> 2589      T11MI <7,11> 3522

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 301Scale 301: Raga Audav Tukhari, Ian Ring Music TheoryRaga Audav Tukhari
Scale 299Scale 299: Raga Chitthakarshini, Ian Ring Music TheoryRaga Chitthakarshini
Scale 295Scale 295: Gyritonic, Ian Ring Music TheoryGyritonic
Scale 311Scale 311: Stagimic, Ian Ring Music TheoryStagimic
Scale 319Scale 319: Epodian, Ian Ring Music TheoryEpodian
Scale 271Scale 271: Bodian, Ian Ring Music TheoryBodian
Scale 287Scale 287: Gynimic, Ian Ring Music TheoryGynimic
Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic
Scale 367Scale 367: Aerodian, Ian Ring Music TheoryAerodian
Scale 431Scale 431: Epyrian, Ian Ring Music TheoryEpyrian
Scale 47Scale 47: Agoian, Ian Ring Music TheoryAgoian
Scale 175Scale 175: Bewian, Ian Ring Music TheoryBewian
Scale 559Scale 559: Lylimic, Ian Ring Music TheoryLylimic
Scale 815Scale 815: Bolian, Ian Ring Music TheoryBolian
Scale 1327Scale 1327: Zalian, Ian Ring Music TheoryZalian
Scale 2351Scale 2351: Gynian, Ian Ring Music TheoryGynian

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.