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Scale 745: "Kolimic"

Scale 745: Kolimic, 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
Kolimic
Dozenal
EPPian

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

 {0,3,5,6,7,9}

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-Z45

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.

 [0]

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

 yes

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.

 4

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.

 6

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

 no
prime: 605

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.

 [3, 2, 1, 1, 2, 3]

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, 3, 4, 2, 2, 2>

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.

<0.4, 0.5, 0.8, 0, 0.4, 0.667>

Interval Spectrum

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

 p2m2n4s3d2t2

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

Spectra Variation

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

 2.667

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.

2.366

Polygon Perimeter

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

5.864

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.

 0.288675

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.

 [0]

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.

(16, 17, 62)

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.

0.492

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.

0.173

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 TriadsF{5,9,0}231.5
Minor Triadscm{0,3,7}231.5
Diminished Triadsc°{0,3,6}231.5
d♯°{3,6,9}231.5
f♯°{6,9,0}231.5
a°{9,0,3}231.5
Parsimonious Voice Leading Between Common Triads of Scale 745. Created by Ian Ring ©2019 c° c° cm cm c°->cm d#° d#° c°->d#° a° a° cm->a° f#° f#° d#°->f#° F F F->f#° F->a°

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
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 745 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 605		Scale 605: Dycrimic, Ian Ring Music TheoryDycrimicThis is the prime mode
3rd mode:
Scale 1175		Scale 1175: Epycrimic, Ian Ring Music TheoryEpycrimic
4th mode:
Scale 2635		Scale 2635: Gocrimic, Ian Ring Music TheoryGocrimic
5th mode:
Scale 3365		Scale 3365: Katolimic, Ian Ring Music TheoryKatolimic
6th mode:
Scale 1865		Scale 1865: Thagimic, Ian Ring Music TheoryThagimic

Prime

The prime form of this scale is Scale 605

Scale 605Scale 605: Dycrimic, Ian Ring Music TheoryDycrimic

Complement

The hexatonic modal family [745, 605, 1175, 2635, 3365, 1865] (Forte: 6-Z45) is the complement of the hexatonic modal family [365, 1115, 1675, 1745, 2605, 2885] (Forte: 6-Z23)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 745 is itself, because it is a palindromic scale!

Scale 745Scale 745: Kolimic, Ian Ring Music TheoryKolimic

Interval Matrix

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

Contradictions (16)

Ambiguities(17)

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
11100101110100745k = 1h = 1
21100101110100745k = 2h = 1
33([0]1[0])111([0]1[0])([0]1[0])745k = 3h = 3
43([0]1[0])111([0]1[0])([0]1[0])745k = 4h = 3
53([0]1[0])111([0]1[0])([0]1[0])745k = 5h = 3

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, 0.288675)
Distance from Center0.288675
Angle in degrees

measured clockwise starting from the root.

180
Angle in cents

100 cents = 1 semitone.

600

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> 745       T0I <11,0> 745
T1 <1,1> 1490      T1I <11,1> 1490
T2 <1,2> 2980      T2I <11,2> 2980
T3 <1,3> 1865      T3I <11,3> 1865
T4 <1,4> 3730      T4I <11,4> 3730
T5 <1,5> 3365      T5I <11,5> 3365
T6 <1,6> 2635      T6I <11,6> 2635
T7 <1,7> 1175      T7I <11,7> 1175
T8 <1,8> 2350      T8I <11,8> 2350
T9 <1,9> 605      T9I <11,9> 605
T10 <1,10> 1210      T10I <11,10> 1210
T11 <1,11> 2420      T11I <11,11> 2420
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2635      T0MI <7,0> 2635
T1M <5,1> 1175      T1MI <7,1> 1175
T2M <5,2> 2350      T2MI <7,2> 2350
T3M <5,3> 605      T3MI <7,3> 605
T4M <5,4> 1210      T4MI <7,4> 1210
T5M <5,5> 2420      T5MI <7,5> 2420
T6M <5,6> 745       T6MI <7,6> 745
T7M <5,7> 1490      T7MI <7,7> 1490
T8M <5,8> 2980      T8MI <7,8> 2980
T9M <5,9> 1865      T9MI <7,9> 1865
T10M <5,10> 3730      T10MI <7,10> 3730
T11M <5,11> 3365      T11MI <7,11> 3365

The transformations that map this set to itself are: T0, T0I, T6M, T6MI

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 747Scale 747: Locrian 𝄫67, Ian Ring Music TheoryLocrian 67
Scale 749Scale 749: Locrian ♮2𝄫67, Ian Ring Music TheoryLocrian 267
Scale 737Scale 737: TRUian, Ian Ring Music TheoryTRUian
Scale 741Scale 741: Gathimic, Ian Ring Music TheoryGathimic
Scale 753Scale 753: Kytrimic, Ian Ring Music TheoryKytrimic
Scale 761Scale 761: Major Locrian ♯2𝄫67, Ian Ring Music TheoryMajor Locrian 267
Scale 713Scale 713: Thoptitonic, Ian Ring Music TheoryThoptitonic
Scale 729Scale 729: Stygimic, Ian Ring Music TheoryStygimic
Scale 681Scale 681: Minor Added Sixth Pentatonic, Ian Ring Music TheoryMinor Added Sixth Pentatonic
Scale 617Scale 617: Katycritonic, Ian Ring Music TheoryKatycritonic
Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic
Scale 1001Scale 1001: Lydian ♯23♭6𝄫7, Ian Ring Music TheoryLydian 2367
Scale 233Scale 233: BIGian, Ian Ring Music TheoryBIGian
Scale 489Scale 489: Phrathimic, Ian Ring Music TheoryPhrathimic
Scale 1257Scale 1257: Minor Blues, Ian Ring Music TheoryMinor Blues
Scale 1769Scale 1769: Lydian 7♯23, Ian Ring Music TheoryLydian 723
Scale 2793Scale 2793: Lydian ♯23, Ian Ring Music TheoryLydian 23

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, and ©2023 Robert Bedwell, 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.