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Scale 977: "Kocrimic"

Scale 977: Kocrimic, 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
Kocrimic
Dozenal
Galian

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

{0,4,6,7,8,9}

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-Z39

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: 377

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.

4

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.

no
prime: 317

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.

[4, 2, 1, 1, 1, 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.

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

Interval Spectrum

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

p2m3n3s3d3t

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> = {2,3,4,6,7}
<3> = {3,4,5,7,8,9}
<4> = {5,6,8,9,10}
<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, 19, 67)

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}131.5
Minor Triadsam{9,0,4}221
Augmented TriadsC+{0,4,8}221
Diminished Triadsf♯°{6,9,0}131.5
Parsimonious Voice Leading Between Common Triads of Scale 977. Created by Ian Ring ©2019 C C C+ C+ C->C+ am am C+->am f#° f#° f#°->am

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+, am
Peripheral VerticesC, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 977 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 317
Scale 317: Korimic, Ian Ring Music TheoryKorimicThis is the prime mode
3rd mode:
Scale 1103
Scale 1103: Lynimic, Ian Ring Music TheoryLynimic
4th mode:
Scale 2599
Scale 2599: Malimic, Ian Ring Music TheoryMalimic
5th mode:
Scale 3347
Scale 3347: Synimic, Ian Ring Music TheorySynimic
6th mode:
Scale 3721
Scale 3721: Phragimic, Ian Ring Music TheoryPhragimic

Prime

The prime form of this scale is Scale 317

Scale 317Scale 317: Korimic, Ian Ring Music TheoryKorimic

Complement

The hexatonic modal family [977, 317, 1103, 2599, 3347, 3721] (Forte: 6-Z39) is the complement of the hexatonic modal family [187, 1559, 1889, 2141, 2827, 3461] (Forte: 6-Z10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 977 is 377

Scale 377Scale 377: Kathimic, Ian Ring Music TheoryKathimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 977 is chiral, and its enantiomorph is scale 377

Scale 377Scale 377: Kathimic, Ian Ring Music TheoryKathimic

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> 977       T0I <11,0> 377
T1 <1,1> 1954      T1I <11,1> 754
T2 <1,2> 3908      T2I <11,2> 1508
T3 <1,3> 3721      T3I <11,3> 3016
T4 <1,4> 3347      T4I <11,4> 1937
T5 <1,5> 2599      T5I <11,5> 3874
T6 <1,6> 1103      T6I <11,6> 3653
T7 <1,7> 2206      T7I <11,7> 3211
T8 <1,8> 317      T8I <11,8> 2327
T9 <1,9> 634      T9I <11,9> 559
T10 <1,10> 1268      T10I <11,10> 1118
T11 <1,11> 2536      T11I <11,11> 2236
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2897      T0MI <7,0> 347
T1M <5,1> 1699      T1MI <7,1> 694
T2M <5,2> 3398      T2MI <7,2> 1388
T3M <5,3> 2701      T3MI <7,3> 2776
T4M <5,4> 1307      T4MI <7,4> 1457
T5M <5,5> 2614      T5MI <7,5> 2914
T6M <5,6> 1133      T6MI <7,6> 1733
T7M <5,7> 2266      T7MI <7,7> 3466
T8M <5,8> 437      T8MI <7,8> 2837
T9M <5,9> 874      T9MI <7,9> 1579
T10M <5,10> 1748      T10MI <7,10> 3158
T11M <5,11> 3496      T11MI <7,11> 2221

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 979Scale 979: Mela Dhavalambari, Ian Ring Music TheoryMela Dhavalambari
Scale 981Scale 981: Mela Kantamani, Ian Ring Music TheoryMela Kantamani
Scale 985Scale 985: Mela Sucaritra, Ian Ring Music TheoryMela Sucaritra
Scale 961Scale 961: Gabian, Ian Ring Music TheoryGabian
Scale 969Scale 969: Ionogimic, Ian Ring Music TheoryIonogimic
Scale 993Scale 993: Gavian, Ian Ring Music TheoryGavian
Scale 1009Scale 1009: Katyptian, Ian Ring Music TheoryKatyptian
Scale 913Scale 913: Aeolyritonic, Ian Ring Music TheoryAeolyritonic
Scale 945Scale 945: Raga Saravati, Ian Ring Music TheoryRaga Saravati
Scale 849Scale 849: Aerynitonic, Ian Ring Music TheoryAerynitonic
Scale 721Scale 721: Raga Dhavalashri, Ian Ring Music TheoryRaga Dhavalashri
Scale 465Scale 465: Zoditonic, Ian Ring Music TheoryZoditonic
Scale 1489Scale 1489: Raga Jyoti, Ian Ring Music TheoryRaga Jyoti
Scale 2001Scale 2001: Gydian, Ian Ring Music TheoryGydian
Scale 3025Scale 3025: Epycrian, Ian Ring Music TheoryEpycrian

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.