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Scale 3187: "Koptian"

Scale 3187: Koptian, 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
Koptian

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

{0,1,4,5,6,10,11}

Forte Number

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

7-7

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: 2503

Hemitonia

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

5 (multihemitonic)

Cohemitonia

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

3 (tricohemitonic)

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.

2

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: 463

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Formula

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

[1, 3, 1, 1, 4, 1, 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.

<5, 3, 2, 3, 5, 3>

Interval Spectrum

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

p5m3n2s3d5t3

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

Spectra Variation

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

2.571

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.

2.183

Polygon Perimeter

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

5.734

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

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 TriadsF♯{6,10,1}121
Minor Triadsa♯m{10,1,5}210.67
Diminished Triadsa♯°{10,1,4}121

The following pitch classes are not present in any of the common triads: {0,11}

Parsimonious Voice Leading Between Common Triads of Scale 3187. Created by Ian Ring ©2019 F# F# a#m a#m F#->a#m a#° a#° a#°->a#m

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 Verticesa♯m
Peripheral VerticesF♯, a♯°

Modes

Modes are the rotational transformation of this scale. Scale 3187 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3641
Scale 3641: Thocrian, Ian Ring Music TheoryThocrian
3rd mode:
Scale 967
Scale 967: Mela Salaga, Ian Ring Music TheoryMela Salaga
4th mode:
Scale 2531
Scale 2531: Danian, Ian Ring Music TheoryDanian
5th mode:
Scale 3313
Scale 3313: Aeolacrian, Ian Ring Music TheoryAeolacrian
6th mode:
Scale 463
Scale 463: Zythian, Ian Ring Music TheoryZythianThis is the prime mode
7th mode:
Scale 2279
Scale 2279: Dyrian, Ian Ring Music TheoryDyrian

Prime

The prime form of this scale is Scale 463

Scale 463Scale 463: Zythian, Ian Ring Music TheoryZythian

Complement

The heptatonic modal family [3187, 3641, 967, 2531, 3313, 463, 2279] (Forte: 7-7) is the complement of the pentatonic modal family [199, 451, 2147, 2273, 3121] (Forte: 5-7)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3187 is 2503

Scale 2503Scale 2503: Mela Jhalavarali, Ian Ring Music TheoryMela Jhalavarali

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3187 is chiral, and its enantiomorph is scale 2503

Scale 2503Scale 2503: Mela Jhalavarali, Ian Ring Music TheoryMela Jhalavarali

Transformations:

T0 3187  T0I 2503
T1 2279  T1I 911
T2 463  T2I 1822
T3 926  T3I 3644
T4 1852  T4I 3193
T5 3704  T5I 2291
T6 3313  T6I 487
T7 2531  T7I 974
T8 967  T8I 1948
T9 1934  T9I 3896
T10 3868  T10I 3697
T11 3641  T11I 3299

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 3185Scale 3185: Messiaen Mode 5 Inverse, Ian Ring Music TheoryMessiaen Mode 5 Inverse
Scale 3189Scale 3189: Aeolonian, Ian Ring Music TheoryAeolonian
Scale 3191Scale 3191: Bynyllic, Ian Ring Music TheoryBynyllic
Scale 3195Scale 3195: Raryllic, Ian Ring Music TheoryRaryllic
Scale 3171Scale 3171: Zythimic, Ian Ring Music TheoryZythimic
Scale 3179Scale 3179: Daptian, Ian Ring Music TheoryDaptian
Scale 3155Scale 3155: Ladimic, Ian Ring Music TheoryLadimic
Scale 3123Scale 3123, Ian Ring Music Theory
Scale 3251Scale 3251: Mela Hatakambari, Ian Ring Music TheoryMela Hatakambari
Scale 3315Scale 3315: Tcherepnin Octatonic Mode 1, Ian Ring Music TheoryTcherepnin Octatonic Mode 1
Scale 3443Scale 3443: Verdi's Scala Enigmatica, Ian Ring Music TheoryVerdi's Scala Enigmatica
Scale 3699Scale 3699: Galyllic, Ian Ring Music TheoryGalyllic
Scale 2163Scale 2163, Ian Ring Music Theory
Scale 2675Scale 2675: Chromatic Lydian, Ian Ring Music TheoryChromatic Lydian
Scale 1139Scale 1139: Aerygimic, Ian Ring Music TheoryAerygimic

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. 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.