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# Scale 3187: "Koptian" ### 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).

Zeitler
Koptian
Dozenal
TUZian

## 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 Starr/Rahn algorithm.

no
prime: 463

#### 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.

[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>

#### 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.

<0.75, 0.25, 0, 0, 0.75, 1>

#### 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 Distribution Spectra have 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.

(22, 27, 84)

#### 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.

0.65

#### 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.

0.333

## 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. ## Generator

This scale has no generator.

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

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

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.

Diameter 2 1 no a♯m F♯, 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 Thocrian 3rd mode:Scale 967 Mela Salaga 4th mode:Scale 2531 Danian 5th mode:Scale 3313 Aeolacrian 6th mode:Scale 463 Zythian This is the prime mode 7th mode:Scale 2279 Dyrian

## Prime

The prime form of this scale is Scale 463

 Scale 463 Zythian

## 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 2503 Mela Jhalavarali

## Enantiomorph

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

 Scale 2503 Mela Jhalavarali

## 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> 3187       T0I <11,0> 2503
T1 <1,1> 2279      T1I <11,1> 911
T2 <1,2> 463      T2I <11,2> 1822
T3 <1,3> 926      T3I <11,3> 3644
T4 <1,4> 1852      T4I <11,4> 3193
T5 <1,5> 3704      T5I <11,5> 2291
T6 <1,6> 3313      T6I <11,6> 487
T7 <1,7> 2531      T7I <11,7> 974
T8 <1,8> 967      T8I <11,8> 1948
T9 <1,9> 1934      T9I <11,9> 3896
T10 <1,10> 3868      T10I <11,10> 3697
T11 <1,11> 3641      T11I <11,11> 3299
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 487      T0MI <7,0> 3313
T1M <5,1> 974      T1MI <7,1> 2531
T2M <5,2> 1948      T2MI <7,2> 967
T3M <5,3> 3896      T3MI <7,3> 1934
T4M <5,4> 3697      T4MI <7,4> 3868
T5M <5,5> 3299      T5MI <7,5> 3641
T6M <5,6> 2503      T6MI <7,6> 3187
T7M <5,7> 911      T7MI <7,7> 2279
T8M <5,8> 1822      T8MI <7,8> 463
T9M <5,9> 3644      T9MI <7,9> 926
T10M <5,10> 3193      T10MI <7,10> 1852
T11M <5,11> 2291      T11MI <7,11> 3704

The transformations that map this set to itself are: T0, 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 3185 Messiaen Mode 5 Rotation 1 Scale 3189 Aeolonian Scale 3191 Bynyllic Scale 3195 Raryllic Scale 3171 Zythimic Scale 3179 Daptian Scale 3155 Ladimic Scale 3123 TOMian Scale 3251 Mela Hatakambari Scale 3315 Tcherepnin Octatonic Mode 1 Scale 3443 Verdi's Scala Enigmatica Scale 3699 Aeolylyllic Scale 2163 NEBian Scale 2675 Chromatic Lydian Scale 1139 Aerygimic

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.

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.