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Scale 417: "Copian"

Scale 417: Copian, 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




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

4 (tetratonic)

Pitch Class Set

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


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


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.



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



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

enantiomorph: 177


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

1 (unhemitonic)


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

0 (ancohemitonic)


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.



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.


Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

prime: 141


Indicates if the scale can be constructed using a generator, and an origin.


Deep Scale

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


Interval Structure

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

[5, 2, 1, 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.

<1, 1, 1, 1, 2, 0>

Interval Spectrum

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


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

Spectra Variation

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


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


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.


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.


Polygon Perimeter

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


Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.



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.


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.



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".


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.

(4, 2, 18)

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
Minor Triadsfm{5,8,0}000

The following pitch classes are not present in any of the common triads: {7}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.


Modes are the rotational transformation of this scale. Scale 417 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 141
Scale 141: Babian, Ian Ring Music TheoryBabianThis is the prime mode
3rd mode:
Scale 1059
Scale 1059: Gikian, Ian Ring Music TheoryGikian
4th mode:
Scale 2577
Scale 2577: Punian, Ian Ring Music TheoryPunian


The prime form of this scale is Scale 141

Scale 141Scale 141: Babian, Ian Ring Music TheoryBabian


The tetratonic modal family [417, 141, 1059, 2577] (Forte: 4-14) is the complement of the octatonic modal family [759, 1839, 1977, 2427, 2967, 3261, 3531, 3813] (Forte: 8-14)


The inverse of a scale is a reflection using the root as its axis. The inverse of 417 is 177

Scale 177Scale 177: Bexian, Ian Ring Music TheoryBexian


Only scales that are chiral will have an enantiomorph. Scale 417 is chiral, and its enantiomorph is scale 177

Scale 177Scale 177: Bexian, Ian Ring Music TheoryBexian


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> 417       T0I <11,0> 177
T1 <1,1> 834      T1I <11,1> 354
T2 <1,2> 1668      T2I <11,2> 708
T3 <1,3> 3336      T3I <11,3> 1416
T4 <1,4> 2577      T4I <11,4> 2832
T5 <1,5> 1059      T5I <11,5> 1569
T6 <1,6> 2118      T6I <11,6> 3138
T7 <1,7> 141      T7I <11,7> 2181
T8 <1,8> 282      T8I <11,8> 267
T9 <1,9> 564      T9I <11,9> 534
T10 <1,10> 1128      T10I <11,10> 1068
T11 <1,11> 2256      T11I <11,11> 2136
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2067      T0MI <7,0> 2307
T1M <5,1> 39      T1MI <7,1> 519
T2M <5,2> 78      T2MI <7,2> 1038
T3M <5,3> 156      T3MI <7,3> 2076
T4M <5,4> 312      T4MI <7,4> 57
T5M <5,5> 624      T5MI <7,5> 114
T6M <5,6> 1248      T6MI <7,6> 228
T7M <5,7> 2496      T7MI <7,7> 456
T8M <5,8> 897      T8MI <7,8> 912
T9M <5,9> 1794      T9MI <7,9> 1824
T10M <5,10> 3588      T10MI <7,10> 3648
T11M <5,11> 3081      T11MI <7,11> 3201

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 419Scale 419: Hon-kumoi-joshi, Ian Ring Music TheoryHon-kumoi-joshi
Scale 421Scale 421: Han-kumoi, Ian Ring Music TheoryHan-kumoi
Scale 425Scale 425: Raga Kokil Pancham, Ian Ring Music TheoryRaga Kokil Pancham
Scale 433Scale 433: Raga Zilaf, Ian Ring Music TheoryRaga Zilaf
Scale 385Scale 385: Civian, Ian Ring Music TheoryCivian
Scale 401Scale 401: Epogic, Ian Ring Music TheoryEpogic
Scale 449Scale 449: Cujian, Ian Ring Music TheoryCujian
Scale 481Scale 481: Dabian, Ian Ring Music TheoryDabian
Scale 289Scale 289: Valian, Ian Ring Music TheoryValian
Scale 353Scale 353: Cebian, Ian Ring Music TheoryCebian
Scale 161Scale 161: Raga Sarvasri, Ian Ring Music TheoryRaga Sarvasri
Scale 673Scale 673: Estian, Ian Ring Music TheoryEstian
Scale 929Scale 929: Fujian, Ian Ring Music TheoryFujian
Scale 1441Scale 1441: Jabian, Ian Ring Music TheoryJabian
Scale 2465Scale 2465: Raga Devaranjani, Ian Ring Music TheoryRaga Devaranjani

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 ( 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.