The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 645: "Duyian"

Scale 645: Duyian, 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.



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

0 (anhemitonic)


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: 165


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

generator: 5
origin: 9

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.

[2, 5, 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.

<0, 2, 1, 0, 3, 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> = {2,3,5}
<2> = {5,7}
<3> = {7,9,10}

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.

(0, 4, 14)


This scale has a generator of 5, originating on 9.

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.


Modes are the rotational transformation of this scale. Scale 645 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 1185
Scale 1185: Genus Primum Inverse, Ian Ring Music TheoryGenus Primum Inverse
3rd mode:
Scale 165
Scale 165: Genus Primum, Ian Ring Music TheoryGenus PrimumThis is the prime mode
4th mode:
Scale 1065
Scale 1065: Gonian, Ian Ring Music TheoryGonian


The prime form of this scale is Scale 165

Scale 165Scale 165: Genus Primum, Ian Ring Music TheoryGenus Primum


The tetratonic modal family [645, 1185, 165, 1065] (Forte: 4-23) is the complement of the octatonic modal family [1455, 1515, 1725, 1965, 2775, 2805, 3435, 3765] (Forte: 8-23)


The inverse of a scale is a reflection using the root as its axis. The inverse of 645 is 1065

Scale 1065Scale 1065: Gonian, Ian Ring Music TheoryGonian


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> 645       T0I <11,0> 1065
T1 <1,1> 1290      T1I <11,1> 2130
T2 <1,2> 2580      T2I <11,2> 165
T3 <1,3> 1065      T3I <11,3> 330
T4 <1,4> 2130      T4I <11,4> 660
T5 <1,5> 165      T5I <11,5> 1320
T6 <1,6> 330      T6I <11,6> 2640
T7 <1,7> 660      T7I <11,7> 1185
T8 <1,8> 1320      T8I <11,8> 2370
T9 <1,9> 2640      T9I <11,9> 645
T10 <1,10> 1185      T10I <11,10> 1290
T11 <1,11> 2370      T11I <11,11> 2580
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3585      T0MI <7,0> 15
T1M <5,1> 3075      T1MI <7,1> 30
T2M <5,2> 2055      T2MI <7,2> 60
T3M <5,3> 15      T3MI <7,3> 120
T4M <5,4> 30      T4MI <7,4> 240
T5M <5,5> 60      T5MI <7,5> 480
T6M <5,6> 120      T6MI <7,6> 960
T7M <5,7> 240      T7MI <7,7> 1920
T8M <5,8> 480      T8MI <7,8> 3840
T9M <5,9> 960      T9MI <7,9> 3585
T10M <5,10> 1920      T10MI <7,10> 3075
T11M <5,11> 3840      T11MI <7,11> 2055

The transformations that map this set to itself are: T0, T9I

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 647Scale 647: Duzian, Ian Ring Music TheoryDuzian
Scale 641Scale 641: Duwian, Ian Ring Music TheoryDuwian
Scale 643Scale 643: Duxian, Ian Ring Music TheoryDuxian
Scale 649Scale 649: Byptic, Ian Ring Music TheoryByptic
Scale 653Scale 653: Dorian Pentatonic, Ian Ring Music TheoryDorian Pentatonic
Scale 661Scale 661: Major Pentatonic, Ian Ring Music TheoryMajor Pentatonic
Scale 677Scale 677: Scottish Pentatonic, Ian Ring Music TheoryScottish Pentatonic
Scale 709Scale 709: Raga Shri Kalyan, Ian Ring Music TheoryRaga Shri Kalyan
Scale 517Scale 517: Aluian, Ian Ring Music TheoryAluian
Scale 581Scale 581: Eporic 2, Ian Ring Music TheoryEporic 2
Scale 773Scale 773: Esuian, Ian Ring Music TheoryEsuian
Scale 901Scale 901: Bofian, Ian Ring Music TheoryBofian
Scale 133Scale 133: Suspended Second Triad, Ian Ring Music TheorySuspended Second Triad
Scale 389Scale 389: Cixian, Ian Ring Music TheoryCixian
Scale 1157Scale 1157: Alkian, Ian Ring Music TheoryAlkian
Scale 1669Scale 1669: Raga Matha Kokila, Ian Ring Music TheoryRaga Matha Kokila
Scale 2693Scale 2693: Rajian, Ian Ring Music TheoryRajian

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.