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Scale 3109: "Tidian"

Scale 3109: Tidian, 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.

5 (pentatonic)

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: 1159


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

2 (dihemitonic)


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

1 (uncohemitonic)


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: 151


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.

[2, 3, 5, 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.

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

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> = {1,2,3,5}
<2> = {2,3,5,6,8}
<3> = {4,6,7,9,10}
<4> = {7,9,10,11}

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.

(13, 7, 38)

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 TriadsA♯{10,2,5}110.5
Diminished Triads{11,2,5}110.5

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

Parsimonious Voice Leading Between Common Triads of Scale 3109. Created by Ian Ring ©2019 A# A# A#->b°

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.



Modes are the rotational transformation of this scale. Scale 3109 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1801
Scale 1801: Lanian, Ian Ring Music TheoryLanian
3rd mode:
Scale 737
Scale 737: Truian, Ian Ring Music TheoryTruian
4th mode:
Scale 151
Scale 151: Bahian, Ian Ring Music TheoryBahianThis is the prime mode
5th mode:
Scale 2123
Scale 2123: Nacian, Ian Ring Music TheoryNacian


The prime form of this scale is Scale 151

Scale 151Scale 151: Bahian, Ian Ring Music TheoryBahian


The pentatonic modal family [3109, 1801, 737, 151, 2123] (Forte: 5-Z36) is the complement of the heptatonic modal family [367, 1777, 1931, 2231, 3013, 3163, 3629] (Forte: 7-Z36)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3109 is 1159

Scale 1159Scale 1159: Hasian, Ian Ring Music TheoryHasian


Only scales that are chiral will have an enantiomorph. Scale 3109 is chiral, and its enantiomorph is scale 1159

Scale 1159Scale 1159: Hasian, Ian Ring Music TheoryHasian


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> 3109       T0I <11,0> 1159
T1 <1,1> 2123      T1I <11,1> 2318
T2 <1,2> 151      T2I <11,2> 541
T3 <1,3> 302      T3I <11,3> 1082
T4 <1,4> 604      T4I <11,4> 2164
T5 <1,5> 1208      T5I <11,5> 233
T6 <1,6> 2416      T6I <11,6> 466
T7 <1,7> 737      T7I <11,7> 932
T8 <1,8> 1474      T8I <11,8> 1864
T9 <1,9> 2948      T9I <11,9> 3728
T10 <1,10> 1801      T10I <11,10> 3361
T11 <1,11> 3602      T11I <11,11> 2627
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1159      T0MI <7,0> 3109
T1M <5,1> 2318      T1MI <7,1> 2123
T2M <5,2> 541      T2MI <7,2> 151
T3M <5,3> 1082      T3MI <7,3> 302
T4M <5,4> 2164      T4MI <7,4> 604
T5M <5,5> 233      T5MI <7,5> 1208
T6M <5,6> 466      T6MI <7,6> 2416
T7M <5,7> 932      T7MI <7,7> 737
T8M <5,8> 1864      T8MI <7,8> 1474
T9M <5,9> 3728      T9MI <7,9> 2948
T10M <5,10> 3361      T10MI <7,10> 1801
T11M <5,11> 2627      T11MI <7,11> 3602

The transformations that map this set to itself are: T0, T0MI

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 3111Scale 3111: Tifian, Ian Ring Music TheoryTifian
Scale 3105Scale 3105: Tibian, Ian Ring Music TheoryTibian
Scale 3107Scale 3107: Tician, Ian Ring Music TheoryTician
Scale 3113Scale 3113: Tigian, Ian Ring Music TheoryTigian
Scale 3117Scale 3117: Tijian, Ian Ring Music TheoryTijian
Scale 3125Scale 3125: Tonian, Ian Ring Music TheoryTonian
Scale 3077Scale 3077: Tekian, Ian Ring Music TheoryTekian
Scale 3093Scale 3093: Buqian, Ian Ring Music TheoryBuqian
Scale 3141Scale 3141: Kanitonic, Ian Ring Music TheoryKanitonic
Scale 3173Scale 3173: Zarimic, Ian Ring Music TheoryZarimic
Scale 3237Scale 3237: Raga Brindabani Sarang, Ian Ring Music TheoryRaga Brindabani Sarang
Scale 3365Scale 3365: Katolimic, Ian Ring Music TheoryKatolimic
Scale 3621Scale 3621: Gylimic, Ian Ring Music TheoryGylimic
Scale 2085Scale 2085: Mogian, Ian Ring Music TheoryMogian
Scale 2597Scale 2597: Raga Rasranjani, Ian Ring Music TheoryRaga Rasranjani
Scale 1061Scale 1061: Gilian, Ian Ring Music TheoryGilian

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.