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Scale 3111: "TIFian"

Scale 3111: TIFian, 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).



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

6 (hexatonic)

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: 3207


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

4 (multihemitonic)


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

3 (tricohemitonic)


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 Starr/Rahn algorithm.

prime: 159


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, an indicator of maximum hierarchization.


Interval Structure

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

[1, 1, 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.

<4, 3, 3, 2, 2, 1>

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.

<0.8, 0.5, 0.6, 0, 0.4, 0.333>

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

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 Distribution Spectra have 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.

(41, 9, 55)

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.


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.



This scale has no generator.

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}210.67
Minor Triadsa♯m{10,1,5}121
Diminished Triads{11,2,5}121

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

Parsimonious Voice Leading Between Common Triads of Scale 3111. Created by Ian Ring ©2019 a#m a#m A# A# a#m->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.

Central VerticesA♯
Peripheral Verticesa♯m, b°


Modes are the rotational transformation of this scale. Scale 3111 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3603
Scale 3603: WOMian, Ian Ring Music TheoryWOMian
3rd mode:
Scale 3849
Scale 3849: YIKian, Ian Ring Music TheoryYIKian
4th mode:
Scale 993
Scale 993: GAVian, Ian Ring Music TheoryGAVian
5th mode:
Scale 159
Scale 159: BAMian, Ian Ring Music TheoryBAMianThis is the prime mode
6th mode:
Scale 2127
Scale 2127: NAFian, Ian Ring Music TheoryNAFian


The prime form of this scale is Scale 159

Scale 159Scale 159: BAMian, Ian Ring Music TheoryBAMian


The hexatonic modal family [3111, 3603, 3849, 993, 159, 2127] (Forte: 6-Z36) is the complement of the hexatonic modal family [111, 1923, 2103, 3009, 3099, 3597] (Forte: 6-Z3)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3111 is 3207

Scale 3207Scale 3207: UCOian, Ian Ring Music TheoryUCOian


Only scales that are chiral will have an enantiomorph. Scale 3111 is chiral, and its enantiomorph is scale 3207

Scale 3207Scale 3207: UCOian, Ian Ring Music TheoryUCOian


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> 3111       T0I <11,0> 3207
T1 <1,1> 2127      T1I <11,1> 2319
T2 <1,2> 159      T2I <11,2> 543
T3 <1,3> 318      T3I <11,3> 1086
T4 <1,4> 636      T4I <11,4> 2172
T5 <1,5> 1272      T5I <11,5> 249
T6 <1,6> 2544      T6I <11,6> 498
T7 <1,7> 993      T7I <11,7> 996
T8 <1,8> 1986      T8I <11,8> 1992
T9 <1,9> 3972      T9I <11,9> 3984
T10 <1,10> 3849      T10I <11,10> 3873
T11 <1,11> 3603      T11I <11,11> 3651
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1191      T0MI <7,0> 3237
T1M <5,1> 2382      T1MI <7,1> 2379
T2M <5,2> 669      T2MI <7,2> 663
T3M <5,3> 1338      T3MI <7,3> 1326
T4M <5,4> 2676      T4MI <7,4> 2652
T5M <5,5> 1257      T5MI <7,5> 1209
T6M <5,6> 2514      T6MI <7,6> 2418
T7M <5,7> 933      T7MI <7,7> 741
T8M <5,8> 1866      T8MI <7,8> 1482
T9M <5,9> 3732      T9MI <7,9> 2964
T10M <5,10> 3369      T10MI <7,10> 1833
T11M <5,11> 2643      T11MI <7,11> 3666

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 3109Scale 3109: TIDian, Ian Ring Music TheoryTIDian
Scale 3107Scale 3107: TICian, Ian Ring Music TheoryTICian
Scale 3115Scale 3115: TIHian, Ian Ring Music TheoryTIHian
Scale 3119Scale 3119: TIKian, Ian Ring Music TheoryTIKian
Scale 3127Scale 3127: TOPian, Ian Ring Music TheoryTOPian
Scale 3079Scale 3079: Pentatonic Chromatic 3, Ian Ring Music TheoryPentatonic Chromatic 3
Scale 3095Scale 3095: TIVian, Ian Ring Music TheoryTIVian
Scale 3143Scale 3143: Polimic, Ian Ring Music TheoryPolimic
Scale 3175Scale 3175: Eponian, Ian Ring Music TheoryEponian
Scale 3239Scale 3239: Mela Tanarupi, Ian Ring Music TheoryMela Tanarupi
Scale 3367Scale 3367: Moptian, Ian Ring Music TheoryMoptian
Scale 3623Scale 3623: Aerocrian, Ian Ring Music TheoryAerocrian
Scale 2087Scale 2087: MUHian, Ian Ring Music TheoryMUHian
Scale 2599Scale 2599: Malimic, Ian Ring Music TheoryMalimic
Scale 1063Scale 1063: GOMian, Ian Ring Music TheoryGOMian

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 ( 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.