The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 3091: "TISian"

Scale 3091: TISian, Ian Ring Music Theory

Warning: mysqli::__construct(): (HY000/2002): Connection refused in /var/www/html/ on line 93

Warning: mysqli::set_charset(): Couldn't fetch mysqli in /var/www/html/ on line 94

Warning: mysqli::query(): Couldn't fetch mysqli in /var/www/html/ on line 96

Warning: mysqli::close(): Couldn't fetch mysqli in /var/www/html/ on line 110

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.

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: 2311


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

3 (trihemitonic)


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

2 (dicohemitonic)


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: 79


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, 3, 6, 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.

<3, 2, 2, 1, 1, 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.

<0.75, 0.5, 0.5, 0, 0.25, 0.5>

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

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.

(20, 3, 32)

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
Diminished Triadsa♯°{10,1,4}000

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

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 3091 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 3593
Scale 3593: WIGian, Ian Ring Music TheoryWIGian
3rd mode:
Scale 961
Scale 961: GABian, Ian Ring Music TheoryGABian
4th mode:
Scale 79
Scale 79: APPian, Ian Ring Music TheoryAPPianThis is the prime mode
5th mode:
Scale 2087
Scale 2087: MUHian, Ian Ring Music TheoryMUHian


The prime form of this scale is Scale 79

Scale 79Scale 79: APPian, Ian Ring Music TheoryAPPian


The pentatonic modal family [3091, 3593, 961, 79, 2087] (Forte: 5-4) is the complement of the heptatonic modal family [223, 1987, 2159, 3041, 3127, 3611, 3853] (Forte: 7-4)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3091 is 2311

Scale 2311Scale 2311: Raga Kumarapriya, Ian Ring Music TheoryRaga Kumarapriya


Only scales that are chiral will have an enantiomorph. Scale 3091 is chiral, and its enantiomorph is scale 2311

Scale 2311Scale 2311: Raga Kumarapriya, Ian Ring Music TheoryRaga Kumarapriya


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> 3091       T0I <11,0> 2311
T1 <1,1> 2087      T1I <11,1> 527
T2 <1,2> 79      T2I <11,2> 1054
T3 <1,3> 158      T3I <11,3> 2108
T4 <1,4> 316      T4I <11,4> 121
T5 <1,5> 632      T5I <11,5> 242
T6 <1,6> 1264      T6I <11,6> 484
T7 <1,7> 2528      T7I <11,7> 968
T8 <1,8> 961      T8I <11,8> 1936
T9 <1,9> 1922      T9I <11,9> 3872
T10 <1,10> 3844      T10I <11,10> 3649
T11 <1,11> 3593      T11I <11,11> 3203
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 421      T0MI <7,0> 1201
T1M <5,1> 842      T1MI <7,1> 2402
T2M <5,2> 1684      T2MI <7,2> 709
T3M <5,3> 3368      T3MI <7,3> 1418
T4M <5,4> 2641      T4MI <7,4> 2836
T5M <5,5> 1187      T5MI <7,5> 1577
T6M <5,6> 2374      T6MI <7,6> 3154
T7M <5,7> 653      T7MI <7,7> 2213
T8M <5,8> 1306      T8MI <7,8> 331
T9M <5,9> 2612      T9MI <7,9> 662
T10M <5,10> 1129      T10MI <7,10> 1324
T11M <5,11> 2258      T11MI <7,11> 2648

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 3089Scale 3089: TIRian, Ian Ring Music TheoryTIRian
Scale 3093Scale 3093: BUQian, Ian Ring Music TheoryBUQian
Scale 3095Scale 3095: TIVian, Ian Ring Music TheoryTIVian
Scale 3099Scale 3099: TIXian, Ian Ring Music TheoryTIXian
Scale 3075Scale 3075: Tetratonic Chromatic 3, Ian Ring Music TheoryTetratonic Chromatic 3
Scale 3083Scale 3083: REHian, Ian Ring Music TheoryREHian
Scale 3107Scale 3107: TICian, Ian Ring Music TheoryTICian
Scale 3123Scale 3123: TOMian, Ian Ring Music TheoryTOMian
Scale 3155Scale 3155: Ladimic, Ian Ring Music TheoryLadimic
Scale 3219Scale 3219: Ionaphimic, Ian Ring Music TheoryIonaphimic
Scale 3347Scale 3347: Synimic, Ian Ring Music TheorySynimic
Scale 3603Scale 3603: WOMian, Ian Ring Music TheoryWOMian
Scale 2067Scale 2067: MOVian, Ian Ring Music TheoryMOVian
Scale 2579Scale 2579: PUPian, Ian Ring Music TheoryPUPian
Scale 1043Scale 1043: GIZian, Ian Ring Music TheoryGIZian

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.