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Scale 3095: "Tivian"

Scale 3095: Tivian, 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.

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: 3335


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 Rahn/Ring formula.

prime: 95


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.

[1, 1, 2, 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.

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

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.

(34, 13, 55)

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: {0,2,11}

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 3095 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3595
Scale 3595: Wihian, Ian Ring Music TheoryWihian
3rd mode:
Scale 3845
Scale 3845: Yihian, Ian Ring Music TheoryYihian
4th mode:
Scale 1985
Scale 1985: Mewian, Ian Ring Music TheoryMewian
5th mode:
Scale 95
Scale 95: Arkian, Ian Ring Music TheoryArkianThis is the prime mode
6th mode:
Scale 2095
Scale 2095: Mumian, Ian Ring Music TheoryMumian


The prime form of this scale is Scale 95

Scale 95Scale 95: Arkian, Ian Ring Music TheoryArkian


The hexatonic modal family [3095, 3595, 3845, 1985, 95, 2095] (Forte: 6-2) is the complement of the hexatonic modal family [95, 1985, 2095, 3095, 3595, 3845] (Forte: 6-2)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3095 is 3335

Scale 3335Scale 3335: Vadian, Ian Ring Music TheoryVadian


Only scales that are chiral will have an enantiomorph. Scale 3095 is chiral, and its enantiomorph is scale 3335

Scale 3335Scale 3335: Vadian, Ian Ring Music TheoryVadian


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> 3095       T0I <11,0> 3335
T1 <1,1> 2095      T1I <11,1> 2575
T2 <1,2> 95      T2I <11,2> 1055
T3 <1,3> 190      T3I <11,3> 2110
T4 <1,4> 380      T4I <11,4> 125
T5 <1,5> 760      T5I <11,5> 250
T6 <1,6> 1520      T6I <11,6> 500
T7 <1,7> 3040      T7I <11,7> 1000
T8 <1,8> 1985      T8I <11,8> 2000
T9 <1,9> 3970      T9I <11,9> 4000
T10 <1,10> 3845      T10I <11,10> 3905
T11 <1,11> 3595      T11I <11,11> 3715
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1445      T0MI <7,0> 1205
T1M <5,1> 2890      T1MI <7,1> 2410
T2M <5,2> 1685      T2MI <7,2> 725
T3M <5,3> 3370      T3MI <7,3> 1450
T4M <5,4> 2645      T4MI <7,4> 2900
T5M <5,5> 1195      T5MI <7,5> 1705
T6M <5,6> 2390      T6MI <7,6> 3410
T7M <5,7> 685      T7MI <7,7> 2725
T8M <5,8> 1370      T8MI <7,8> 1355
T9M <5,9> 2740      T9MI <7,9> 2710
T10M <5,10> 1385      T10MI <7,10> 1325
T11M <5,11> 2770      T11MI <7,11> 2650

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 3093Scale 3093: Buqian, Ian Ring Music TheoryBuqian
Scale 3091Scale 3091: Tisian, Ian Ring Music TheoryTisian
Scale 3099Scale 3099: Tixian, Ian Ring Music TheoryTixian
Scale 3103Scale 3103: Heptatonic Chromatic 3, Ian Ring Music TheoryHeptatonic Chromatic 3
Scale 3079Scale 3079: Pentatonic Chromatic 3, Ian Ring Music TheoryPentatonic Chromatic 3
Scale 3087Scale 3087: Hexatonic Chromatic 3, Ian Ring Music TheoryHexatonic Chromatic 3
Scale 3111Scale 3111: Tifian, Ian Ring Music TheoryTifian
Scale 3127Scale 3127: Topian, Ian Ring Music TheoryTopian
Scale 3159Scale 3159: Stocrian, Ian Ring Music TheoryStocrian
Scale 3223Scale 3223: Thyphian, Ian Ring Music TheoryThyphian
Scale 3351Scale 3351: Crater Scale, Ian Ring Music TheoryCrater Scale
Scale 3607Scale 3607: Wopian, Ian Ring Music TheoryWopian
Scale 2071Scale 2071: Moxian, Ian Ring Music TheoryMoxian
Scale 2583Scale 2583: Purian, Ian Ring Music TheoryPurian
Scale 1047Scale 1047: Gician, Ian Ring Music TheoryGician

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.