The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3099: "Tixian"

Scale 3099: Tixian, 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

Dozenal
Tixian

Analysis

Cardinality

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

{0,1,3,4,10,11}

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

6-Z3

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

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.

none

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

yes
enantiomorph: 2823

Hemitonia

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

4 (multihemitonic)

Cohemitonia

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

2 (dicohemitonic)

Imperfections

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.

4

Modes

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.

5

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 111

Generator

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

none

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Structure

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

[1, 2, 1, 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, 3, 3, 2, 2, 1>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p2m2n3s3d4t

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}
<3> = {3,4,8,9}
<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.

4.333

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

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.

no

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.

1.433

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.071

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

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.

no

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.

none

Propriety

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".

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, 9, 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: {0,3,11}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 3099 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3597
Scale 3597: Wijian, Ian Ring Music TheoryWijian
3rd mode:
Scale 1923
Scale 1923: Lulian, Ian Ring Music TheoryLulian
4th mode:
Scale 3009
Scale 3009: Suvian, Ian Ring Music TheorySuvian
5th mode:
Scale 111
Scale 111: Aroian, Ian Ring Music TheoryAroianThis is the prime mode
6th mode:
Scale 2103
Scale 2103: Murian, Ian Ring Music TheoryMurian

Prime

The prime form of this scale is Scale 111

Scale 111Scale 111: Aroian, Ian Ring Music TheoryAroian

Complement

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

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3099 is 2823

Scale 2823Scale 2823: Rulian, Ian Ring Music TheoryRulian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3099 is chiral, and its enantiomorph is scale 2823

Scale 2823Scale 2823: Rulian, Ian Ring Music TheoryRulian

Transformations:

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> 3099       T0I <11,0> 2823
T1 <1,1> 2103      T1I <11,1> 1551
T2 <1,2> 111      T2I <11,2> 3102
T3 <1,3> 222      T3I <11,3> 2109
T4 <1,4> 444      T4I <11,4> 123
T5 <1,5> 888      T5I <11,5> 246
T6 <1,6> 1776      T6I <11,6> 492
T7 <1,7> 3552      T7I <11,7> 984
T8 <1,8> 3009      T8I <11,8> 1968
T9 <1,9> 1923      T9I <11,9> 3936
T10 <1,10> 3846      T10I <11,10> 3777
T11 <1,11> 3597      T11I <11,11> 3459
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 429      T0MI <7,0> 1713
T1M <5,1> 858      T1MI <7,1> 3426
T2M <5,2> 1716      T2MI <7,2> 2757
T3M <5,3> 3432      T3MI <7,3> 1419
T4M <5,4> 2769      T4MI <7,4> 2838
T5M <5,5> 1443      T5MI <7,5> 1581
T6M <5,6> 2886      T6MI <7,6> 3162
T7M <5,7> 1677      T7MI <7,7> 2229
T8M <5,8> 3354      T8MI <7,8> 363
T9M <5,9> 2613      T9MI <7,9> 726
T10M <5,10> 1131      T10MI <7,10> 1452
T11M <5,11> 2262      T11MI <7,11> 2904

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 3097Scale 3097: Tiwian, Ian Ring Music TheoryTiwian
Scale 3101Scale 3101: Tiyian, Ian Ring Music TheoryTiyian
Scale 3103Scale 3103: Heptatonic Chromatic 3, Ian Ring Music TheoryHeptatonic Chromatic 3
Scale 3091Scale 3091: Tisian, Ian Ring Music TheoryTisian
Scale 3095Scale 3095: Tivian, Ian Ring Music TheoryTivian
Scale 3083Scale 3083: Rehian, Ian Ring Music TheoryRehian
Scale 3115Scale 3115: Tihian, Ian Ring Music TheoryTihian
Scale 3131Scale 3131: Torian, Ian Ring Music TheoryTorian
Scale 3163Scale 3163: Rogian, Ian Ring Music TheoryRogian
Scale 3227Scale 3227: Aeolocrian, Ian Ring Music TheoryAeolocrian
Scale 3355Scale 3355: Bagian, Ian Ring Music TheoryBagian
Scale 3611Scale 3611: Worian, Ian Ring Music TheoryWorian
Scale 2075Scale 2075: Mozian, Ian Ring Music TheoryMozian
Scale 2587Scale 2587: Putian, Ian Ring Music TheoryPutian
Scale 1051Scale 1051: Gifian, Ian Ring Music TheoryGifian

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 (http://allthescales.org) 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.