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Scale 2055: "Tetratonic Chromatic 2"

Scale 2055: Tetratonic Chromatic 2, 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

Western Modern
Tetratonic Chromatic 2

Analysis

Cardinality

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

4 (tetratonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,2,11}

Forte Number

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

4-1

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.

[0.5]

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.

no

Hemitonia

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

3 (trihemitonic)

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.

3

Prime Form

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

no
prime: 15

Generator

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

generator: 1
origin: 11

Deep Scale

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

no

Interval Formula

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

[1, 1, 9, 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, 1, 0, 0, 0>

Interval Spectrum

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

ns2d3

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> = {1,9}
<2> = {2,10}
<3> = {3,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

6

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.

0.25

Polygon Perimeter

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

2.967

Myhill Property

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

yes

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.

[1]

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.

(5, 0, 10)

Generator

This scale has a generator of 1, originating on 11.

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes

Modes are the rotational transformation of this scale. Scale 2055 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 3075
Scale 3075: Tetratonic Chromatic 3, Ian Ring Music TheoryTetratonic Chromatic 3
3rd mode:
Scale 3585
Scale 3585: Tetratonic Chromatic Descending, Ian Ring Music TheoryTetratonic Chromatic Descending
4th mode:
Scale 15
Scale 15: Tetratonic Chromatic, Ian Ring Music TheoryTetratonic ChromaticThis is the prime mode

Prime

The prime form of this scale is Scale 15

Scale 15Scale 15: Tetratonic Chromatic, Ian Ring Music TheoryTetratonic Chromatic

Complement

The tetratonic modal family [2055, 3075, 3585, 15] (Forte: 4-1) is the complement of the octatonic modal family [255, 2175, 3135, 3615, 3855, 3975, 4035, 4065] (Forte: 8-1)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2055 is 3075

Scale 3075Scale 3075: Tetratonic Chromatic 3, Ian Ring Music TheoryTetratonic Chromatic 3

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> 2055       T0I <11,0> 3075
T1 <1,1> 15      T1I <11,1> 2055
T2 <1,2> 30      T2I <11,2> 15
T3 <1,3> 60      T3I <11,3> 30
T4 <1,4> 120      T4I <11,4> 60
T5 <1,5> 240      T5I <11,5> 120
T6 <1,6> 480      T6I <11,6> 240
T7 <1,7> 960      T7I <11,7> 480
T8 <1,8> 1920      T8I <11,8> 960
T9 <1,9> 3840      T9I <11,9> 1920
T10 <1,10> 3585      T10I <11,10> 3840
T11 <1,11> 3075      T11I <11,11> 3585
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1185      T0MI <7,0> 165
T1M <5,1> 2370      T1MI <7,1> 330
T2M <5,2> 645      T2MI <7,2> 660
T3M <5,3> 1290      T3MI <7,3> 1320
T4M <5,4> 2580      T4MI <7,4> 2640
T5M <5,5> 1065      T5MI <7,5> 1185
T6M <5,6> 2130      T6MI <7,6> 2370
T7M <5,7> 165      T7MI <7,7> 645
T8M <5,8> 330      T8MI <7,8> 1290
T9M <5,9> 660      T9MI <7,9> 2580
T10M <5,10> 1320      T10MI <7,10> 1065
T11M <5,11> 2640      T11MI <7,11> 2130

The transformations that map this set to itself are: T0, T1I

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 2053Scale 2053, Ian Ring Music Theory
Scale 2051Scale 2051: Tritonic Chromatic 2, Ian Ring Music TheoryTritonic Chromatic 2
Scale 2059Scale 2059, Ian Ring Music Theory
Scale 2063Scale 2063: Pentatonic Chromatic 2, Ian Ring Music TheoryPentatonic Chromatic 2
Scale 2071Scale 2071, Ian Ring Music Theory
Scale 2087Scale 2087, Ian Ring Music Theory
Scale 2119Scale 2119, Ian Ring Music Theory
Scale 2183Scale 2183, Ian Ring Music Theory
Scale 2311Scale 2311: Raga Kumarapriya, Ian Ring Music TheoryRaga Kumarapriya
Scale 2567Scale 2567, Ian Ring Music Theory
Scale 3079Scale 3079: Pentatonic Chromatic 3, Ian Ring Music TheoryPentatonic Chromatic 3
Scale 7Scale 7: Tritonic Chromatic, Ian Ring Music TheoryTritonic Chromatic
Scale 1031Scale 1031, Ian Ring Music Theory

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.