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Scale 2069

Scale 2069, 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).

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

{0,2,4,11}

Forte Number

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

4-11

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: 1283

Hemitonia

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

1 (unhemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

3

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: 43

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 Formula

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

[2, 2, 7, 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.

<1, 2, 1, 1, 1, 0>

Interval Spectrum

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

pmns2d

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

Spectra Variation

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

4.5

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.

0.866

Polygon Perimeter

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

4.449

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.

(5, 0, 16)

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 2069 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 1541
Scale 1541, Ian Ring Music Theory
3rd mode:
Scale 1409
Scale 1409, Ian Ring Music Theory
4th mode:
Scale 43
Scale 43, Ian Ring Music TheoryThis is the prime mode

Prime

The prime form of this scale is Scale 43

Scale 43Scale 43, Ian Ring Music Theory

Complement

The tetratonic modal family [2069, 1541, 1409, 43] (Forte: 4-11) is the complement of the octatonic modal family [703, 1529, 2021, 2399, 3247, 3671, 3883, 3989] (Forte: 8-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2069 is 1283

Scale 1283Scale 1283, Ian Ring Music Theory

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2069 is chiral, and its enantiomorph is scale 1283

Scale 1283Scale 1283, Ian Ring Music Theory

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> 2069       T0I <11,0> 1283
T1 <1,1> 43      T1I <11,1> 2566
T2 <1,2> 86      T2I <11,2> 1037
T3 <1,3> 172      T3I <11,3> 2074
T4 <1,4> 344      T4I <11,4> 53
T5 <1,5> 688      T5I <11,5> 106
T6 <1,6> 1376      T6I <11,6> 212
T7 <1,7> 2752      T7I <11,7> 424
T8 <1,8> 1409      T8I <11,8> 848
T9 <1,9> 2818      T9I <11,9> 1696
T10 <1,10> 1541      T10I <11,10> 3392
T11 <1,11> 3082      T11I <11,11> 2689
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1409      T0MI <7,0> 53
T1M <5,1> 2818      T1MI <7,1> 106
T2M <5,2> 1541      T2MI <7,2> 212
T3M <5,3> 3082      T3MI <7,3> 424
T4M <5,4> 2069       T4MI <7,4> 848
T5M <5,5> 43      T5MI <7,5> 1696
T6M <5,6> 86      T6MI <7,6> 3392
T7M <5,7> 172      T7MI <7,7> 2689
T8M <5,8> 344      T8MI <7,8> 1283
T9M <5,9> 688      T9MI <7,9> 2566
T10M <5,10> 1376      T10MI <7,10> 1037
T11M <5,11> 2752      T11MI <7,11> 2074

The transformations that map this set to itself are: T0, T4M

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 2071Scale 2071, Ian Ring Music Theory
Scale 2065Scale 2065, Ian Ring Music Theory
Scale 2067Scale 2067, Ian Ring Music Theory
Scale 2073Scale 2073, Ian Ring Music Theory
Scale 2077Scale 2077, Ian Ring Music Theory
Scale 2053Scale 2053, Ian Ring Music Theory
Scale 2061Scale 2061, Ian Ring Music Theory
Scale 2085Scale 2085, Ian Ring Music Theory
Scale 2101Scale 2101, Ian Ring Music Theory
Scale 2133Scale 2133: Raga Kumurdaki, Ian Ring Music TheoryRaga Kumurdaki
Scale 2197Scale 2197: Raga Hamsadhvani, Ian Ring Music TheoryRaga Hamsadhvani
Scale 2325Scale 2325: Pynitonic, Ian Ring Music TheoryPynitonic
Scale 2581Scale 2581: Raga Neroshta, Ian Ring Music TheoryRaga Neroshta
Scale 3093Scale 3093, Ian Ring Music Theory
Scale 21Scale 21, Ian Ring Music Theory
Scale 1045Scale 1045, 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.