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Scale 961

Scale 961, 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.

5 (pentatonic)

Pitch Class Set

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

{0,6,7,8,9}

Forte Number

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

5-4

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: 121

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.

4

Prime Form

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

no
prime: 79

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.

[6, 1, 1, 1, 3]

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>

Interval Spectrum

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

pmn2s2d3t

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.

4.8

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.

1.25

Polygon Perimeter

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

4.967

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

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 Triadsf♯°{6,9,0}000

The following pitch classes are not present in any of the common triads: {7,8}

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

2nd mode:
Scale 79
Scale 79, Ian Ring Music TheoryThis is the prime mode
3rd mode:
Scale 2087
Scale 2087, Ian Ring Music Theory
4th mode:
Scale 3091
Scale 3091, Ian Ring Music Theory
5th mode:
Scale 3593
Scale 3593, Ian Ring Music Theory

Prime

The prime form of this scale is Scale 79

Scale 79Scale 79, Ian Ring Music Theory

Complement

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

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 961 is 121

Scale 121Scale 121, Ian Ring Music Theory

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 961 is chiral, and its enantiomorph is scale 121

Scale 121Scale 121, Ian Ring Music Theory

Transformations:

T0 961  T0I 121
T1 1922  T1I 242
T2 3844  T2I 484
T3 3593  T3I 968
T4 3091  T4I 1936
T5 2087  T5I 3872
T6 79  T6I 3649
T7 158  T7I 3203
T8 316  T8I 2311
T9 632  T9I 527
T10 1264  T10I 1054
T11 2528  T11I 2108

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 963Scale 963, Ian Ring Music Theory
Scale 965Scale 965: Ionothimic, Ian Ring Music TheoryIonothimic
Scale 969Scale 969: Ionogimic, Ian Ring Music TheoryIonogimic
Scale 977Scale 977: Kocrimic, Ian Ring Music TheoryKocrimic
Scale 993Scale 993, Ian Ring Music Theory
Scale 897Scale 897, Ian Ring Music Theory
Scale 929Scale 929, Ian Ring Music Theory
Scale 833Scale 833, Ian Ring Music Theory
Scale 705Scale 705, Ian Ring Music Theory
Scale 449Scale 449, Ian Ring Music Theory
Scale 1473Scale 1473, Ian Ring Music Theory
Scale 1985Scale 1985, Ian Ring Music Theory
Scale 3009Scale 3009, 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. 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.