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Scale 897

Scale 897, 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

41161837294116105072918310504116183
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

{0,7,8,9}

Forte Number

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

4-4

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: 57

Hemitonia

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

2 (dihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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: 39

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.

[7, 1, 1, 3] 9

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.

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

Interval Spectrum

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

pmnsd2

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

Spectra Variation

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

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.

0.75

Polygon Perimeter

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

4.381

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

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

2nd mode:
Scale 39
Scale 39, Ian Ring Music TheoryThis is the prime mode
3rd mode:
Scale 2067
Scale 2067, Ian Ring Music Theory
4th mode:
Scale 3081
Scale 3081, Ian Ring Music Theory

Prime

The prime form of this scale is Scale 39

Scale 39Scale 39, Ian Ring Music Theory

Complement

The tetratonic modal family [897, 39, 2067, 3081] (Forte: 4-4) is the complement of the octatonic modal family [447, 2019, 2271, 3057, 3183, 3639, 3867, 3981] (Forte: 8-4)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 897 is 57

Scale 57Scale 57, Ian Ring Music Theory

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 897 is chiral, and its enantiomorph is scale 57

Scale 57Scale 57, Ian Ring Music Theory

Transformations:

T0 897  T0I 57
T1 1794  T1I 114
T2 3588  T2I 228
T3 3081  T3I 456
T4 2067  T4I 912
T5 39  T5I 1824
T6 78  T6I 3648
T7 156  T7I 3201
T8 312  T8I 2307
T9 624  T9I 519
T10 1248  T10I 1038
T11 2496  T11I 2076

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 899Scale 899, Ian Ring Music Theory
Scale 901Scale 901, Ian Ring Music Theory
Scale 905Scale 905: Bylitonic, Ian Ring Music TheoryBylitonic
Scale 913Scale 913: Aeolyritonic, Ian Ring Music TheoryAeolyritonic
Scale 929Scale 929, Ian Ring Music Theory
Scale 961Scale 961, Ian Ring Music Theory
Scale 769Scale 769, Ian Ring Music Theory
Scale 833Scale 833, Ian Ring Music Theory
Scale 641Scale 641, Ian Ring Music Theory
Scale 385Scale 385, Ian Ring Music Theory
Scale 1409Scale 1409, Ian Ring Music Theory
Scale 1921Scale 1921, Ian Ring Music Theory
Scale 2945Scale 2945, 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.