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Scale 1857

Scale 1857, 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.

5 (pentatonic)

Pitch Class Set

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

{0,6,8,9,10}

Forte Number

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

5-8

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.

[3]

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.

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.

5

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: 93

Deep Scale

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

no

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, 3, 2, 2, 0, 1]

Interval Spectrum

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

m2n2s3d2t

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> = {2,3,8}
<3> = {4,9,10}
<4> = {6,10,11}

Spectra Variation

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

4.4

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.

1.366

Polygon Perimeter

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

5.035

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.

[6]

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: {8,10}

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 1857 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 93
Scale 93, Ian Ring Music TheoryThis is the prime mode
3rd mode:
Scale 1047
Scale 1047, Ian Ring Music Theory
4th mode:
Scale 2571
Scale 2571, Ian Ring Music Theory
5th mode:
Scale 3333
Scale 3333, Ian Ring Music Theory

Prime

The prime form of this scale is Scale 93

Scale 93Scale 93, Ian Ring Music Theory

Complement

The pentatonic modal family [1857, 93, 1047, 2571, 3333] (Forte: 5-8) is the complement of the heptatonic modal family [381, 1119, 2001, 2607, 3351, 3723, 3909] (Forte: 7-8)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1857 is 93

Scale 93Scale 93, Ian Ring Music Theory

Transformations:

T0 1857  T0I 93
T1 3714  T1I 186
T2 3333  T2I 372
T3 2571  T3I 744
T4 1047  T4I 1488
T5 2094  T5I 2976
T6 93  T6I 1857
T7 186  T7I 3714
T8 372  T8I 3333
T9 744  T9I 2571
T10 1488  T10I 1047
T11 2976  T11I 2094

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 1859Scale 1859, Ian Ring Music Theory
Scale 1861Scale 1861: Phrygimic, Ian Ring Music TheoryPhrygimic
Scale 1865Scale 1865: Thagimic, Ian Ring Music TheoryThagimic
Scale 1873Scale 1873: Dathimic, Ian Ring Music TheoryDathimic
Scale 1889Scale 1889, Ian Ring Music Theory
Scale 1793Scale 1793, Ian Ring Music Theory
Scale 1825Scale 1825, Ian Ring Music Theory
Scale 1921Scale 1921, Ian Ring Music Theory
Scale 1985Scale 1985, Ian Ring Music Theory
Scale 1601Scale 1601, Ian Ring Music Theory
Scale 1729Scale 1729, Ian Ring Music Theory
Scale 1345Scale 1345, Ian Ring Music Theory
Scale 833Scale 833, Ian Ring Music Theory
Scale 2881Scale 2881, Ian Ring Music Theory
Scale 3905Scale 3905, Ian Ring Music Theory

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. 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.