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Scale 187


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.

6 (hexatonic)

Pitch Class Set

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

{0,1,3,4,5,7}

Forte Number

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

6-Z10

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: 2977

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.

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.

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.

5

Prime Form

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

yes

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.

[3, 3, 3, 3, 2, 1]

Interval Spectrum

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

p2m3n3s3d3t

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> = {1,2,5}
<2> = {2,3,6,7}
<3> = {4,8}
<4> = {5,6,9,10}
<5> = {7,10,11}

Spectra Variation

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

3.667

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.

1.866

Polygon Perimeter

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

5.485

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
Major TriadsC{0,4,7}210.67
Minor Triadscm{0,3,7}121
Diminished Triadsc♯°{1,4,7}121

The following pitch classes are not present in any of the common triads: {5}

Parsimonious Voice Leading Between Common Triads of Scale 187. Created by Ian Ring ©2019 cm cm C C cm->C c#° c#° C->c#°

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter2
Radius1
Self-Centeredno
Central VerticesC
Peripheral Verticescm, c♯°

Modes

Modes are the rotational transformation of this scale. Scale 187 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2141
Scale 2141, Ian Ring Music Theory
3rd mode:
Scale 1559
Scale 1559, Ian Ring Music Theory
4th mode:
Scale 2827
Scale 2827, Ian Ring Music Theory
5th mode:
Scale 3461
Scale 3461, Ian Ring Music Theory
6th mode:
Scale 1889
Scale 1889, Ian Ring Music Theory

Prime

This is the prime form of this scale.

Complement

The hexatonic modal family [187, 2141, 1559, 2827, 3461, 1889] (Forte: 6-Z10) is the complement of the hexatonic modal family [317, 977, 1103, 2599, 3347, 3721] (Forte: 6-Z39)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 187 is 2977

Scale 2977Scale 2977, Ian Ring Music Theory

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 187 is chiral, and its enantiomorph is scale 2977

Scale 2977Scale 2977, Ian Ring Music Theory

Transformations:

T0 187  T0I 2977
T1 374  T1I 1859
T2 748  T2I 3718
T3 1496  T3I 3341
T4 2992  T4I 2587
T5 1889  T5I 1079
T6 3778  T6I 2158
T7 3461  T7I 221
T8 2827  T8I 442
T9 1559  T9I 884
T10 3118  T10I 1768
T11 2141  T11I 3536

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 185Scale 185, Ian Ring Music Theory
Scale 189Scale 189, Ian Ring Music Theory
Scale 191Scale 191, Ian Ring Music Theory
Scale 179Scale 179, Ian Ring Music Theory
Scale 183Scale 183, Ian Ring Music Theory
Scale 171Scale 171, Ian Ring Music Theory
Scale 155Scale 155, Ian Ring Music Theory
Scale 219Scale 219: Istrian, Ian Ring Music TheoryIstrian
Scale 251Scale 251, Ian Ring Music Theory
Scale 59Scale 59, Ian Ring Music Theory
Scale 123Scale 123, Ian Ring Music Theory
Scale 315Scale 315: Stodimic, Ian Ring Music TheoryStodimic
Scale 443Scale 443: Kothian, Ian Ring Music TheoryKothian
Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian
Scale 1211Scale 1211: Zadian, Ian Ring Music TheoryZadian
Scale 2235Scale 2235: Bathian, Ian Ring Music TheoryBathian

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.