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Scale 871: "Locrian Double-flat 3 Double-flat 7"

Scale 871: Locrian Double-flat 3 Double-flat 7, 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).

Common Names

Western Altered
Locrian Double-flat 3 Double-flat 7
Dozenal
Fiyian
Zeitler
Epadian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

7 (heptatonic)

Pitch Class Set

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

{0,1,2,5,6,8,9}

Forte Number

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

7-22

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.

[1]

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.

4 (multihemitonic)

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.

6

Prime Form

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

yes

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 Structure

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

[1, 1, 3, 1, 2, 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.

<4, 2, 4, 5, 4, 2>

Interval Spectrum

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

p4m5n4s2d4t2

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

Spectra Variation

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

1.714

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.

2.433

Polygon Perimeter

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

5.899

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.

yes

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.

[2]

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.

(4, 22, 84)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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♯{1,5,8}331.67
D{2,6,9}242
F{5,9,0}331.67
Minor Triadsdm{2,5,9}331.67
fm{5,8,0}242
f♯m{6,9,1}331.67
Augmented TriadsC♯+{1,5,9}421.33
Diminished Triads{2,5,8}242
f♯°{6,9,0}242
Parsimonious Voice Leading Between Common Triads of Scale 871. Created by Ian Ring ©2019 C# C# C#+ C#+ C#->C#+ C#->d° fm fm C#->fm dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m d°->dm D D dm->D D->f#m fm->F f#° f#° F->f#° f#°->f#m

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.

Diameter4
Radius2
Self-Centeredno
Central VerticesC♯+
Peripheral Verticesd°, D, fm, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 871 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2483
Scale 2483: Double Harmonic, Ian Ring Music TheoryDouble Harmonic
3rd mode:
Scale 3289
Scale 3289: Lydian Sharp 2 Sharp 6, Ian Ring Music TheoryLydian Sharp 2 Sharp 6
4th mode:
Scale 923
Scale 923: Ultraphrygian, Ian Ring Music TheoryUltraphrygian
5th mode:
Scale 2509
Scale 2509: Double Harmonic Minor, Ian Ring Music TheoryDouble Harmonic Minor
6th mode:
Scale 1651
Scale 1651: Asian, Ian Ring Music TheoryAsian
7th mode:
Scale 2873
Scale 2873: Ionian Augmented Sharp 2, Ian Ring Music TheoryIonian Augmented Sharp 2

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [871, 2483, 3289, 923, 2509, 1651, 2873] (Forte: 7-22) is the complement of the pentatonic modal family [403, 611, 793, 2249, 2353] (Forte: 5-22)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 871 is 3289

Scale 3289Scale 3289: Lydian Sharp 2 Sharp 6, Ian Ring Music TheoryLydian Sharp 2 Sharp 6

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> 871       T0I <11,0> 3289
T1 <1,1> 1742      T1I <11,1> 2483
T2 <1,2> 3484      T2I <11,2> 871
T3 <1,3> 2873      T3I <11,3> 1742
T4 <1,4> 1651      T4I <11,4> 3484
T5 <1,5> 3302      T5I <11,5> 2873
T6 <1,6> 2509      T6I <11,6> 1651
T7 <1,7> 923      T7I <11,7> 3302
T8 <1,8> 1846      T8I <11,8> 2509
T9 <1,9> 3692      T9I <11,9> 923
T10 <1,10> 3289      T10I <11,10> 1846
T11 <1,11> 2483      T11I <11,11> 3692
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1651      T0MI <7,0> 2509
T1M <5,1> 3302      T1MI <7,1> 923
T2M <5,2> 2509      T2MI <7,2> 1846
T3M <5,3> 923      T3MI <7,3> 3692
T4M <5,4> 1846      T4MI <7,4> 3289
T5M <5,5> 3692      T5MI <7,5> 2483
T6M <5,6> 3289      T6MI <7,6> 871
T7M <5,7> 2483      T7MI <7,7> 1742
T8M <5,8> 871       T8MI <7,8> 3484
T9M <5,9> 1742      T9MI <7,9> 2873
T10M <5,10> 3484      T10MI <7,10> 1651
T11M <5,11> 2873      T11MI <7,11> 3302

The transformations that map this set to itself are: T0, T2I, T8M, T6MI

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 869Scale 869: Kothimic, Ian Ring Music TheoryKothimic
Scale 867Scale 867: Phrocrimic, Ian Ring Music TheoryPhrocrimic
Scale 875Scale 875: Locrian Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 7
Scale 879Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllic
Scale 887Scale 887: Sathyllic, Ian Ring Music TheorySathyllic
Scale 839Scale 839: Ionathimic, Ian Ring Music TheoryIonathimic
Scale 855Scale 855: Porian, Ian Ring Music TheoryPorian
Scale 807Scale 807: Raga Suddha Mukhari, Ian Ring Music TheoryRaga Suddha Mukhari
Scale 935Scale 935: Chromatic Dorian, Ian Ring Music TheoryChromatic Dorian
Scale 999Scale 999: Ionodyllic, Ian Ring Music TheoryIonodyllic
Scale 615Scale 615: Schoenberg Hexachord, Ian Ring Music TheorySchoenberg Hexachord
Scale 743Scale 743: Chromatic Hypophrygian Inverse, Ian Ring Music TheoryChromatic Hypophrygian Inverse
Scale 359Scale 359: Bothimic, Ian Ring Music TheoryBothimic
Scale 1383Scale 1383: Pynian, Ian Ring Music TheoryPynian
Scale 1895Scale 1895: Salyllic, Ian Ring Music TheorySalyllic
Scale 2919Scale 2919: Molyllic, Ian Ring Music TheoryMolyllic

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, and George Howlett for assistance with the Carnatic ragas.