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Scale 875: "Locrian Double-flat 7"

Scale 875: Locrian 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 7
Dozenal
Fibian
Jazz and Blues
Diminished Blues Flat 9
Zeitler
Stothian

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

{0,1,3,5,6,8,9}

Forte Number

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

7-32

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: 2777

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.

0 (ancohemitonic)

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.

no
prime: 859

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, 2, 2, 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.

<3, 3, 5, 4, 4, 2>

Interval Spectrum

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

p4m4n5s3d3t2

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

Spectra Variation

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

1.429

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.

2.549

Polygon Perimeter

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

5.967

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".

Proper

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.

(0, 18, 82)

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}232
F{5,9,0}431.6
G♯{8,0,3}331.8
Minor Triadsfm{5,8,0}331.7
f♯m{6,9,1}331.8
Augmented TriadsC♯+{1,5,9}331.7
Diminished Triads{0,3,6}232
d♯°{3,6,9}232
f♯°{6,9,0}231.9
{9,0,3}231.9
Parsimonious Voice Leading Between Common Triads of Scale 875. Created by Ian Ring ©2019 d#° d#° c°->d#° G# G# c°->G# C# C# C#+ C#+ C#->C#+ fm fm C#->fm F F C#+->F f#m f#m C#+->f#m d#°->f#m fm->F fm->G# f#° f#° F->f#° F->a° f#°->f#m G#->a°

view full size

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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 875 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2485
Scale 2485: Harmonic Major, Ian Ring Music TheoryHarmonic Major
3rd mode:
Scale 1645
Scale 1645: Dorian Flat 5, Ian Ring Music TheoryDorian Flat 5
4th mode:
Scale 1435
Scale 1435: Makam Huzzam, Ian Ring Music TheoryMakam Huzzam
5th mode:
Scale 2765
Scale 2765: Lydian Diminished, Ian Ring Music TheoryLydian Diminished
6th mode:
Scale 1715
Scale 1715: Harmonic Minor Inverse, Ian Ring Music TheoryHarmonic Minor Inverse
7th mode:
Scale 2905
Scale 2905: Aeolian Flat 1, Ian Ring Music TheoryAeolian Flat 1

Prime

The prime form of this scale is Scale 859

Scale 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian

Complement

The heptatonic modal family [875, 2485, 1645, 1435, 2765, 1715, 2905] (Forte: 7-32) is the complement of the pentatonic modal family [595, 665, 805, 1225, 2345] (Forte: 5-32)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 875 is 2777

Scale 2777Scale 2777: Aeolian Harmonic, Ian Ring Music TheoryAeolian Harmonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 875 is chiral, and its enantiomorph is scale 2777

Scale 2777Scale 2777: Aeolian Harmonic, Ian Ring Music TheoryAeolian Harmonic

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> 875       T0I <11,0> 2777
T1 <1,1> 1750      T1I <11,1> 1459
T2 <1,2> 3500      T2I <11,2> 2918
T3 <1,3> 2905      T3I <11,3> 1741
T4 <1,4> 1715      T4I <11,4> 3482
T5 <1,5> 3430      T5I <11,5> 2869
T6 <1,6> 2765      T6I <11,6> 1643
T7 <1,7> 1435      T7I <11,7> 3286
T8 <1,8> 2870      T8I <11,8> 2477
T9 <1,9> 1645      T9I <11,9> 859
T10 <1,10> 3290      T10I <11,10> 1718
T11 <1,11> 2485      T11I <11,11> 3436
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 635      T0MI <7,0> 3017
T1M <5,1> 1270      T1MI <7,1> 1939
T2M <5,2> 2540      T2MI <7,2> 3878
T3M <5,3> 985      T3MI <7,3> 3661
T4M <5,4> 1970      T4MI <7,4> 3227
T5M <5,5> 3940      T5MI <7,5> 2359
T6M <5,6> 3785      T6MI <7,6> 623
T7M <5,7> 3475      T7MI <7,7> 1246
T8M <5,8> 2855      T8MI <7,8> 2492
T9M <5,9> 1615      T9MI <7,9> 889
T10M <5,10> 3230      T10MI <7,10> 1778
T11M <5,11> 2365      T11MI <7,11> 3556

The transformations that map this set to itself are: T0

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 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic
Scale 877Scale 877: Moravian Pistalkova, Ian Ring Music TheoryMoravian Pistalkova
Scale 879Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllic
Scale 867Scale 867: Phrocrimic, Ian Ring Music TheoryPhrocrimic
Scale 871Scale 871: Locrian Double-flat 3 Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 3 Double-flat 7
Scale 883Scale 883: Ralian, Ian Ring Music TheoryRalian
Scale 891Scale 891: Ionilyllic, Ian Ring Music TheoryIonilyllic
Scale 843Scale 843: Molimic, Ian Ring Music TheoryMolimic
Scale 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian
Scale 811Scale 811: Radimic, Ian Ring Music TheoryRadimic
Scale 939Scale 939: Mela Senavati, Ian Ring Music TheoryMela Senavati
Scale 1003Scale 1003: Ionyryllic, Ian Ring Music TheoryIonyryllic
Scale 619Scale 619: Double-Phrygian Hexatonic, Ian Ring Music TheoryDouble-Phrygian Hexatonic
Scale 747Scale 747: Lynian, Ian Ring Music TheoryLynian
Scale 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic
Scale 1387Scale 1387: Locrian, Ian Ring Music TheoryLocrian
Scale 1899Scale 1899: Moptyllic, Ian Ring Music TheoryMoptyllic
Scale 2923Scale 2923: Baryllic, Ian Ring Music TheoryBaryllic

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.