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Scale 2867: "Socrian"

Scale 2867: Socrian, 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

Zeitler
Socrian
Dozenal
Salian

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

{0,1,4,5,8,9,11}

Forte Number

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

7-21

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: 2459

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.

no
prime: 823

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

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, 6, 4, 1>

Interval Spectrum

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

p4m6n4s2d4t

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> = {4,5,6,7}
<4> = {5,6,7,8}
<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.

2

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.

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

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.

(7, 33, 90)

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}331.7
E{4,8,11}242.1
F{5,9,0}331.7
A{9,1,4}341.9
Minor Triadsc♯m{1,4,8}331.7
fm{5,8,0}431.5
am{9,0,4}331.7
Augmented TriadsC+{0,4,8}431.5
C♯+{1,5,9}341.9
Diminished Triads{5,8,11}242.1
Parsimonious Voice Leading Between Common Triads of Scale 2867. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m E E C+->E fm fm C+->fm am am C+->am C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->fm F F C#+->F C#+->A E->f° f°->fm fm->F F->am am->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.

Diameter4
Radius3
Self-Centeredno
Central VerticesC+, c♯m, C♯, fm, F, am
Peripheral VerticesC♯+, E, f°, A

Modes

Modes are the rotational transformation of this scale. Scale 2867 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3481
Scale 3481: Katathian, Ian Ring Music TheoryKatathian
3rd mode:
Scale 947
Scale 947: Mela Gayakapriya, Ian Ring Music TheoryMela Gayakapriya
4th mode:
Scale 2521
Scale 2521: Mela Dhatuvardhani, Ian Ring Music TheoryMela Dhatuvardhani
5th mode:
Scale 827
Scale 827: Mixolocrian, Ian Ring Music TheoryMixolocrian
6th mode:
Scale 2461
Scale 2461: Sagian, Ian Ring Music TheorySagian
7th mode:
Scale 1639
Scale 1639: Aeolothian, Ian Ring Music TheoryAeolothian

Prime

The prime form of this scale is Scale 823

Scale 823Scale 823: Stodian, Ian Ring Music TheoryStodian

Complement

The heptatonic modal family [2867, 3481, 947, 2521, 827, 2461, 1639] (Forte: 7-21) is the complement of the pentatonic modal family [307, 787, 817, 2201, 2441] (Forte: 5-21)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2867 is 2459

Scale 2459Scale 2459: Ionocrian, Ian Ring Music TheoryIonocrian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2867 is chiral, and its enantiomorph is scale 2459

Scale 2459Scale 2459: Ionocrian, Ian Ring Music TheoryIonocrian

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> 2867       T0I <11,0> 2459
T1 <1,1> 1639      T1I <11,1> 823
T2 <1,2> 3278      T2I <11,2> 1646
T3 <1,3> 2461      T3I <11,3> 3292
T4 <1,4> 827      T4I <11,4> 2489
T5 <1,5> 1654      T5I <11,5> 883
T6 <1,6> 3308      T6I <11,6> 1766
T7 <1,7> 2521      T7I <11,7> 3532
T8 <1,8> 947      T8I <11,8> 2969
T9 <1,9> 1894      T9I <11,9> 1843
T10 <1,10> 3788      T10I <11,10> 3686
T11 <1,11> 3481      T11I <11,11> 3277
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 947      T0MI <7,0> 2489
T1M <5,1> 1894      T1MI <7,1> 883
T2M <5,2> 3788      T2MI <7,2> 1766
T3M <5,3> 3481      T3MI <7,3> 3532
T4M <5,4> 2867       T4MI <7,4> 2969
T5M <5,5> 1639      T5MI <7,5> 1843
T6M <5,6> 3278      T6MI <7,6> 3686
T7M <5,7> 2461      T7MI <7,7> 3277
T8M <5,8> 827      T8MI <7,8> 2459
T9M <5,9> 1654      T9MI <7,9> 823
T10M <5,10> 3308      T10MI <7,10> 1646
T11M <5,11> 2521      T11MI <7,11> 3292

The transformations that map this set to itself are: T0, T4M

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 2865Scale 2865: Solimic, Ian Ring Music TheorySolimic
Scale 2869Scale 2869: Major Augmented, Ian Ring Music TheoryMajor Augmented
Scale 2871Scale 2871: Stanyllic, Ian Ring Music TheoryStanyllic
Scale 2875Scale 2875: Ganyllic, Ian Ring Music TheoryGanyllic
Scale 2851Scale 2851: Katoptimic, Ian Ring Music TheoryKatoptimic
Scale 2859Scale 2859: Phrycrian, Ian Ring Music TheoryPhrycrian
Scale 2835Scale 2835: Ionygimic, Ian Ring Music TheoryIonygimic
Scale 2899Scale 2899: Kagian, Ian Ring Music TheoryKagian
Scale 2931Scale 2931: Zathyllic, Ian Ring Music TheoryZathyllic
Scale 2995Scale 2995: Raga Saurashtra, Ian Ring Music TheoryRaga Saurashtra
Scale 2611Scale 2611: Raga Vasanta, Ian Ring Music TheoryRaga Vasanta
Scale 2739Scale 2739: Mela Suryakanta, Ian Ring Music TheoryMela Suryakanta
Scale 2355Scale 2355: Raga Lalita, Ian Ring Music TheoryRaga Lalita
Scale 3379Scale 3379: Verdi's Scala Enigmatica Descending, Ian Ring Music TheoryVerdi's Scala Enigmatica Descending
Scale 3891Scale 3891: Ryryllic, Ian Ring Music TheoryRyryllic
Scale 819Scale 819: Augmented Inverse, Ian Ring Music TheoryAugmented Inverse
Scale 1843Scale 1843: Ionygian, Ian Ring Music TheoryIonygian

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.