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Scale 1127: "Eparimic"

Scale 1127: Eparimic, 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
Eparimic
Dozenal
Guzian

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

{0,1,2,5,6,10}

Forte Number

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

6-16

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: 3269

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.

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.

5

Prime Form

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

no
prime: 371

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, 4, 2]

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, 2, 2, 4, 3, 1>

Interval Spectrum

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

p3m4n2s2d3t

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

Spectra Variation

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

3

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.

2.116

Polygon Perimeter

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

5.699

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.

(14, 19, 65)

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 TriadsF♯{6,10,1}221
A♯{10,2,5}221
Minor Triadsa♯m{10,1,5}221
Augmented TriadsD+{2,6,10}221

The following pitch classes are not present in any of the common triads: {0}

Parsimonious Voice Leading Between Common Triads of Scale 1127. Created by Ian Ring ©2019 D+ D+ F# F# D+->F# A# A# D+->A# a#m a#m F#->a#m a#m->A#

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
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1127 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2611
Scale 2611: Raga Vasanta, Ian Ring Music TheoryRaga Vasanta
3rd mode:
Scale 3353
Scale 3353: Phraptimic, Ian Ring Music TheoryPhraptimic
4th mode:
Scale 931
Scale 931: Raga Kalakanthi, Ian Ring Music TheoryRaga Kalakanthi
5th mode:
Scale 2513
Scale 2513: Aerycrimic, Ian Ring Music TheoryAerycrimic
6th mode:
Scale 413
Scale 413: Ganimic, Ian Ring Music TheoryGanimic

Prime

The prime form of this scale is Scale 371

Scale 371Scale 371: Rythimic, Ian Ring Music TheoryRythimic

Complement

The hexatonic modal family [1127, 2611, 3353, 931, 2513, 413] (Forte: 6-16) is the complement of the hexatonic modal family [371, 791, 1841, 2233, 2443, 3269] (Forte: 6-16)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1127 is 3269

Scale 3269Scale 3269: Raga Malarani, Ian Ring Music TheoryRaga Malarani

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1127 is chiral, and its enantiomorph is scale 3269

Scale 3269Scale 3269: Raga Malarani, Ian Ring Music TheoryRaga Malarani

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> 1127       T0I <11,0> 3269
T1 <1,1> 2254      T1I <11,1> 2443
T2 <1,2> 413      T2I <11,2> 791
T3 <1,3> 826      T3I <11,3> 1582
T4 <1,4> 1652      T4I <11,4> 3164
T5 <1,5> 3304      T5I <11,5> 2233
T6 <1,6> 2513      T6I <11,6> 371
T7 <1,7> 931      T7I <11,7> 742
T8 <1,8> 1862      T8I <11,8> 1484
T9 <1,9> 3724      T9I <11,9> 2968
T10 <1,10> 3353      T10I <11,10> 1841
T11 <1,11> 2611      T11I <11,11> 3682
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1127       T0MI <7,0> 3269
T1M <5,1> 2254      T1MI <7,1> 2443
T2M <5,2> 413      T2MI <7,2> 791
T3M <5,3> 826      T3MI <7,3> 1582
T4M <5,4> 1652      T4MI <7,4> 3164
T5M <5,5> 3304      T5MI <7,5> 2233
T6M <5,6> 2513      T6MI <7,6> 371
T7M <5,7> 931      T7MI <7,7> 742
T8M <5,8> 1862      T8MI <7,8> 1484
T9M <5,9> 3724      T9MI <7,9> 2968
T10M <5,10> 3353      T10MI <7,10> 1841
T11M <5,11> 2611      T11MI <7,11> 3682

The transformations that map this set to itself are: T0, T0M

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 1125Scale 1125: Ionaritonic, Ian Ring Music TheoryIonaritonic
Scale 1123Scale 1123: Iwato, Ian Ring Music TheoryIwato
Scale 1131Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form
Scale 1135Scale 1135: Katolian, Ian Ring Music TheoryKatolian
Scale 1143Scale 1143: Styrian, Ian Ring Music TheoryStyrian
Scale 1095Scale 1095: Phrythitonic, Ian Ring Music TheoryPhrythitonic
Scale 1111Scale 1111: Sycrimic, Ian Ring Music TheorySycrimic
Scale 1063Scale 1063: Gomian, Ian Ring Music TheoryGomian
Scale 1191Scale 1191: Pyrimic, Ian Ring Music TheoryPyrimic
Scale 1255Scale 1255: Chromatic Mixolydian, Ian Ring Music TheoryChromatic Mixolydian
Scale 1383Scale 1383: Pynian, Ian Ring Music TheoryPynian
Scale 1639Scale 1639: Aeolothian, Ian Ring Music TheoryAeolothian
Scale 103Scale 103: Apuian, Ian Ring Music TheoryApuian
Scale 615Scale 615: Schoenberg Hexachord, Ian Ring Music TheorySchoenberg Hexachord
Scale 2151Scale 2151: Natian, Ian Ring Music TheoryNatian
Scale 3175Scale 3175: Eponian, Ian Ring Music TheoryEponian

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.