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Scale 3895: "Eparygic"

Scale 3895: Eparygic, 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
Eparygic
Dozenal
YUMian

Analysis

Cardinality

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

 9 (enneatonic)

Pitch Class Set

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

 {0,1,2,4,5,8,9,10,11}

Forte Number

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

 9-3

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: 3487

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

 7 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

 5 (multicohemitonic)

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 includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

 9

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

 no
prime: 895

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, an indicator of maximum hierarchization.

 no

Interval Structure

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

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

 <7, 6, 7, 7, 6, 3>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.5, 0, 0.5, 0.333, 0, 0>

Interval Spectrum

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

 p6m7n7s6d7t3

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

Spectra Variation

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

 2.222

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.

2.683

Polygon Perimeter

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

6.038

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

(64, 107, 194)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.63

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.326

Generator

This scale has no generator.

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}442.12
E{4,8,11}342.53
F{5,9,0}342.24
A{9,1,4}442.24
A♯{10,2,5}342.53
Minor Triadsc♯m{1,4,8}342.24
dm{2,5,9}342.35
fm{5,8,0}442.24
am{9,0,4}342.35
a♯m{10,1,5}342.35
Augmented TriadsC+{0,4,8}442.24
C♯+{1,5,9}542
Diminished Triadsd°{2,5,8}242.59
f°{5,8,11}252.71
g♯°{8,11,2}242.76
a♯°{10,1,4}252.71
b°{11,2,5}242.76
Parsimonious Voice Leading Between Common Triads of Scale 3895. 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#+ d° d° C#->d° C#->fm dm dm C#+->dm F F C#+->F C#+->A a#m a#m C#+->a#m d°->dm A# A# dm->A# f° f° E->f° g#° g#° E->g#° f°->fm fm->F F->am b° b° g#°->b° am->A a#° a#° A->a#° a#°->a#m a#m->A# A#->b°

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.

Diameter5
Radius4
Self-Centeredno
Central VerticesC+, c♯m, C♯, C♯+, d°, dm, E, fm, F, g♯°, am, A, a♯m, A♯, b°
Peripheral Verticesf°, a♯°

Modes

Modes are the rotational transformation of this scale. Scale 3895 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3995		Scale 3995: Ionygic, Ian Ring Music TheoryIonygic
3rd mode:
Scale 4045		Scale 4045: Gyptygic, Ian Ring Music TheoryGyptygic
4th mode:
Scale 2035		Scale 2035: Aerythygic, Ian Ring Music TheoryAerythygic
5th mode:
Scale 3065		Scale 3065: Zothygic, Ian Ring Music TheoryZothygic
6th mode:
Scale 895		Scale 895: Aeolathygic, Ian Ring Music TheoryAeolathygicThis is the prime mode
7th mode:
Scale 2495		Scale 2495: Aeolocrygic, Ian Ring Music TheoryAeolocrygic
8th mode:
Scale 3295		Scale 3295: Phroptygic, Ian Ring Music TheoryPhroptygic
9th mode:
Scale 3695		Scale 3695: Kodygic, Ian Ring Music TheoryKodygic

Prime

The prime form of this scale is Scale 895

Scale 895Scale 895: Aeolathygic, Ian Ring Music TheoryAeolathygic

Complement

The enneatonic modal family [3895, 3995, 4045, 2035, 3065, 895, 2495, 3295, 3695] (Forte: 9-3) is the complement of the tritonic modal family [19, 769, 2057] (Forte: 3-3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3895 is 3487

Scale 3487Scale 3487: Byptygic, Ian Ring Music TheoryByptygic

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
111110110011113895k = 1h = 1
211110110011113895k = 2h = 1
32(1)(1)(1)0(1)(1)00(1)(1)(1)(1)3895k = 3h = 2
42(11)10(11)00(11)(11)3895k = 4h = 2
52(11)10(11)00(11)(11)3895k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3895 is chiral, and its enantiomorph is scale 3487

Scale 3487Scale 3487: Byptygic, Ian Ring Music TheoryByptygic

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 3895       T0I <11,0> 3487
T1 <1,1> 3695      T1I <11,1> 2879
T2 <1,2> 3295      T2I <11,2> 1663
T3 <1,3> 2495      T3I <11,3> 3326
T4 <1,4> 895      T4I <11,4> 2557
T5 <1,5> 1790      T5I <11,5> 1019
T6 <1,6> 3580      T6I <11,6> 2038
T7 <1,7> 3065      T7I <11,7> 4076
T8 <1,8> 2035      T8I <11,8> 4057
T9 <1,9> 4070      T9I <11,9> 4019
T10 <1,10> 4045      T10I <11,10> 3943
T11 <1,11> 3995      T11I <11,11> 3791
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1975      T0MI <7,0> 3517
T1M <5,1> 3950      T1MI <7,1> 2939
T2M <5,2> 3805      T2MI <7,2> 1783
T3M <5,3> 3515      T3MI <7,3> 3566
T4M <5,4> 2935      T4MI <7,4> 3037
T5M <5,5> 1775      T5MI <7,5> 1979
T6M <5,6> 3550      T6MI <7,6> 3958
T7M <5,7> 3005      T7MI <7,7> 3821
T8M <5,8> 1915      T8MI <7,8> 3547
T9M <5,9> 3830      T9MI <7,9> 2999
T10M <5,10> 3565      T10MI <7,10> 1903
T11M <5,11> 3035      T11MI <7,11> 3806

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 3893Scale 3893: Phrocryllic, Ian Ring Music TheoryPhrocryllic
Scale 3891Scale 3891: Ryryllic, Ian Ring Music TheoryRyryllic
Scale 3899Scale 3899: Katorygic, Ian Ring Music TheoryKatorygic
Scale 3903Scale 3903: Decatonic Chromatic 5, Ian Ring Music TheoryDecatonic Chromatic 5
Scale 3879Scale 3879: Pathyllic, Ian Ring Music TheoryPathyllic
Scale 3887Scale 3887: Phrathygic, Ian Ring Music TheoryPhrathygic
Scale 3863Scale 3863: Eparyllic, Ian Ring Music TheoryEparyllic
Scale 3927Scale 3927: Monygic, Ian Ring Music TheoryMonygic
Scale 3959Scale 3959: Katagyllian, Ian Ring Music TheoryKatagyllian
Scale 4023Scale 4023: Styptyllian, Ian Ring Music TheoryStyptyllian
Scale 3639Scale 3639: Paptyllic, Ian Ring Music TheoryPaptyllic
Scale 3767Scale 3767: Chromatic Bebop, Ian Ring Music TheoryChromatic Bebop
Scale 3383Scale 3383: Daptyllic, Ian Ring Music TheoryDaptyllic
Scale 2871Scale 2871: Stanyllic, Ian Ring Music TheoryStanyllic
Scale 1847Scale 1847: Thacryllic, Ian Ring Music TheoryThacryllic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.