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Scale 3785: "Epagian"

Scale 3785: Epagian, 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
Epagian
Dozenal
Yewian

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

{0,3,6,7,9,10,11}

Forte Number

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

7-16

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: 623

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.

2 (dicohemitonic)

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.

4

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: 623

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.

[3, 3, 1, 2, 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.

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

Interval Spectrum

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

p3m4n5s3d4t2

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

Spectra Variation

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

2.857

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.

(32, 38, 102)

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 TriadsD♯{3,7,10}231.89
B{11,3,6}331.67
Minor Triadscm{0,3,7}331.67
d♯m{3,6,10}331.67
Augmented TriadsD♯+{3,7,11}331.67
Diminished Triads{0,3,6}231.89
d♯°{3,6,9}231.89
f♯°{6,9,0}232
{9,0,3}231.89
Parsimonious Voice Leading Between Common Triads of Scale 3785. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ cm->a° d#° d#° d#m d#m d#°->d#m f#° f#° d#°->f#° D# D# d#m->D# d#m->B D#->D#+ D#+->B f#°->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 3785 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 985
Scale 985: Mela Sucaritra, Ian Ring Music TheoryMela Sucaritra
3rd mode:
Scale 635
Scale 635: Epolian, Ian Ring Music TheoryEpolian
4th mode:
Scale 2365
Scale 2365: Sythian, Ian Ring Music TheorySythian
5th mode:
Scale 1615
Scale 1615: Sydian, Ian Ring Music TheorySydian
6th mode:
Scale 2855
Scale 2855: Epocrain, Ian Ring Music TheoryEpocrain
7th mode:
Scale 3475
Scale 3475: Kylian, Ian Ring Music TheoryKylian

Prime

The prime form of this scale is Scale 623

Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian

Complement

The heptatonic modal family [3785, 985, 635, 2365, 1615, 2855, 3475] (Forte: 7-16) is the complement of the pentatonic modal family [155, 865, 1555, 2125, 2825] (Forte: 5-16)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3785 is 623

Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3785 is chiral, and its enantiomorph is scale 623

Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian

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

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 3787Scale 3787: Kagyllic, Ian Ring Music TheoryKagyllic
Scale 3789Scale 3789: Eporyllic, Ian Ring Music TheoryEporyllic
Scale 3777Scale 3777: Yarian, Ian Ring Music TheoryYarian
Scale 3781Scale 3781: Gyphian, Ian Ring Music TheoryGyphian
Scale 3793Scale 3793: Aeopian, Ian Ring Music TheoryAeopian
Scale 3801Scale 3801: Maptyllic, Ian Ring Music TheoryMaptyllic
Scale 3817Scale 3817: Zoryllic, Ian Ring Music TheoryZoryllic
Scale 3721Scale 3721: Phragimic, Ian Ring Music TheoryPhragimic
Scale 3753Scale 3753: Phraptian, Ian Ring Music TheoryPhraptian
Scale 3657Scale 3657: Epynimic, Ian Ring Music TheoryEpynimic
Scale 3913Scale 3913: Bonian, Ian Ring Music TheoryBonian
Scale 4041Scale 4041: Zaryllic, Ian Ring Music TheoryZaryllic
Scale 3273Scale 3273: Raga Jivantini, Ian Ring Music TheoryRaga Jivantini
Scale 3529Scale 3529: Stalian, Ian Ring Music TheoryStalian
Scale 2761Scale 2761: Dagimic, Ian Ring Music TheoryDagimic
Scale 1737Scale 1737: Raga Madhukauns, Ian Ring Music TheoryRaga Madhukauns

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.