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Scale 1607: "Epytimic"

Scale 1607: Epytimic, 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
Epytimic
Dozenal
Juzian

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

{0,1,2,6,9,10}

Forte Number

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

6-15

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: 3149

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.

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.

5

Prime Form

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

no
prime: 311

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, 4, 3, 1, 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, 3, 4, 2, 1>

Interval Spectrum

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

p2m4n3s2d3t

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

3.333

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.

(21, 18, 64)

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{2,6,9}221.2
F♯{6,10,1}221.2
Minor Triadsf♯m{6,9,1}321
Augmented TriadsD+{2,6,10}231.4
Diminished Triadsf♯°{6,9,0}131.6
Parsimonious Voice Leading Between Common Triads of Scale 1607. Created by Ian Ring ©2019 D D D+ D+ D->D+ f#m f#m D->f#m F# F# D+->F# f#° f#° f#°->f#m f#m->F#

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
Radius2
Self-Centeredno
Central VerticesD, f♯m, F♯
Peripheral VerticesD+, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 1607 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2851
Scale 2851: Katoptimic, Ian Ring Music TheoryKatoptimic
3rd mode:
Scale 3473
Scale 3473: Lathimic, Ian Ring Music TheoryLathimic
4th mode:
Scale 473
Scale 473: Aeralimic, Ian Ring Music TheoryAeralimic
5th mode:
Scale 571
Scale 571: Kynimic, Ian Ring Music TheoryKynimic
6th mode:
Scale 2333
Scale 2333: Stynimic, Ian Ring Music TheoryStynimic

Prime

The prime form of this scale is Scale 311

Scale 311Scale 311: Stagimic, Ian Ring Music TheoryStagimic

Complement

The hexatonic modal family [1607, 2851, 3473, 473, 571, 2333] (Forte: 6-15) is the complement of the hexatonic modal family [311, 881, 1811, 2203, 2953, 3149] (Forte: 6-15)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1607 is 3149

Scale 3149Scale 3149: Phrycrimic, Ian Ring Music TheoryPhrycrimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1607 is chiral, and its enantiomorph is scale 3149

Scale 3149Scale 3149: Phrycrimic, Ian Ring Music TheoryPhrycrimic

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> 1607       T0I <11,0> 3149
T1 <1,1> 3214      T1I <11,1> 2203
T2 <1,2> 2333      T2I <11,2> 311
T3 <1,3> 571      T3I <11,3> 622
T4 <1,4> 1142      T4I <11,4> 1244
T5 <1,5> 2284      T5I <11,5> 2488
T6 <1,6> 473      T6I <11,6> 881
T7 <1,7> 946      T7I <11,7> 1762
T8 <1,8> 1892      T8I <11,8> 3524
T9 <1,9> 3784      T9I <11,9> 2953
T10 <1,10> 3473      T10I <11,10> 1811
T11 <1,11> 2851      T11I <11,11> 3622
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1637      T0MI <7,0> 1229
T1M <5,1> 3274      T1MI <7,1> 2458
T2M <5,2> 2453      T2MI <7,2> 821
T3M <5,3> 811      T3MI <7,3> 1642
T4M <5,4> 1622      T4MI <7,4> 3284
T5M <5,5> 3244      T5MI <7,5> 2473
T6M <5,6> 2393      T6MI <7,6> 851
T7M <5,7> 691      T7MI <7,7> 1702
T8M <5,8> 1382      T8MI <7,8> 3404
T9M <5,9> 2764      T9MI <7,9> 2713
T10M <5,10> 1433      T10MI <7,10> 1331
T11M <5,11> 2866      T11MI <7,11> 2662

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 1605Scale 1605: Zanitonic, Ian Ring Music TheoryZanitonic
Scale 1603Scale 1603: Juxian, Ian Ring Music TheoryJuxian
Scale 1611Scale 1611: Dacrimic, Ian Ring Music TheoryDacrimic
Scale 1615Scale 1615: Sydian, Ian Ring Music TheorySydian
Scale 1623Scale 1623: Lothian, Ian Ring Music TheoryLothian
Scale 1639Scale 1639: Aeolothian, Ian Ring Music TheoryAeolothian
Scale 1543Scale 1543: Jomian, Ian Ring Music TheoryJomian
Scale 1575Scale 1575: Zycrimic, Ian Ring Music TheoryZycrimic
Scale 1671Scale 1671: Kemian, Ian Ring Music TheoryKemian
Scale 1735Scale 1735: Mela Navanitam, Ian Ring Music TheoryMela Navanitam
Scale 1863Scale 1863: Pycrian, Ian Ring Music TheoryPycrian
Scale 1095Scale 1095: Phrythitonic, Ian Ring Music TheoryPhrythitonic
Scale 1351Scale 1351: Aeraptimic, Ian Ring Music TheoryAeraptimic
Scale 583Scale 583: Aeritonic, Ian Ring Music TheoryAeritonic
Scale 2631Scale 2631: Macrimic, Ian Ring Music TheoryMacrimic
Scale 3655Scale 3655: Mathian, Ian Ring Music TheoryMathian

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.