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Scale 3653: "Sathimic"

Scale 3653: Sathimic, 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
Sathimic
Dozenal
Wurian

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

{0,2,6,9,10,11}

Forte Number

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

6-Z39

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: 1103

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.

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.

5

Prime Form

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

no
prime: 317

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.

[2, 4, 3, 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.

<3, 3, 3, 3, 2, 1>

Interval Spectrum

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

p2m3n3s3d3t

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

3.667

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.

(29, 19, 67)

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
Minor Triadsbm{11,2,6}131.5
Augmented TriadsD+{2,6,10}221
Diminished Triadsf♯°{6,9,0}131.5
Parsimonious Voice Leading Between Common Triads of Scale 3653. Created by Ian Ring ©2019 D D D+ D+ D->D+ f#° f#° D->f#° bm bm D+->bm

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, D+
Peripheral Verticesf♯°, bm

Modes

Modes are the rotational transformation of this scale. Scale 3653 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1937
Scale 1937: Galimic, Ian Ring Music TheoryGalimic
3rd mode:
Scale 377
Scale 377: Kathimic, Ian Ring Music TheoryKathimic
4th mode:
Scale 559
Scale 559: Lylimic, Ian Ring Music TheoryLylimic
5th mode:
Scale 2327
Scale 2327: Epalimic, Ian Ring Music TheoryEpalimic
6th mode:
Scale 3211
Scale 3211: Epacrimic, Ian Ring Music TheoryEpacrimic

Prime

The prime form of this scale is Scale 317

Scale 317Scale 317: Korimic, Ian Ring Music TheoryKorimic

Complement

The hexatonic modal family [3653, 1937, 377, 559, 2327, 3211] (Forte: 6-Z39) is the complement of the hexatonic modal family [187, 1559, 1889, 2141, 2827, 3461] (Forte: 6-Z10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3653 is 1103

Scale 1103Scale 1103: Lynimic, Ian Ring Music TheoryLynimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3653 is chiral, and its enantiomorph is scale 1103

Scale 1103Scale 1103: Lynimic, Ian Ring Music TheoryLynimic

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> 3653       T0I <11,0> 1103
T1 <1,1> 3211      T1I <11,1> 2206
T2 <1,2> 2327      T2I <11,2> 317
T3 <1,3> 559      T3I <11,3> 634
T4 <1,4> 1118      T4I <11,4> 1268
T5 <1,5> 2236      T5I <11,5> 2536
T6 <1,6> 377      T6I <11,6> 977
T7 <1,7> 754      T7I <11,7> 1954
T8 <1,8> 1508      T8I <11,8> 3908
T9 <1,9> 3016      T9I <11,9> 3721
T10 <1,10> 1937      T10I <11,10> 3347
T11 <1,11> 3874      T11I <11,11> 2599
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1733      T0MI <7,0> 1133
T1M <5,1> 3466      T1MI <7,1> 2266
T2M <5,2> 2837      T2MI <7,2> 437
T3M <5,3> 1579      T3MI <7,3> 874
T4M <5,4> 3158      T4MI <7,4> 1748
T5M <5,5> 2221      T5MI <7,5> 3496
T6M <5,6> 347      T6MI <7,6> 2897
T7M <5,7> 694      T7MI <7,7> 1699
T8M <5,8> 1388      T8MI <7,8> 3398
T9M <5,9> 2776      T9MI <7,9> 2701
T10M <5,10> 1457      T10MI <7,10> 1307
T11M <5,11> 2914      T11MI <7,11> 2614

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 3655Scale 3655: Mathian, Ian Ring Music TheoryMathian
Scale 3649Scale 3649: Wupian, Ian Ring Music TheoryWupian
Scale 3651Scale 3651: Wuqian, Ian Ring Music TheoryWuqian
Scale 3657Scale 3657: Epynimic, Ian Ring Music TheoryEpynimic
Scale 3661Scale 3661: Mixodorian, Ian Ring Music TheoryMixodorian
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian
Scale 3685Scale 3685: Kodian, Ian Ring Music TheoryKodian
Scale 3589Scale 3589: Widian, Ian Ring Music TheoryWidian
Scale 3621Scale 3621: Gylimic, Ian Ring Music TheoryGylimic
Scale 3717Scale 3717: Xidian, Ian Ring Music TheoryXidian
Scale 3781Scale 3781: Gyphian, Ian Ring Music TheoryGyphian
Scale 3909Scale 3909: Rydian, Ian Ring Music TheoryRydian
Scale 3141Scale 3141: Kanitonic, Ian Ring Music TheoryKanitonic
Scale 3397Scale 3397: Sydimic, Ian Ring Music TheorySydimic
Scale 2629Scale 2629: Raga Shubravarni, Ian Ring Music TheoryRaga Shubravarni
Scale 1605Scale 1605: Zanitonic, Ian Ring Music TheoryZanitonic

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.