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Scale 315: "Stodimic"

Scale 315: Stodimic, 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
Stodimic
Dozenal
Cadian

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

{0,1,3,4,5,8}

Forte Number

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

6-14

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: 2961

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.

yes

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, 2, 1, 1, 3, 4]

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, 3, 0>

Interval Spectrum

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

p3m4n3s2d3

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> = {4,5,7,8}
<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.

(19, 22, 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 TriadsC♯{1,5,8}231.4
G♯{8,0,3}131.6
Minor Triadsc♯m{1,4,8}221.2
fm{5,8,0}221.2
Augmented TriadsC+{0,4,8}321
Parsimonious Voice Leading Between Common Triads of Scale 315. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m fm fm C+->fm G# G# C+->G# C# C# c#m->C# C#->fm

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 VerticesC+, c♯m, fm
Peripheral VerticesC♯, G♯

Modes

Modes are the rotational transformation of this scale. Scale 315 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2205
Scale 2205: Ionocrimic, Ian Ring Music TheoryIonocrimic
3rd mode:
Scale 1575
Scale 1575: Zycrimic, Ian Ring Music TheoryZycrimic
4th mode:
Scale 2835
Scale 2835: Ionygimic, Ian Ring Music TheoryIonygimic
5th mode:
Scale 3465
Scale 3465: Katathimic, Ian Ring Music TheoryKatathimic
6th mode:
Scale 945
Scale 945: Raga Saravati, Ian Ring Music TheoryRaga Saravati

Prime

This is the prime form of this scale.

Complement

The hexatonic modal family [315, 2205, 1575, 2835, 3465, 945] (Forte: 6-14) is the complement of the hexatonic modal family [315, 945, 1575, 2205, 2835, 3465] (Forte: 6-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 315 is 2961

Scale 2961Scale 2961: Bygimic, Ian Ring Music TheoryBygimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 315 is chiral, and its enantiomorph is scale 2961

Scale 2961Scale 2961: Bygimic, Ian Ring Music TheoryBygimic

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> 315       T0I <11,0> 2961
T1 <1,1> 630      T1I <11,1> 1827
T2 <1,2> 1260      T2I <11,2> 3654
T3 <1,3> 2520      T3I <11,3> 3213
T4 <1,4> 945      T4I <11,4> 2331
T5 <1,5> 1890      T5I <11,5> 567
T6 <1,6> 3780      T6I <11,6> 1134
T7 <1,7> 3465      T7I <11,7> 2268
T8 <1,8> 2835      T8I <11,8> 441
T9 <1,9> 1575      T9I <11,9> 882
T10 <1,10> 3150      T10I <11,10> 1764
T11 <1,11> 2205      T11I <11,11> 3528
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 315       T0MI <7,0> 2961
T1M <5,1> 630      T1MI <7,1> 1827
T2M <5,2> 1260      T2MI <7,2> 3654
T3M <5,3> 2520      T3MI <7,3> 3213
T4M <5,4> 945      T4MI <7,4> 2331
T5M <5,5> 1890      T5MI <7,5> 567
T6M <5,6> 3780      T6MI <7,6> 1134
T7M <5,7> 3465      T7MI <7,7> 2268
T8M <5,8> 2835      T8MI <7,8> 441
T9M <5,9> 1575      T9MI <7,9> 882
T10M <5,10> 3150      T10MI <7,10> 1764
T11M <5,11> 2205      T11MI <7,11> 3528

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 313Scale 313: Goritonic, Ian Ring Music TheoryGoritonic
Scale 317Scale 317: Korimic, Ian Ring Music TheoryKorimic
Scale 319Scale 319: Epodian, Ian Ring Music TheoryEpodian
Scale 307Scale 307: Raga Megharanjani, Ian Ring Music TheoryRaga Megharanjani
Scale 311Scale 311: Stagimic, Ian Ring Music TheoryStagimic
Scale 299Scale 299: Raga Chitthakarshini, Ian Ring Music TheoryRaga Chitthakarshini
Scale 283Scale 283: Aerylitonic, Ian Ring Music TheoryAerylitonic
Scale 347Scale 347: Barimic, Ian Ring Music TheoryBarimic
Scale 379Scale 379: Aeragian, Ian Ring Music TheoryAeragian
Scale 443Scale 443: Kothian, Ian Ring Music TheoryKothian
Scale 59Scale 59: Ahuian, Ian Ring Music TheoryAhuian
Scale 187Scale 187: Bedian, Ian Ring Music TheoryBedian
Scale 571Scale 571: Kynimic, Ian Ring Music TheoryKynimic
Scale 827Scale 827: Mixolocrian, Ian Ring Music TheoryMixolocrian
Scale 1339Scale 1339: Kycrian, Ian Ring Music TheoryKycrian
Scale 2363Scale 2363: Kataptian, Ian Ring Music TheoryKataptian

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.