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Scale 937: "Stothimic"

Scale 937: Stothimic, 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
Stothimic
Dozenal
Spuian

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

{0,3,5,7,8,9}

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-Z46

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: 697

Hemitonia

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

2 (dihemitonic)

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.

no
prime: 599

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, 2, 2, 1, 1, 3]

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.

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

Interval Spectrum

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

p3m3n3s3d2t

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

Spectra Variation

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

2.667

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.

2.366

Polygon Perimeter

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

5.864

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.

(12, 17, 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 TriadsF{5,9,0}231.4
G♯{8,0,3}321
Minor Triadscm{0,3,7}131.6
fm{5,8,0}221.2
Diminished Triads{9,0,3}221.2
Parsimonious Voice Leading Between Common Triads of Scale 937. Created by Ian Ring ©2019 cm cm G# G# cm->G# fm fm F F fm->F fm->G# F->a° G#->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
Radius2
Self-Centeredno
Central Verticesfm, G♯, a°
Peripheral Verticescm, F

Modes

Modes are the rotational transformation of this scale. Scale 937 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 629
Scale 629: Aeronimic, Ian Ring Music TheoryAeronimic
3rd mode:
Scale 1181
Scale 1181: Katagimic, Ian Ring Music TheoryKatagimic
4th mode:
Scale 1319
Scale 1319: Phronimic, Ian Ring Music TheoryPhronimic
5th mode:
Scale 2707
Scale 2707: Banimic, Ian Ring Music TheoryBanimic
6th mode:
Scale 3401
Scale 3401: Palimic, Ian Ring Music TheoryPalimic

Prime

The prime form of this scale is Scale 599

Scale 599Scale 599: Thyrimic, Ian Ring Music TheoryThyrimic

Complement

The hexatonic modal family [937, 629, 1181, 1319, 2707, 3401] (Forte: 6-Z46) is the complement of the hexatonic modal family [347, 1457, 1579, 1733, 2221, 2837] (Forte: 6-Z24)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 937 is 697

Scale 697Scale 697: Lagimic, Ian Ring Music TheoryLagimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 937 is chiral, and its enantiomorph is scale 697

Scale 697Scale 697: Lagimic, Ian Ring Music TheoryLagimic

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> 937       T0I <11,0> 697
T1 <1,1> 1874      T1I <11,1> 1394
T2 <1,2> 3748      T2I <11,2> 2788
T3 <1,3> 3401      T3I <11,3> 1481
T4 <1,4> 2707      T4I <11,4> 2962
T5 <1,5> 1319      T5I <11,5> 1829
T6 <1,6> 2638      T6I <11,6> 3658
T7 <1,7> 1181      T7I <11,7> 3221
T8 <1,8> 2362      T8I <11,8> 2347
T9 <1,9> 629      T9I <11,9> 599
T10 <1,10> 1258      T10I <11,10> 1198
T11 <1,11> 2516      T11I <11,11> 2396
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2587      T0MI <7,0> 2827
T1M <5,1> 1079      T1MI <7,1> 1559
T2M <5,2> 2158      T2MI <7,2> 3118
T3M <5,3> 221      T3MI <7,3> 2141
T4M <5,4> 442      T4MI <7,4> 187
T5M <5,5> 884      T5MI <7,5> 374
T6M <5,6> 1768      T6MI <7,6> 748
T7M <5,7> 3536      T7MI <7,7> 1496
T8M <5,8> 2977      T8MI <7,8> 2992
T9M <5,9> 1859      T9MI <7,9> 1889
T10M <5,10> 3718      T10MI <7,10> 3778
T11M <5,11> 3341      T11MI <7,11> 3461

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 939Scale 939: Mela Senavati, Ian Ring Music TheoryMela Senavati
Scale 941Scale 941: Mela Jhankaradhvani, Ian Ring Music TheoryMela Jhankaradhvani
Scale 929Scale 929: Fujian, Ian Ring Music TheoryFujian
Scale 933Scale 933: Dadimic, Ian Ring Music TheoryDadimic
Scale 945Scale 945: Raga Saravati, Ian Ring Music TheoryRaga Saravati
Scale 953Scale 953: Mela Yagapriya, Ian Ring Music TheoryMela Yagapriya
Scale 905Scale 905: Bylitonic, Ian Ring Music TheoryBylitonic
Scale 921Scale 921: Bogimic, Ian Ring Music TheoryBogimic
Scale 969Scale 969: Ionogimic, Ian Ring Music TheoryIonogimic
Scale 1001Scale 1001: Badian, Ian Ring Music TheoryBadian
Scale 809Scale 809: Dogitonic, Ian Ring Music TheoryDogitonic
Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic
Scale 681Scale 681: Kyemyonjo, Ian Ring Music TheoryKyemyonjo
Scale 425Scale 425: Raga Kokil Pancham, Ian Ring Music TheoryRaga Kokil Pancham
Scale 1449Scale 1449: Raga Gopikavasantam, Ian Ring Music TheoryRaga Gopikavasantam
Scale 1961Scale 1961: Soptian, Ian Ring Music TheorySoptian
Scale 2985Scale 2985: Epacrian, Ian Ring Music TheoryEpacrian

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.