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Scale 3665: "Stalimic"

Scale 3665: Stalimic, 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
Stalimic
Dozenal
Wuyian

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

{0,4,6,9,10,11}

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-Z41

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: 335

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.

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: 335

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.

[4, 2, 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, 2, 2, 3, 2>

Interval Spectrum

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

p3m2n2s3d3t2

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

(20, 17, 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
Minor Triadsam{9,0,4}110.5
Diminished Triadsf♯°{6,9,0}110.5

The following pitch classes are not present in any of the common triads: {10,11}

Parsimonious Voice Leading Between Common Triads of Scale 3665. Created by Ian Ring ©2019 f#° f#° am am f#°->am

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3665 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 485
Scale 485: Stoptimic, Ian Ring Music TheoryStoptimic
3rd mode:
Scale 1145
Scale 1145: Zygimic, Ian Ring Music TheoryZygimic
4th mode:
Scale 655
Scale 655: Kataptimic, Ian Ring Music TheoryKataptimic
5th mode:
Scale 2375
Scale 2375: Aeolaptimic, Ian Ring Music TheoryAeolaptimic
6th mode:
Scale 3235
Scale 3235: Pothimic, Ian Ring Music TheoryPothimic

Prime

The prime form of this scale is Scale 335

Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic

Complement

The hexatonic modal family [3665, 485, 1145, 655, 2375, 3235] (Forte: 6-Z41) is the complement of the hexatonic modal family [215, 1475, 1805, 2155, 2785, 3125] (Forte: 6-Z12)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3665 is 335

Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3665 is chiral, and its enantiomorph is scale 335

Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic

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> 3665       T0I <11,0> 335
T1 <1,1> 3235      T1I <11,1> 670
T2 <1,2> 2375      T2I <11,2> 1340
T3 <1,3> 655      T3I <11,3> 2680
T4 <1,4> 1310      T4I <11,4> 1265
T5 <1,5> 2620      T5I <11,5> 2530
T6 <1,6> 1145      T6I <11,6> 965
T7 <1,7> 2290      T7I <11,7> 1930
T8 <1,8> 485      T8I <11,8> 3860
T9 <1,9> 970      T9I <11,9> 3625
T10 <1,10> 1940      T10I <11,10> 3155
T11 <1,11> 3880      T11I <11,11> 2215
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 965      T0MI <7,0> 1145
T1M <5,1> 1930      T1MI <7,1> 2290
T2M <5,2> 3860      T2MI <7,2> 485
T3M <5,3> 3625      T3MI <7,3> 970
T4M <5,4> 3155      T4MI <7,4> 1940
T5M <5,5> 2215      T5MI <7,5> 3880
T6M <5,6> 335      T6MI <7,6> 3665
T7M <5,7> 670      T7MI <7,7> 3235
T8M <5,8> 1340      T8MI <7,8> 2375
T9M <5,9> 2680      T9MI <7,9> 655
T10M <5,10> 1265      T10MI <7,10> 1310
T11M <5,11> 2530      T11MI <7,11> 2620

The transformations that map this set to itself are: T0, T6MI

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 3667Scale 3667: Kaptian, Ian Ring Music TheoryKaptian
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian
Scale 3673Scale 3673: Ranian, Ian Ring Music TheoryRanian
Scale 3649Scale 3649: Wupian, Ian Ring Music TheoryWupian
Scale 3657Scale 3657: Epynimic, Ian Ring Music TheoryEpynimic
Scale 3681Scale 3681: Xahian, Ian Ring Music TheoryXahian
Scale 3697Scale 3697: Ionarian, Ian Ring Music TheoryIonarian
Scale 3601Scale 3601: Wilian, Ian Ring Music TheoryWilian
Scale 3633Scale 3633: Daptimic, Ian Ring Music TheoryDaptimic
Scale 3729Scale 3729: Starimic, Ian Ring Music TheoryStarimic
Scale 3793Scale 3793: Aeopian, Ian Ring Music TheoryAeopian
Scale 3921Scale 3921: Pythian, Ian Ring Music TheoryPythian
Scale 3153Scale 3153: Zathitonic, Ian Ring Music TheoryZathitonic
Scale 3409Scale 3409: Katanimic, Ian Ring Music TheoryKatanimic
Scale 2641Scale 2641: Raga Hindol, Ian Ring Music TheoryRaga Hindol
Scale 1617Scale 1617: Phronitonic, Ian Ring Music TheoryPhronitonic

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.