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Scale 3729: "Starimic"

Scale 3729: Starimic, 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
Starimic
Dozenal
Xolian

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

{0,4,7,9,10,11}

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-Z40

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: 303

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: 303

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

Interval Spectrum

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

p3m2n3s3d3t

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> = {2,3,5,7}
<3> = {3,4,6,8,9}
<4> = {5,7,9,10}
<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, 16, 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 TriadsC{0,4,7}221
Minor Triadsem{4,7,11}221
am{9,0,4}131.5
Diminished Triads{4,7,10}131.5
Parsimonious Voice Leading Between Common Triads of Scale 3729. Created by Ian Ring ©2019 C C em em C->em am am C->am e°->em

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, em
Peripheral Verticese°, am

Modes

Modes are the rotational transformation of this scale. Scale 3729 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 489
Scale 489: Phrathimic, Ian Ring Music TheoryPhrathimic
3rd mode:
Scale 573
Scale 573: Saptimic, Ian Ring Music TheorySaptimic
4th mode:
Scale 1167
Scale 1167: Aerodimic, Ian Ring Music TheoryAerodimic
5th mode:
Scale 2631
Scale 2631: Macrimic, Ian Ring Music TheoryMacrimic
6th mode:
Scale 3363
Scale 3363: Rogimic, Ian Ring Music TheoryRogimic

Prime

The prime form of this scale is Scale 303

Scale 303Scale 303: Golimic, Ian Ring Music TheoryGolimic

Complement

The hexatonic modal family [3729, 489, 573, 1167, 2631, 3363] (Forte: 6-Z40) is the complement of the hexatonic modal family [183, 1761, 1803, 2139, 2949, 3117] (Forte: 6-Z11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3729 is 303

Scale 303Scale 303: Golimic, Ian Ring Music TheoryGolimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3729 is chiral, and its enantiomorph is scale 303

Scale 303Scale 303: Golimic, Ian Ring Music TheoryGolimic

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> 3729       T0I <11,0> 303
T1 <1,1> 3363      T1I <11,1> 606
T2 <1,2> 2631      T2I <11,2> 1212
T3 <1,3> 1167      T3I <11,3> 2424
T4 <1,4> 2334      T4I <11,4> 753
T5 <1,5> 573      T5I <11,5> 1506
T6 <1,6> 1146      T6I <11,6> 3012
T7 <1,7> 2292      T7I <11,7> 1929
T8 <1,8> 489      T8I <11,8> 3858
T9 <1,9> 978      T9I <11,9> 3621
T10 <1,10> 1956      T10I <11,10> 3147
T11 <1,11> 3912      T11I <11,11> 2199
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2949      T0MI <7,0> 1083
T1M <5,1> 1803      T1MI <7,1> 2166
T2M <5,2> 3606      T2MI <7,2> 237
T3M <5,3> 3117      T3MI <7,3> 474
T4M <5,4> 2139      T4MI <7,4> 948
T5M <5,5> 183      T5MI <7,5> 1896
T6M <5,6> 366      T6MI <7,6> 3792
T7M <5,7> 732      T7MI <7,7> 3489
T8M <5,8> 1464      T8MI <7,8> 2883
T9M <5,9> 2928      T9MI <7,9> 1671
T10M <5,10> 1761      T10MI <7,10> 3342
T11M <5,11> 3522      T11MI <7,11> 2589

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 3731Scale 3731: Aeryrian, Ian Ring Music TheoryAeryrian
Scale 3733Scale 3733: Gycrian, Ian Ring Music TheoryGycrian
Scale 3737Scale 3737: Phrocrian, Ian Ring Music TheoryPhrocrian
Scale 3713Scale 3713: Xibian, Ian Ring Music TheoryXibian
Scale 3721Scale 3721: Phragimic, Ian Ring Music TheoryPhragimic
Scale 3745Scale 3745: Xuvian, Ian Ring Music TheoryXuvian
Scale 3761Scale 3761: Raga Madhuri, Ian Ring Music TheoryRaga Madhuri
Scale 3793Scale 3793: Aeopian, Ian Ring Music TheoryAeopian
Scale 3601Scale 3601: Wilian, Ian Ring Music TheoryWilian
Scale 3665Scale 3665: Stalimic, Ian Ring Music TheoryStalimic
Scale 3857Scale 3857: Ponimic, Ian Ring Music TheoryPonimic
Scale 3985Scale 3985: Thadian, Ian Ring Music TheoryThadian
Scale 3217Scale 3217: Molitonic, Ian Ring Music TheoryMolitonic
Scale 3473Scale 3473: Lathimic, Ian Ring Music TheoryLathimic
Scale 2705Scale 2705: Raga Mamata, Ian Ring Music TheoryRaga Mamata
Scale 1681Scale 1681: Raga Valaji, Ian Ring Music TheoryRaga Valaji

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.