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Scale 3341: "Vahian"

Scale 3341: Vahian, 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

Dozenal
Vahian

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

{0,2,3,8,10,11}

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-Z10

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: 1559

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.

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: 187

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, 1, 5, 2, 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> = {1,2,5}
<2> = {2,3,6,7}
<3> = {4,8}
<4> = {5,6,9,10}
<5> = {7,10,11}

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.

1.866

Polygon Perimeter

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

5.485

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, 7, 55)

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 TriadsG♯{8,0,3}121
Minor Triadsg♯m{8,11,3}210.67
Diminished Triadsg♯°{8,11,2}121

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

Parsimonious Voice Leading Between Common Triads of Scale 3341. Created by Ian Ring ©2019 g#° g#° g#m g#m g#°->g#m G# G# g#m->G#

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.

Diameter2
Radius1
Self-Centeredno
Central Verticesg♯m
Peripheral Verticesg♯°, G♯

Modes

Modes are the rotational transformation of this scale. Scale 3341 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1859
Scale 1859: Lixian, Ian Ring Music TheoryLixian
3rd mode:
Scale 2977
Scale 2977: Sobian, Ian Ring Music TheorySobian
4th mode:
Scale 221
Scale 221: Biyian, Ian Ring Music TheoryBiyian
5th mode:
Scale 1079
Scale 1079: Gowian, Ian Ring Music TheoryGowian
6th mode:
Scale 2587
Scale 2587: Putian, Ian Ring Music TheoryPutian

Prime

The prime form of this scale is Scale 187

Scale 187Scale 187: Bedian, Ian Ring Music TheoryBedian

Complement

The hexatonic modal family [3341, 1859, 2977, 221, 1079, 2587] (Forte: 6-Z10) is the complement of the hexatonic modal family [317, 977, 1103, 2599, 3347, 3721] (Forte: 6-Z39)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3341 is 1559

Scale 1559Scale 1559: Jowian, Ian Ring Music TheoryJowian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3341 is chiral, and its enantiomorph is scale 1559

Scale 1559Scale 1559: Jowian, Ian Ring Music TheoryJowian

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

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 3343Scale 3343: Vajian, Ian Ring Music TheoryVajian
Scale 3337Scale 3337: Vafian, Ian Ring Music TheoryVafian
Scale 3339Scale 3339: Smuian, Ian Ring Music TheorySmuian
Scale 3333Scale 3333: Vacian, Ian Ring Music TheoryVacian
Scale 3349Scale 3349: Aeolocrimic, Ian Ring Music TheoryAeolocrimic
Scale 3357Scale 3357: Phrodian, Ian Ring Music TheoryPhrodian
Scale 3373Scale 3373: Lodian, Ian Ring Music TheoryLodian
Scale 3405Scale 3405: Stynian, Ian Ring Music TheoryStynian
Scale 3469Scale 3469: Monian, Ian Ring Music TheoryMonian
Scale 3085Scale 3085: Tepian, Ian Ring Music TheoryTepian
Scale 3213Scale 3213: Eponimic, Ian Ring Music TheoryEponimic
Scale 3597Scale 3597: Wijian, Ian Ring Music TheoryWijian
Scale 3853Scale 3853: Yomian, Ian Ring Music TheoryYomian
Scale 2317Scale 2317: Odoian, Ian Ring Music TheoryOdoian
Scale 2829Scale 2829: Rupian, Ian Ring Music TheoryRupian
Scale 1293Scale 1293: Huxian, Ian Ring Music TheoryHuxian

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.