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Scale 2663: "Lalian"

Scale 2663: Lalian, 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
Lalian
Dozenal
Qonian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

7 (heptatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,2,5,6,9,11}

Forte Number

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

7-Z18

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: 3275

Hemitonia

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

4 (multihemitonic)

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.

6

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 755

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, 1, 3, 1, 3, 2, 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.

<4, 3, 4, 4, 4, 2>

Interval Spectrum

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

p4m4n4s3d4t2

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

Spectra Variation

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

2.571

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.

2.433

Polygon Perimeter

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

5.899

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.

(23, 34, 100)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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 TriadsD{2,6,9}331.5
F{5,9,0}242
Minor Triadsdm{2,5,9}331.5
f♯m{6,9,1}331.5
bm{11,2,6}242
Augmented TriadsC♯+{1,5,9}331.5
Diminished Triadsf♯°{6,9,0}242
{11,2,5}242
Parsimonious Voice Leading Between Common Triads of Scale 2663. Created by Ian Ring ©2019 C#+ C#+ dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m D D dm->D dm->b° D->f#m bm bm D->bm f#° f#° F->f#° f#°->f#m b°->bm

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.

Diameter4
Radius3
Self-Centeredno
Central VerticesC♯+, dm, D, f♯m
Peripheral VerticesF, f♯°, b°, bm

Modes

Modes are the rotational transformation of this scale. Scale 2663 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3379
Scale 3379: Verdi's Scala Enigmatica Descending, Ian Ring Music TheoryVerdi's Scala Enigmatica Descending
3rd mode:
Scale 3737
Scale 3737: Phrocrian, Ian Ring Music TheoryPhrocrian
4th mode:
Scale 979
Scale 979: Mela Dhavalambari, Ian Ring Music TheoryMela Dhavalambari
5th mode:
Scale 2537
Scale 2537: Laptian, Ian Ring Music TheoryLaptian
6th mode:
Scale 829
Scale 829: Lygian, Ian Ring Music TheoryLygian
7th mode:
Scale 1231
Scale 1231: Logian, Ian Ring Music TheoryLogian

Prime

The prime form of this scale is Scale 755

Scale 755Scale 755: Phrythian, Ian Ring Music TheoryPhrythian

Complement

The heptatonic modal family [2663, 3379, 3737, 979, 2537, 829, 1231] (Forte: 7-Z18) is the complement of the pentatonic modal family [179, 779, 1633, 2137, 2437] (Forte: 5-Z18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2663 is 3275

Scale 3275Scale 3275: Mela Divyamani, Ian Ring Music TheoryMela Divyamani

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2663 is chiral, and its enantiomorph is scale 3275

Scale 3275Scale 3275: Mela Divyamani, Ian Ring Music TheoryMela Divyamani

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> 2663       T0I <11,0> 3275
T1 <1,1> 1231      T1I <11,1> 2455
T2 <1,2> 2462      T2I <11,2> 815
T3 <1,3> 829      T3I <11,3> 1630
T4 <1,4> 1658      T4I <11,4> 3260
T5 <1,5> 3316      T5I <11,5> 2425
T6 <1,6> 2537      T6I <11,6> 755
T7 <1,7> 979      T7I <11,7> 1510
T8 <1,8> 1958      T8I <11,8> 3020
T9 <1,9> 3916      T9I <11,9> 1945
T10 <1,10> 3737      T10I <11,10> 3890
T11 <1,11> 3379      T11I <11,11> 3685
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1763      T0MI <7,0> 2285
T1M <5,1> 3526      T1MI <7,1> 475
T2M <5,2> 2957      T2MI <7,2> 950
T3M <5,3> 1819      T3MI <7,3> 1900
T4M <5,4> 3638      T4MI <7,4> 3800
T5M <5,5> 3181      T5MI <7,5> 3505
T6M <5,6> 2267      T6MI <7,6> 2915
T7M <5,7> 439      T7MI <7,7> 1735
T8M <5,8> 878      T8MI <7,8> 3470
T9M <5,9> 1756      T9MI <7,9> 2845
T10M <5,10> 3512      T10MI <7,10> 1595
T11M <5,11> 2929      T11MI <7,11> 3190

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 2661Scale 2661: Stydimic, Ian Ring Music TheoryStydimic
Scale 2659Scale 2659: Katynimic, Ian Ring Music TheoryKatynimic
Scale 2667Scale 2667: Byrian, Ian Ring Music TheoryByrian
Scale 2671Scale 2671: Aerolyllic, Ian Ring Music TheoryAerolyllic
Scale 2679Scale 2679: Rathyllic, Ian Ring Music TheoryRathyllic
Scale 2631Scale 2631: Macrimic, Ian Ring Music TheoryMacrimic
Scale 2647Scale 2647: Dadian, Ian Ring Music TheoryDadian
Scale 2599Scale 2599: Malimic, Ian Ring Music TheoryMalimic
Scale 2727Scale 2727: Mela Manavati, Ian Ring Music TheoryMela Manavati
Scale 2791Scale 2791: Mixothyllic, Ian Ring Music TheoryMixothyllic
Scale 2919Scale 2919: Molyllic, Ian Ring Music TheoryMolyllic
Scale 2151Scale 2151: Natian, Ian Ring Music TheoryNatian
Scale 2407Scale 2407: Zylian, Ian Ring Music TheoryZylian
Scale 3175Scale 3175: Eponian, Ian Ring Music TheoryEponian
Scale 3687Scale 3687: Zonyllic, Ian Ring Music TheoryZonyllic
Scale 615Scale 615: Schoenberg Hexachord, Ian Ring Music TheorySchoenberg Hexachord
Scale 1639Scale 1639: Aeolothian, Ian Ring Music TheoryAeolothian

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.