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Scale 1343: "Zalyllic"

Scale 1343: Zalyllic, 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
Zalyllic
Dozenal
Irtian

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,1,2,3,4,5,8,10}

Forte Number

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

8-11

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: 3989

Hemitonia

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

5 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

4 (multicohemitonic)

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.

7

Prime Form

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

no
prime: 703

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, 1, 1, 1, 3, 2, 2]

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.

<5, 6, 5, 5, 5, 2>

Interval Spectrum

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

p5m5n5s6d5t2

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

Spectra Variation

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

2.75

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.

2.616

Polygon Perimeter

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

6.002

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.

(59, 69, 149)

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♯{1,5,8}431.44
G♯{8,0,3}152.67
A♯{10,2,5}252.33
Minor Triadsc♯m{1,4,8}331.56
fm{5,8,0}231.78
a♯m{10,1,5}341.78
Augmented TriadsC+{0,4,8}341.89
Diminished Triads{2,5,8}242
a♯°{10,1,4}231.89
Parsimonious Voice Leading Between Common Triads of Scale 1343. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m fm fm C+->fm G# G# C+->G# C# C# c#m->C# a#° a#° c#m->a#° C#->d° C#->fm a#m a#m C#->a#m A# A# d°->A# a#°->a#m a#m->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.

Diameter5
Radius3
Self-Centeredno
Central Verticesc♯m, C♯, fm, a♯°
Peripheral VerticesG♯, A♯

Modes

Modes are the rotational transformation of this scale. Scale 1343 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2719
Scale 2719: Zocryllic, Ian Ring Music TheoryZocryllic
3rd mode:
Scale 3407
Scale 3407: Katocryllic, Ian Ring Music TheoryKatocryllic
4th mode:
Scale 3751
Scale 3751: Aerathyllic, Ian Ring Music TheoryAerathyllic
5th mode:
Scale 3923
Scale 3923: Stoptyllic, Ian Ring Music TheoryStoptyllic
6th mode:
Scale 4009
Scale 4009: Phranyllic, Ian Ring Music TheoryPhranyllic
7th mode:
Scale 1013
Scale 1013: Stydyllic, Ian Ring Music TheoryStydyllic
8th mode:
Scale 1277
Scale 1277: Zadyllic, Ian Ring Music TheoryZadyllic

Prime

The prime form of this scale is Scale 703

Scale 703Scale 703: Aerocryllic, Ian Ring Music TheoryAerocryllic

Complement

The octatonic modal family [1343, 2719, 3407, 3751, 3923, 4009, 1013, 1277] (Forte: 8-11) is the complement of the tetratonic modal family [43, 1409, 1541, 2069] (Forte: 4-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1343 is 3989

Scale 3989Scale 3989: Sythyllic, Ian Ring Music TheorySythyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1343 is chiral, and its enantiomorph is scale 3989

Scale 3989Scale 3989: Sythyllic, Ian Ring Music TheorySythyllic

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> 1343       T0I <11,0> 3989
T1 <1,1> 2686      T1I <11,1> 3883
T2 <1,2> 1277      T2I <11,2> 3671
T3 <1,3> 2554      T3I <11,3> 3247
T4 <1,4> 1013      T4I <11,4> 2399
T5 <1,5> 2026      T5I <11,5> 703
T6 <1,6> 4052      T6I <11,6> 1406
T7 <1,7> 4009      T7I <11,7> 2812
T8 <1,8> 3923      T8I <11,8> 1529
T9 <1,9> 3751      T9I <11,9> 3058
T10 <1,10> 3407      T10I <11,10> 2021
T11 <1,11> 2719      T11I <11,11> 4042
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1343       T0MI <7,0> 3989
T1M <5,1> 2686      T1MI <7,1> 3883
T2M <5,2> 1277      T2MI <7,2> 3671
T3M <5,3> 2554      T3MI <7,3> 3247
T4M <5,4> 1013      T4MI <7,4> 2399
T5M <5,5> 2026      T5MI <7,5> 703
T6M <5,6> 4052      T6MI <7,6> 1406
T7M <5,7> 4009      T7MI <7,7> 2812
T8M <5,8> 3923      T8MI <7,8> 1529
T9M <5,9> 3751      T9MI <7,9> 3058
T10M <5,10> 3407      T10MI <7,10> 2021
T11M <5,11> 2719      T11MI <7,11> 4042

The transformations that map this set to itself are: T0, T0M

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 1341Scale 1341: Madian, Ian Ring Music TheoryMadian
Scale 1339Scale 1339: Kycrian, Ian Ring Music TheoryKycrian
Scale 1335Scale 1335: Elephant Scale, Ian Ring Music TheoryElephant Scale
Scale 1327Scale 1327: Zalian, Ian Ring Music TheoryZalian
Scale 1311Scale 1311: Bynian, Ian Ring Music TheoryBynian
Scale 1375Scale 1375: Bothyllic, Ian Ring Music TheoryBothyllic
Scale 1407Scale 1407: Tharygic, Ian Ring Music TheoryTharygic
Scale 1471Scale 1471: Radygic, Ian Ring Music TheoryRadygic
Scale 1087Scale 1087: Gobian, Ian Ring Music TheoryGobian
Scale 1215Scale 1215: Hibian, Ian Ring Music TheoryHibian
Scale 1599Scale 1599: Pocryllic, Ian Ring Music TheoryPocryllic
Scale 1855Scale 1855: Gaptygic, Ian Ring Music TheoryGaptygic
Scale 319Scale 319: Epodian, Ian Ring Music TheoryEpodian
Scale 831Scale 831: Rodyllic, Ian Ring Music TheoryRodyllic
Scale 2367Scale 2367: Laryllic, Ian Ring Music TheoryLaryllic
Scale 3391Scale 3391: Aeolynygic, Ian Ring Music TheoryAeolynygic

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.