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Scale 1907: "Lynyllic"

Scale 1907: Lynyllic, 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
Lynyllic
Dozenal
LOBian

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

 {0,1,4,5,6,8,9,10}

Forte Number

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

 8-19

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: 2525

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.

 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 includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

 8

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

 no
prime: 887

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, an indicator of maximum hierarchization.

 no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

 [1, 3, 1, 1, 2, 1, 1, 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, 4, 5, 7, 5, 2>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.333, 0, 0.25, 1, 0.333, 0>

Interval Spectrum

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

 p5m7n5s4d5t2

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

 1.75

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 Distribution Spectra have 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.

(10, 53, 129)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.763

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.342

Generator

This scale has no generator.

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}331.92
F{5,9,0}431.77
F♯{6,10,1}252.62
A{9,1,4}431.77
Minor Triadsc♯m{1,4,8}342.15
fm{5,8,0}342.15
f♯m{6,9,1}342.08
am{9,0,4}342
a♯m{10,1,5}342.08
Augmented TriadsC+{0,4,8}352.38
C♯+{1,5,9}531.54
Diminished Triadsf♯°{6,9,0}242.31
a♯°{10,1,4}242.31
Parsimonious Voice Leading Between Common Triads of Scale 1907. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m fm fm C+->fm am am C+->am C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->fm F F C#+->F f#m f#m C#+->f#m C#+->A a#m a#m C#+->a#m fm->F f#° f#° F->f#° F->am f#°->f#m F# F# f#m->F# F#->a#m am->A a#° a#° A->a#° a#°->a#m

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♯, C♯+, F, A
Peripheral VerticesC+, F♯

Modes

Modes are the rotational transformation of this scale. Scale 1907 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3001		Scale 3001: Lonyllic, Ian Ring Music TheoryLonyllic
3rd mode:
Scale 887		Scale 887: Sathyllic, Ian Ring Music TheorySathyllicThis is the prime mode
4th mode:
Scale 2491		Scale 2491: Layllic, Ian Ring Music TheoryLayllic
5th mode:
Scale 3293		Scale 3293: Saryllic, Ian Ring Music TheorySaryllic
6th mode:
Scale 1847		Scale 1847: Thacryllic, Ian Ring Music TheoryThacryllic
7th mode:
Scale 2971		Scale 2971: Aeolynyllic, Ian Ring Music TheoryAeolynyllic
8th mode:
Scale 3533		Scale 3533: Thadyllic, Ian Ring Music TheoryThadyllic

Prime

The prime form of this scale is Scale 887

Scale 887Scale 887: Sathyllic, Ian Ring Music TheorySathyllic

Complement

The octatonic modal family [1907, 3001, 887, 2491, 3293, 1847, 2971, 3533] (Forte: 8-19) is the complement of the tetratonic modal family [275, 305, 785, 2185] (Forte: 4-19)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1907 is 2525

Scale 2525Scale 2525: Aeolaryllic, Ian Ring Music TheoryAeolaryllic

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
111100111011101907k = 1h = 1
211100111011101907k = 2h = 1
33([1][1]0)01([1][1]0)1([1][1]0)1907k = 3h = 3
43110(0[1][1][1])(0[1][1][1])01907k = 4h = 3
53110(0[1][1][1])(0[1][1][1])01907k = 5h = 3

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1907 is chiral, and its enantiomorph is scale 2525

Scale 2525Scale 2525: Aeolaryllic, Ian Ring Music TheoryAeolaryllic

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 1907       T0I <11,0> 2525
T1 <1,1> 3814      T1I <11,1> 955
T2 <1,2> 3533      T2I <11,2> 1910
T3 <1,3> 2971      T3I <11,3> 3820
T4 <1,4> 1847      T4I <11,4> 3545
T5 <1,5> 3694      T5I <11,5> 2995
T6 <1,6> 3293      T6I <11,6> 1895
T7 <1,7> 2491      T7I <11,7> 3790
T8 <1,8> 887      T8I <11,8> 3485
T9 <1,9> 1774      T9I <11,9> 2875
T10 <1,10> 3548      T10I <11,10> 1655
T11 <1,11> 3001      T11I <11,11> 3310
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 887      T0MI <7,0> 3545
T1M <5,1> 1774      T1MI <7,1> 2995
T2M <5,2> 3548      T2MI <7,2> 1895
T3M <5,3> 3001      T3MI <7,3> 3790
T4M <5,4> 1907       T4MI <7,4> 3485
T5M <5,5> 3814      T5MI <7,5> 2875
T6M <5,6> 3533      T6MI <7,6> 1655
T7M <5,7> 2971      T7MI <7,7> 3310
T8M <5,8> 1847      T8MI <7,8> 2525
T9M <5,9> 3694      T9MI <7,9> 955
T10M <5,10> 3293      T10MI <7,10> 1910
T11M <5,11> 2491      T11MI <7,11> 3820

The transformations that map this set to itself are: T0, T4M

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 1905Scale 1905: Katacrian, Ian Ring Music TheoryKatacrian
Scale 1909Scale 1909: Epicryllic, Ian Ring Music TheoryEpicryllic
Scale 1911Scale 1911: Messiaen Mode 3 Rotation 1, Ian Ring Music TheoryMessiaen Mode 3 Rotation 1
Scale 1915Scale 1915: Thydygic, Ian Ring Music TheoryThydygic
Scale 1891Scale 1891: Thalian, Ian Ring Music TheoryThalian
Scale 1899Scale 1899: Moptyllic, Ian Ring Music TheoryMoptyllic
Scale 1875Scale 1875: Persichetti Scale, Ian Ring Music TheoryPersichetti Scale
Scale 1843Scale 1843: Ionygian, Ian Ring Music TheoryIonygian
Scale 1971Scale 1971: Aerynyllic, Ian Ring Music TheoryAerynyllic
Scale 2035Scale 2035: Aerythygic, Ian Ring Music TheoryAerythygic
Scale 1651Scale 1651: Asian, Ian Ring Music TheoryAsian
Scale 1779Scale 1779: Aerythyllic, Ian Ring Music TheoryAerythyllic
Scale 1395Scale 1395: Locrian Dominant, Ian Ring Music TheoryLocrian Dominant
Scale 883Scale 883: Ralian, Ian Ring Music TheoryRalian
Scale 2931Scale 2931: Zathyllic, Ian Ring Music TheoryZathyllic
Scale 3955Scale 3955: Galygic, Ian Ring Music TheoryGalygic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.