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Scale 3293: "Saryllic"

Scale 3293: Saryllic, 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
Saryllic
Dozenal
Upuian

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

{0,2,3,4,6,7,10,11}

Forte Number

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

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: 1895

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 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: 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.

no

Interval Structure

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

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

<5, 4, 5, 7, 5, 2>

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 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.

(10, 53, 129)

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}252.62
D♯{3,7,10}431.77
G{7,11,2}331.92
B{11,3,6}431.77
Minor Triadscm{0,3,7}342.08
d♯m{3,6,10}342
em{4,7,11}342.08
gm{7,10,2}342.15
bm{11,2,6}342.15
Augmented TriadsD+{2,6,10}352.38
D♯+{3,7,11}531.54
Diminished Triads{0,3,6}242.31
{4,7,10}242.31
Parsimonious Voice Leading Between Common Triads of Scale 3293. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B C C cm->C D#+ D#+ cm->D#+ em em C->em D+ D+ d#m d#m D+->d#m gm gm D+->gm bm bm D+->bm D# D# d#m->D# d#m->B D#->D#+ D#->e° D#->gm D#+->em Parsimonious Voice Leading Between Common Triads of Scale 3293. Created by Ian Ring ©2019 G D#+->G D#+->B e°->em gm->G G->bm bm->B

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 VerticesD♯, D♯+, G, B
Peripheral VerticesC, D+

Modes

Modes are the rotational transformation of this scale. Scale 3293 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1847
Scale 1847: Thacryllic, Ian Ring Music TheoryThacryllic
3rd mode:
Scale 2971
Scale 2971: Aeolynyllic, Ian Ring Music TheoryAeolynyllic
4th mode:
Scale 3533
Scale 3533: Thadyllic, Ian Ring Music TheoryThadyllic
5th mode:
Scale 1907
Scale 1907: Lynyllic, Ian Ring Music TheoryLynyllic
6th mode:
Scale 3001
Scale 3001: Lonyllic, Ian Ring Music TheoryLonyllic
7th mode:
Scale 887
Scale 887: Sathyllic, Ian Ring Music TheorySathyllicThis is the prime mode
8th mode:
Scale 2491
Scale 2491: Layllic, Ian Ring Music TheoryLayllic

Prime

The prime form of this scale is Scale 887

Scale 887Scale 887: Sathyllic, Ian Ring Music TheorySathyllic

Complement

The octatonic modal family [3293, 1847, 2971, 3533, 1907, 3001, 887, 2491] (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 3293 is 1895

Scale 1895Scale 1895: Salyllic, Ian Ring Music TheorySalyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3293 is chiral, and its enantiomorph is scale 1895

Scale 1895Scale 1895: Salyllic, Ian Ring Music TheorySalyllic

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

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 3295Scale 3295: Phroptygic, Ian Ring Music TheoryPhroptygic
Scale 3289Scale 3289: Lydian Sharp 2 Sharp 6, Ian Ring Music TheoryLydian Sharp 2 Sharp 6
Scale 3291Scale 3291: Lygyllic, Ian Ring Music TheoryLygyllic
Scale 3285Scale 3285: Mela Citrambari, Ian Ring Music TheoryMela Citrambari
Scale 3277Scale 3277: Mela Nitimati, Ian Ring Music TheoryMela Nitimati
Scale 3309Scale 3309: Bycryllic, Ian Ring Music TheoryBycryllic
Scale 3325Scale 3325: Mixolygic, Ian Ring Music TheoryMixolygic
Scale 3229Scale 3229: Aeolaptian, Ian Ring Music TheoryAeolaptian
Scale 3261Scale 3261: Dodyllic, Ian Ring Music TheoryDodyllic
Scale 3165Scale 3165: Mylian, Ian Ring Music TheoryMylian
Scale 3421Scale 3421: Aerothyllic, Ian Ring Music TheoryAerothyllic
Scale 3549Scale 3549: Messiaen Mode 3 Inverse, Ian Ring Music TheoryMessiaen Mode 3 Inverse
Scale 3805Scale 3805: Moptygic, Ian Ring Music TheoryMoptygic
Scale 2269Scale 2269: Pygian, Ian Ring Music TheoryPygian
Scale 2781Scale 2781: Gycryllic, Ian Ring Music TheoryGycryllic
Scale 1245Scale 1245: Lathian, Ian Ring Music TheoryLathian

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.