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Scale 3989: "Sythyllic"

Scale 3989: Sythyllic, 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
Sythyllic
Dozenal
Kodian

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

{0,2,4,7,8,9,10,11}

Forte Number

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

8-11

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: 1343

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.

[2, 2, 3, 1, 1, 1, 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, 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{0,4,7}231.78
E{4,8,11}331.56
G{7,11,2}341.78
Minor Triadsem{4,7,11}431.44
gm{7,10,2}252.33
am{9,0,4}152.67
Augmented TriadsC+{0,4,8}341.89
Diminished Triads{4,7,10}242
g♯°{8,11,2}231.89
Parsimonious Voice Leading Between Common Triads of Scale 3989. Created by Ian Ring ©2019 C C C+ C+ C->C+ em em C->em E E C+->E am am C+->am e°->em gm gm e°->gm em->E Parsimonious Voice Leading Between Common Triads of Scale 3989. Created by Ian Ring ©2019 G em->G g#° g#° E->g#° gm->G G->g#°

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, em, E, g♯°
Peripheral Verticesgm, am

Modes

Modes are the rotational transformation of this scale. Scale 3989 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2021
Scale 2021: Katycryllic, Ian Ring Music TheoryKatycryllic
3rd mode:
Scale 1529
Scale 1529: Kataryllic, Ian Ring Music TheoryKataryllic
4th mode:
Scale 703
Scale 703: Aerocryllic, Ian Ring Music TheoryAerocryllicThis is the prime mode
5th mode:
Scale 2399
Scale 2399: Zanyllic, Ian Ring Music TheoryZanyllic
6th mode:
Scale 3247
Scale 3247: Aeolonyllic, Ian Ring Music TheoryAeolonyllic
7th mode:
Scale 3671
Scale 3671: Aeonyllic, Ian Ring Music TheoryAeonyllic
8th mode:
Scale 3883
Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic

Prime

The prime form of this scale is Scale 703

Scale 703Scale 703: Aerocryllic, Ian Ring Music TheoryAerocryllic

Complement

The octatonic modal family [3989, 2021, 1529, 703, 2399, 3247, 3671, 3883] (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 3989 is 1343

Scale 1343Scale 1343: Zalyllic, Ian Ring Music TheoryZalyllic

Enantiomorph

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

Scale 1343Scale 1343: Zalyllic, Ian Ring Music TheoryZalyllic

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

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 3991Scale 3991: Badygic, Ian Ring Music TheoryBadygic
Scale 3985Scale 3985: Thadian, Ian Ring Music TheoryThadian
Scale 3987Scale 3987: Loryllic, Ian Ring Music TheoryLoryllic
Scale 3993Scale 3993: Ioniptyllic, Ian Ring Music TheoryIoniptyllic
Scale 3997Scale 3997: Dogygic, Ian Ring Music TheoryDogygic
Scale 3973Scale 3973: Zehian, Ian Ring Music TheoryZehian
Scale 3981Scale 3981: Phrycryllic, Ian Ring Music TheoryPhrycryllic
Scale 4005Scale 4005: Zibian, Ian Ring Music TheoryZibian
Scale 4021Scale 4021: Raga Pahadi, Ian Ring Music TheoryRaga Pahadi
Scale 4053Scale 4053: Kyrygic, Ian Ring Music TheoryKyrygic
Scale 3861Scale 3861: Phroptian, Ian Ring Music TheoryPhroptian
Scale 3925Scale 3925: Thyryllic, Ian Ring Music TheoryThyryllic
Scale 3733Scale 3733: Gycrian, Ian Ring Music TheoryGycrian
Scale 3477Scale 3477: Kyptian, Ian Ring Music TheoryKyptian
Scale 2965Scale 2965: Darian, Ian Ring Music TheoryDarian
Scale 1941Scale 1941: Aeranian, Ian Ring Music TheoryAeranian

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.