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Scale 3817: "Zoryllic"

Scale 3817: Zoryllic, 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
Zoryllic
Dozenal
Yeqian

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

{0,3,5,6,7,9,10,11}

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-Z29

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: 751

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.

3 (tricohemitonic)

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: 751

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.

[3, 2, 1, 1, 2, 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, 5, 5, 5, 5, 3>

Interval Spectrum

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

p5m5n5s5d5t3

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

2.25

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.

(30, 60, 141)

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♯{3,7,10}242.1
F{5,9,0}242.3
B{11,3,6}341.9
Minor Triadscm{0,3,7}341.9
d♯m{3,6,10}341.9
Augmented TriadsD♯+{3,7,11}341.9
Diminished Triads{0,3,6}242.1
d♯°{3,6,9}242.1
f♯°{6,9,0}242.3
{9,0,3}242.1
Parsimonious Voice Leading Between Common Triads of Scale 3817. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ cm->a° d#° d#° d#m d#m d#°->d#m f#° f#° d#°->f#° D# D# d#m->D# d#m->B D#->D#+ D#+->B F F F->f#° F->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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3817 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 989
Scale 989: Phrolyllic, Ian Ring Music TheoryPhrolyllic
3rd mode:
Scale 1271
Scale 1271: Kolyllic, Ian Ring Music TheoryKolyllic
4th mode:
Scale 2683
Scale 2683: Thodyllic, Ian Ring Music TheoryThodyllic
5th mode:
Scale 3389
Scale 3389: Socryllic, Ian Ring Music TheorySocryllic
6th mode:
Scale 1871
Scale 1871: Aeolyllic, Ian Ring Music TheoryAeolyllic
7th mode:
Scale 2983
Scale 2983: Zythyllic, Ian Ring Music TheoryZythyllic
8th mode:
Scale 3539
Scale 3539: Aeoryllic, Ian Ring Music TheoryAeoryllic

Prime

The prime form of this scale is Scale 751

Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian

Complement

The octatonic modal family [3817, 989, 1271, 2683, 3389, 1871, 2983, 3539] (Forte: 8-Z29) is the complement of the tetratonic modal family [139, 353, 1553, 2117] (Forte: 4-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3817 is 751

Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3817 is chiral, and its enantiomorph is scale 751

Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian

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> 3817       T0I <11,0> 751
T1 <1,1> 3539      T1I <11,1> 1502
T2 <1,2> 2983      T2I <11,2> 3004
T3 <1,3> 1871      T3I <11,3> 1913
T4 <1,4> 3742      T4I <11,4> 3826
T5 <1,5> 3389      T5I <11,5> 3557
T6 <1,6> 2683      T6I <11,6> 3019
T7 <1,7> 1271      T7I <11,7> 1943
T8 <1,8> 2542      T8I <11,8> 3886
T9 <1,9> 989      T9I <11,9> 3677
T10 <1,10> 1978      T10I <11,10> 3259
T11 <1,11> 3956      T11I <11,11> 2423
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2767      T0MI <7,0> 3691
T1M <5,1> 1439      T1MI <7,1> 3287
T2M <5,2> 2878      T2MI <7,2> 2479
T3M <5,3> 1661      T3MI <7,3> 863
T4M <5,4> 3322      T4MI <7,4> 1726
T5M <5,5> 2549      T5MI <7,5> 3452
T6M <5,6> 1003      T6MI <7,6> 2809
T7M <5,7> 2006      T7MI <7,7> 1523
T8M <5,8> 4012      T8MI <7,8> 3046
T9M <5,9> 3929      T9MI <7,9> 1997
T10M <5,10> 3763      T10MI <7,10> 3994
T11M <5,11> 3431      T11MI <7,11> 3893

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 3819Scale 3819: Aeolanygic, Ian Ring Music TheoryAeolanygic
Scale 3821Scale 3821: Epyrygic, Ian Ring Music TheoryEpyrygic
Scale 3809Scale 3809: Yelian, Ian Ring Music TheoryYelian
Scale 3813Scale 3813: Aeologyllic, Ian Ring Music TheoryAeologyllic
Scale 3825Scale 3825: Pynyllic, Ian Ring Music TheoryPynyllic
Scale 3833Scale 3833: Dycrygic, Ian Ring Music TheoryDycrygic
Scale 3785Scale 3785: Epagian, Ian Ring Music TheoryEpagian
Scale 3801Scale 3801: Maptyllic, Ian Ring Music TheoryMaptyllic
Scale 3753Scale 3753: Phraptian, Ian Ring Music TheoryPhraptian
Scale 3689Scale 3689: Katocrian, Ian Ring Music TheoryKatocrian
Scale 3945Scale 3945: Lydyllic, Ian Ring Music TheoryLydyllic
Scale 4073Scale 4073: Sathygic, Ian Ring Music TheorySathygic
Scale 3305Scale 3305: Chromatic Hypophrygian, Ian Ring Music TheoryChromatic Hypophrygian
Scale 3561Scale 3561: Pothyllic, Ian Ring Music TheoryPothyllic
Scale 2793Scale 2793: Eporian, Ian Ring Music TheoryEporian
Scale 1769Scale 1769: Blues Heptatonic II, Ian Ring Music TheoryBlues Heptatonic II

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.