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Scale 3693: "Stadyllic"

Scale 3693: Stadyllic, 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
Stadyllic
Dozenal
Xepian

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

{0,2,3,5,6,9,10,11}

Forte Number

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

8-18

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: 1743

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: 879

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, 2, 1, 3, 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, 4, 6, 5, 5, 3>

Interval Spectrum

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

p5m5n6s4d5t3

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> = {3,4,5,6}
<4> = {5,6,7}
<5> = {6,7,8,9}
<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.

2

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.

(12, 59, 138)

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{2,6,9}441.92
F{5,9,0}342.23
A♯{10,2,5}342.08
B{11,3,6}342.15
Minor Triadsdm{2,5,9}342
d♯m{3,6,10}342.08
bm{11,2,6}342.08
Augmented TriadsD+{2,6,10}441.85
Diminished Triads{0,3,6}242.38
d♯°{3,6,9}242.31
f♯°{6,9,0}242.31
{9,0,3}242.46
{11,2,5}242.46
Parsimonious Voice Leading Between Common Triads of Scale 3693. Created by Ian Ring ©2019 c°->a° B B c°->B dm dm D D dm->D F F dm->F A# A# dm->A# D+ D+ D->D+ d#° d#° D->d#° f#° f#° D->f#° d#m d#m D+->d#m D+->A# bm bm D+->bm d#°->d#m d#m->B F->f#° F->a° A#->b° b°->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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3693 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1947
Scale 1947: Byptyllic, Ian Ring Music TheoryByptyllic
3rd mode:
Scale 3021
Scale 3021: Stodyllic, Ian Ring Music TheoryStodyllic
4th mode:
Scale 1779
Scale 1779: Zynyllic, Ian Ring Music TheoryZynyllic
5th mode:
Scale 2937
Scale 2937: Phragyllic, Ian Ring Music TheoryPhragyllic
6th mode:
Scale 879
Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllicThis is the prime mode
7th mode:
Scale 2487
Scale 2487: Dothyllic, Ian Ring Music TheoryDothyllic
8th mode:
Scale 3291
Scale 3291: Lygyllic, Ian Ring Music TheoryLygyllic

Prime

The prime form of this scale is Scale 879

Scale 879Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllic

Complement

The octatonic modal family [3693, 1947, 3021, 1779, 2937, 879, 2487, 3291] (Forte: 8-18) is the complement of the tetratonic modal family [147, 609, 777, 2121] (Forte: 4-18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3693 is 1743

Scale 1743Scale 1743: Epigyllic, Ian Ring Music TheoryEpigyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3693 is chiral, and its enantiomorph is scale 1743

Scale 1743Scale 1743: Epigyllic, Ian Ring Music TheoryEpigyllic

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> 3693       T0I <11,0> 1743
T1 <1,1> 3291      T1I <11,1> 3486
T2 <1,2> 2487      T2I <11,2> 2877
T3 <1,3> 879      T3I <11,3> 1659
T4 <1,4> 1758      T4I <11,4> 3318
T5 <1,5> 3516      T5I <11,5> 2541
T6 <1,6> 2937      T6I <11,6> 987
T7 <1,7> 1779      T7I <11,7> 1974
T8 <1,8> 3558      T8I <11,8> 3948
T9 <1,9> 3021      T9I <11,9> 3801
T10 <1,10> 1947      T10I <11,10> 3507
T11 <1,11> 3894      T11I <11,11> 2919
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1743      T0MI <7,0> 3693
T1M <5,1> 3486      T1MI <7,1> 3291
T2M <5,2> 2877      T2MI <7,2> 2487
T3M <5,3> 1659      T3MI <7,3> 879
T4M <5,4> 3318      T4MI <7,4> 1758
T5M <5,5> 2541      T5MI <7,5> 3516
T6M <5,6> 987      T6MI <7,6> 2937
T7M <5,7> 1974      T7MI <7,7> 1779
T8M <5,8> 3948      T8MI <7,8> 3558
T9M <5,9> 3801      T9MI <7,9> 3021
T10M <5,10> 3507      T10MI <7,10> 1947
T11M <5,11> 2919      T11MI <7,11> 3894

The transformations that map this set to itself are: T0, T0MI

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 3695Scale 3695: Kodygic, Ian Ring Music TheoryKodygic
Scale 3689Scale 3689: Katocrian, Ian Ring Music TheoryKatocrian
Scale 3691Scale 3691: Badyllic, Ian Ring Music TheoryBadyllic
Scale 3685Scale 3685: Kodian, Ian Ring Music TheoryKodian
Scale 3701Scale 3701: Bagyllic, Ian Ring Music TheoryBagyllic
Scale 3709Scale 3709: Katynygic, Ian Ring Music TheoryKatynygic
Scale 3661Scale 3661: Mixodorian, Ian Ring Music TheoryMixodorian
Scale 3677Scale 3677: Xafian, Ian Ring Music TheoryXafian
Scale 3629Scale 3629: Boptian, Ian Ring Music TheoryBoptian
Scale 3757Scale 3757: Raga Mian Ki Malhar, Ian Ring Music TheoryRaga Mian Ki Malhar
Scale 3821Scale 3821: Epyrygic, Ian Ring Music TheoryEpyrygic
Scale 3949Scale 3949: Koptygic, Ian Ring Music TheoryKoptygic
Scale 3181Scale 3181: Rolian, Ian Ring Music TheoryRolian
Scale 3437Scale 3437: Vopian, Ian Ring Music TheoryVopian
Scale 2669Scale 2669: Jeths' Mode, Ian Ring Music TheoryJeths' Mode
Scale 1645Scale 1645: Dorian Flat 5, Ian Ring Music TheoryDorian Flat 5

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.