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Scale 3755: "Phryryllic"

Scale 3755: Phryryllic, 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
Phryryllic
Dozenal
Yadian

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

{0,1,3,5,7,9,10,11}

Forte Number

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

8-21

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.

[5]

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.

no

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

4 (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.

4

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: 1375

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.

[1, 2, 2, 2, 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.

<4, 7, 4, 6, 4, 3>

Interval Spectrum

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

p4m6n4s7d4t3

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}
<2> = {2,3,4}
<3> = {3,4,5,6}
<4> = {4,5,6,7,8}
<5> = {6,7,8,9}
<6> = {8,9,10}
<7> = {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.

yes

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.

2.732

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

6.071

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.

[10]

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.

(28, 74, 147)

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
F{5,9,0}242
Minor Triadscm{0,3,7}242
a♯m{10,1,5}242
Augmented TriadsC♯+{1,5,9}242
D♯+{3,7,11}242
Diminished Triads{7,10,1}242
{9,0,3}242
Parsimonious Voice Leading Between Common Triads of Scale 3755. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ cm->a° C#+ C#+ F F C#+->F a#m a#m C#+->a#m D# D# D#->D#+ D#->g° F->a° g°->a#m

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 3755 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3925
Scale 3925: Thyryllic, Ian Ring Music TheoryThyryllic
3rd mode:
Scale 2005
Scale 2005: Gygyllic, Ian Ring Music TheoryGygyllic
4th mode:
Scale 1525
Scale 1525: Sodyllic, Ian Ring Music TheorySodyllic
5th mode:
Scale 1405
Scale 1405: Goryllic, Ian Ring Music TheoryGoryllic
6th mode:
Scale 1375
Scale 1375: Bothyllic, Ian Ring Music TheoryBothyllicThis is the prime mode
7th mode:
Scale 2735
Scale 2735: Gynyllic, Ian Ring Music TheoryGynyllic
8th mode:
Scale 3415
Scale 3415: Ionaptyllic, Ian Ring Music TheoryIonaptyllic

Prime

The prime form of this scale is Scale 1375

Scale 1375Scale 1375: Bothyllic, Ian Ring Music TheoryBothyllic

Complement

The octatonic modal family [3755, 3925, 2005, 1525, 1405, 1375, 2735, 3415] (Forte: 8-21) is the complement of the tetratonic modal family [85, 1045, 1285, 1345] (Forte: 4-21)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3755 is 2735

Scale 2735Scale 2735: Gynyllic, Ian Ring Music TheoryGynyllic

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> 3755       T0I <11,0> 2735
T1 <1,1> 3415      T1I <11,1> 1375
T2 <1,2> 2735      T2I <11,2> 2750
T3 <1,3> 1375      T3I <11,3> 1405
T4 <1,4> 2750      T4I <11,4> 2810
T5 <1,5> 1405      T5I <11,5> 1525
T6 <1,6> 2810      T6I <11,6> 3050
T7 <1,7> 1525      T7I <11,7> 2005
T8 <1,8> 3050      T8I <11,8> 4010
T9 <1,9> 2005      T9I <11,9> 3925
T10 <1,10> 4010      T10I <11,10> 3755
T11 <1,11> 3925      T11I <11,11> 3415
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2735      T0MI <7,0> 3755
T1M <5,1> 1375      T1MI <7,1> 3415
T2M <5,2> 2750      T2MI <7,2> 2735
T3M <5,3> 1405      T3MI <7,3> 1375
T4M <5,4> 2810      T4MI <7,4> 2750
T5M <5,5> 1525      T5MI <7,5> 1405
T6M <5,6> 3050      T6MI <7,6> 2810
T7M <5,7> 2005      T7MI <7,7> 1525
T8M <5,8> 4010      T8MI <7,8> 3050
T9M <5,9> 3925      T9MI <7,9> 2005
T10M <5,10> 3755       T10MI <7,10> 4010
T11M <5,11> 3415      T11MI <7,11> 3925

The transformations that map this set to itself are: T0, T10I, T10M, 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 3753Scale 3753: Phraptian, Ian Ring Music TheoryPhraptian
Scale 3757Scale 3757: Raga Mian Ki Malhar, Ian Ring Music TheoryRaga Mian Ki Malhar
Scale 3759Scale 3759: Darygic, Ian Ring Music TheoryDarygic
Scale 3747Scale 3747: Myrian, Ian Ring Music TheoryMyrian
Scale 3751Scale 3751: Aerathyllic, Ian Ring Music TheoryAerathyllic
Scale 3763Scale 3763: Modyllic, Ian Ring Music TheoryModyllic
Scale 3771Scale 3771: Stophygic, Ian Ring Music TheoryStophygic
Scale 3723Scale 3723: Myptian, Ian Ring Music TheoryMyptian
Scale 3739Scale 3739: Epanyllic, Ian Ring Music TheoryEpanyllic
Scale 3787Scale 3787: Kagyllic, Ian Ring Music TheoryKagyllic
Scale 3819Scale 3819: Aeolanygic, Ian Ring Music TheoryAeolanygic
Scale 3627Scale 3627: Kalian, Ian Ring Music TheoryKalian
Scale 3691Scale 3691: Badyllic, Ian Ring Music TheoryBadyllic
Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic
Scale 4011Scale 4011: Styrygic, Ian Ring Music TheoryStyrygic
Scale 3243Scale 3243: Mela Rupavati, Ian Ring Music TheoryMela Rupavati
Scale 3499Scale 3499: Hamel, Ian Ring Music TheoryHamel
Scale 2731Scale 2731: Neapolitan Major, Ian Ring Music TheoryNeapolitan Major
Scale 1707Scale 1707: Dorian Flat 2, Ian Ring Music TheoryDorian Flat 2

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.