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Scale 3893: "Phrocryllic"

Scale 3893: Phrocryllic, 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
Phrocryllic
Dozenal
Yulian

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

{0,2,4,5,8,9,10,11}

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-Z15

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: 1439

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: 863

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, 1, 3, 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, 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}
<3> = {3,4,5,6}
<4> = {4,5,6,7,8}
<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.

2.25

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, 59, 140)

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 TriadsE{4,8,11}342
F{5,9,0}341.91
A♯{10,2,5}242.27
Minor Triadsdm{2,5,9}342
fm{5,8,0}441.82
am{9,0,4}242.18
Augmented TriadsC+{0,4,8}341.91
Diminished Triads{2,5,8}242.09
{5,8,11}242.09
g♯°{8,11,2}242.27
{11,2,5}242.36
Parsimonious Voice Leading Between Common Triads of Scale 3893. Created by Ian Ring ©2019 C+ C+ E E C+->E fm fm C+->fm am am C+->am dm dm d°->dm d°->fm F F dm->F A# A# dm->A# E->f° g#° g#° E->g#° f°->fm fm->F F->am g#°->b° A#->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 3893 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1997
Scale 1997: Raga Cintamani, Ian Ring Music TheoryRaga Cintamani
3rd mode:
Scale 1523
Scale 1523: Zothyllic, Ian Ring Music TheoryZothyllic
4th mode:
Scale 2809
Scale 2809: Gythyllic, Ian Ring Music TheoryGythyllic
5th mode:
Scale 863
Scale 863: Pyryllic, Ian Ring Music TheoryPyryllicThis is the prime mode
6th mode:
Scale 2479
Scale 2479: Harmonic and Neapolitan Minor Mixed, Ian Ring Music TheoryHarmonic and Neapolitan Minor Mixed
7th mode:
Scale 3287
Scale 3287: Phrathyllic, Ian Ring Music TheoryPhrathyllic
8th mode:
Scale 3691
Scale 3691: Badyllic, Ian Ring Music TheoryBadyllic

Prime

The prime form of this scale is Scale 863

Scale 863Scale 863: Pyryllic, Ian Ring Music TheoryPyryllic

Complement

The octatonic modal family [3893, 1997, 1523, 2809, 863, 2479, 3287, 3691] (Forte: 8-Z15) is the complement of the tetratonic modal family [83, 773, 1217, 2089] (Forte: 4-Z15)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3893 is 1439

Scale 1439Scale 1439: Rolyllic, Ian Ring Music TheoryRolyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3893 is chiral, and its enantiomorph is scale 1439

Scale 1439Scale 1439: Rolyllic, Ian Ring Music TheoryRolyllic

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

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 3895Scale 3895: Eparygic, Ian Ring Music TheoryEparygic
Scale 3889Scale 3889: Parian, Ian Ring Music TheoryParian
Scale 3891Scale 3891: Ryryllic, Ian Ring Music TheoryRyryllic
Scale 3897Scale 3897: Kalyllic, Ian Ring Music TheoryKalyllic
Scale 3901Scale 3901: Bycrygic, Ian Ring Music TheoryBycrygic
Scale 3877Scale 3877: Thanian, Ian Ring Music TheoryThanian
Scale 3885Scale 3885: Styryllic, Ian Ring Music TheoryStyryllic
Scale 3861Scale 3861: Phroptian, Ian Ring Music TheoryPhroptian
Scale 3925Scale 3925: Thyryllic, Ian Ring Music TheoryThyryllic
Scale 3957Scale 3957: Porygic, Ian Ring Music TheoryPorygic
Scale 4021Scale 4021: Raga Pahadi, Ian Ring Music TheoryRaga Pahadi
Scale 3637Scale 3637: Raga Rageshri, Ian Ring Music TheoryRaga Rageshri
Scale 3765Scale 3765: Dominant Bebop, Ian Ring Music TheoryDominant Bebop
Scale 3381Scale 3381: Katanian, Ian Ring Music TheoryKatanian
Scale 2869Scale 2869: Major Augmented, Ian Ring Music TheoryMajor Augmented
Scale 1845Scale 1845: Lagian, Ian Ring Music TheoryLagian

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.