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Scale 3887: "Phrathygic"

Scale 3887: Phrathygic, 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
Phrathygic

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

9 (enneatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,2,3,5,8,9,10,11}

Forte Number

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

9-2

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: 3743

Hemitonia

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

7 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

6 (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.

8

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 767

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 1, 1, 2, 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.

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

Interval Spectrum

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

p6m6n7s7d7t3

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> = {4,5,6,7}
<5> = {5,6,7,8}
<6> = {6,7,8,9}
<7> = {7,8,9,10}
<8> = {9,10,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

2.444

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.

2.683

Polygon Perimeter

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

6.038

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.

(97, 89, 176)

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♯{1,5,8}342.14
F{5,9,0}342.14
G♯{8,0,3}342.29
A♯{10,2,5}342.43
Minor Triadsdm{2,5,9}342.29
fm{5,8,0}442.07
g♯m{8,11,3}342.43
a♯m{10,1,5}242.43
Augmented TriadsC♯+{1,5,9}442.07
Diminished Triads{2,5,8}242.5
{5,8,11}242.43
g♯°{8,11,2}242.57
{9,0,3}242.5
{11,2,5}242.57
Parsimonious Voice Leading Between Common Triads of Scale 3887. Created by Ian Ring ©2019 C# C# C#+ C#+ C#->C#+ C#->d° fm fm C#->fm dm dm C#+->dm F F C#+->F a#m a#m C#+->a#m d°->dm A# A# dm->A# f°->fm g#m g#m f°->g#m fm->F G# G# fm->G# F->a° g#° g#° g#°->g#m g#°->b° g#m->G# G#->a° a#m->A# 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 3887 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3991
Scale 3991: Badygic, Ian Ring Music TheoryBadygic
3rd mode:
Scale 4043
Scale 4043: Phrocrygic, Ian Ring Music TheoryPhrocrygic
4th mode:
Scale 4069
Scale 4069: Starygic, Ian Ring Music TheoryStarygic
5th mode:
Scale 2041
Scale 2041: Aeolacrygic, Ian Ring Music TheoryAeolacrygic
6th mode:
Scale 767
Scale 767: Raptygic, Ian Ring Music TheoryRaptygicThis is the prime mode
7th mode:
Scale 2431
Scale 2431: Gythygic, Ian Ring Music TheoryGythygic
8th mode:
Scale 3263
Scale 3263: Pyrygic, Ian Ring Music TheoryPyrygic
9th mode:
Scale 3679
Scale 3679: Rycrygic, Ian Ring Music TheoryRycrygic

Prime

The prime form of this scale is Scale 767

Scale 767Scale 767: Raptygic, Ian Ring Music TheoryRaptygic

Complement

The enneatonic modal family [3887, 3991, 4043, 4069, 2041, 767, 2431, 3263, 3679] (Forte: 9-2) is the complement of the tritonic modal family [11, 1537, 2053] (Forte: 3-2)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3887 is 3743

Scale 3743Scale 3743: Thadygic, Ian Ring Music TheoryThadygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3887 is chiral, and its enantiomorph is scale 3743

Scale 3743Scale 3743: Thadygic, Ian Ring Music TheoryThadygic

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> 3887       T0I <11,0> 3743
T1 <1,1> 3679      T1I <11,1> 3391
T2 <1,2> 3263      T2I <11,2> 2687
T3 <1,3> 2431      T3I <11,3> 1279
T4 <1,4> 767      T4I <11,4> 2558
T5 <1,5> 1534      T5I <11,5> 1021
T6 <1,6> 3068      T6I <11,6> 2042
T7 <1,7> 2041      T7I <11,7> 4084
T8 <1,8> 4082      T8I <11,8> 4073
T9 <1,9> 4069      T9I <11,9> 4051
T10 <1,10> 4043      T10I <11,10> 4007
T11 <1,11> 3991      T11I <11,11> 3919
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1727      T0MI <7,0> 4013
T1M <5,1> 3454      T1MI <7,1> 3931
T2M <5,2> 2813      T2MI <7,2> 3767
T3M <5,3> 1531      T3MI <7,3> 3439
T4M <5,4> 3062      T4MI <7,4> 2783
T5M <5,5> 2029      T5MI <7,5> 1471
T6M <5,6> 4058      T6MI <7,6> 2942
T7M <5,7> 4021      T7MI <7,7> 1789
T8M <5,8> 3947      T8MI <7,8> 3578
T9M <5,9> 3799      T9MI <7,9> 3061
T10M <5,10> 3503      T10MI <7,10> 2027
T11M <5,11> 2911      T11MI <7,11> 4054

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 3885Scale 3885: Styryllic, Ian Ring Music TheoryStyryllic
Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic
Scale 3879Scale 3879: Pathyllic, Ian Ring Music TheoryPathyllic
Scale 3895Scale 3895: Eparygic, Ian Ring Music TheoryEparygic
Scale 3903Scale 3903: Decatonic Chromatic 5, Ian Ring Music TheoryDecatonic Chromatic 5
Scale 3855Scale 3855: Octatonic Chromatic 5, Ian Ring Music TheoryOctatonic Chromatic 5
Scale 3871Scale 3871: Nonatonic Chromatic 5, Ian Ring Music TheoryNonatonic Chromatic 5
Scale 3919Scale 3919: Lynygic, Ian Ring Music TheoryLynygic
Scale 3951Scale 3951: Mathyllian, Ian Ring Music TheoryMathyllian
Scale 4015Scale 4015: Phradyllian, Ian Ring Music TheoryPhradyllian
Scale 3631Scale 3631: Gydyllic, Ian Ring Music TheoryGydyllic
Scale 3759Scale 3759: Darygic, Ian Ring Music TheoryDarygic
Scale 3375Scale 3375, Ian Ring Music Theory
Scale 2863Scale 2863: Aerogyllic, Ian Ring Music TheoryAerogyllic
Scale 1839Scale 1839: Zogyllic, Ian Ring Music TheoryZogyllic

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.