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Scale 1787: "Mycrygic"

Scale 1787: Mycrygic, 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
Mycrygic
Dozenal
Lafian

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

{0,1,3,4,5,6,7,9,10}

Forte Number

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

9-10

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.

6 (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.

8

Prime Form

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

no
prime: 1759

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, 1, 1, 1, 1, 2, 1, 2]

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.

<6, 6, 8, 6, 6, 4>

Interval Spectrum

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

p6m6n8s6d6t4

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}
<3> = {3,4,5}
<4> = {4,5,6}
<5> = {6,7,8}
<6> = {7,8,9}
<7> = {9,10}
<8> = {10,11}

Spectra Variation

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

1.333

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.

2.799

Polygon Perimeter

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

6.106

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.

(4, 84, 168)

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{0,4,7}442.37
D♯{3,7,10}442.42
F{5,9,0}342.47
F♯{6,10,1}442.32
A{9,1,4}442.37
Minor Triadscm{0,3,7}442.42
d♯m{3,6,10}442.37
f♯m{6,9,1}442.37
am{9,0,4}442.32
a♯m{10,1,5}342.47
Augmented TriadsC♯+{1,5,9}442.32
Diminished Triads{0,3,6}242.63
c♯°{1,4,7}242.63
d♯°{3,6,9}242.63
{4,7,10}242.63
f♯°{6,9,0}242.74
{7,10,1}242.58
{9,0,3}242.58
a♯°{10,1,4}242.74
Parsimonious Voice Leading Between Common Triads of Scale 1787. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m C C cm->C D# D# cm->D# cm->a° c#° c#° C->c#° C->e° am am C->am A A c#°->A C#+ C#+ F F C#+->F f#m f#m C#+->f#m C#+->A a#m a#m C#+->a#m d#° d#° d#°->d#m d#°->f#m d#m->D# F# F# d#m->F# D#->e° D#->g° f#° f#° F->f#° F->am f#°->f#m f#m->F# F#->g° F#->a#m a°->am am->A a#° a#° A->a#° a#°->a#m

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 1787 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 2941
Scale 2941: Laptygic, Ian Ring Music TheoryLaptygic
3rd mode:
Scale 1759
Scale 1759: Pylygic, Ian Ring Music TheoryPylygicThis is the prime mode
4th mode:
Scale 2927
Scale 2927: Rodygic, Ian Ring Music TheoryRodygic
5th mode:
Scale 3511
Scale 3511: Epolygic, Ian Ring Music TheoryEpolygic
6th mode:
Scale 3803
Scale 3803: Epidygic, Ian Ring Music TheoryEpidygic
7th mode:
Scale 3949
Scale 3949: Koptygic, Ian Ring Music TheoryKoptygic
8th mode:
Scale 2011
Scale 2011: Raphygic, Ian Ring Music TheoryRaphygic
9th mode:
Scale 3053
Scale 3053: Zycrygic, Ian Ring Music TheoryZycrygic

Prime

The prime form of this scale is Scale 1759

Scale 1759Scale 1759: Pylygic, Ian Ring Music TheoryPylygic

Complement

The enneatonic modal family [1787, 2941, 1759, 2927, 3511, 3803, 3949, 2011, 3053] (Forte: 9-10) is the complement of the tritonic modal family [73, 521, 577] (Forte: 3-10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1787 is 3053

Scale 3053Scale 3053: Zycrygic, Ian Ring Music TheoryZycrygic

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> 1787       T0I <11,0> 3053
T1 <1,1> 3574      T1I <11,1> 2011
T2 <1,2> 3053      T2I <11,2> 4022
T3 <1,3> 2011      T3I <11,3> 3949
T4 <1,4> 4022      T4I <11,4> 3803
T5 <1,5> 3949      T5I <11,5> 3511
T6 <1,6> 3803      T6I <11,6> 2927
T7 <1,7> 3511      T7I <11,7> 1759
T8 <1,8> 2927      T8I <11,8> 3518
T9 <1,9> 1759      T9I <11,9> 2941
T10 <1,10> 3518      T10I <11,10> 1787
T11 <1,11> 2941      T11I <11,11> 3574
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2927      T0MI <7,0> 3803
T1M <5,1> 1759      T1MI <7,1> 3511
T2M <5,2> 3518      T2MI <7,2> 2927
T3M <5,3> 2941      T3MI <7,3> 1759
T4M <5,4> 1787       T4MI <7,4> 3518
T5M <5,5> 3574      T5MI <7,5> 2941
T6M <5,6> 3053      T6MI <7,6> 1787
T7M <5,7> 2011      T7MI <7,7> 3574
T8M <5,8> 4022      T8MI <7,8> 3053
T9M <5,9> 3949      T9MI <7,9> 2011
T10M <5,10> 3803      T10MI <7,10> 4022
T11M <5,11> 3511      T11MI <7,11> 3949

The transformations that map this set to itself are: T0, T10I, T4M, T6MI

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 1785Scale 1785: Tharyllic, Ian Ring Music TheoryTharyllic
Scale 1789Scale 1789: Blues Enneatonic II, Ian Ring Music TheoryBlues Enneatonic II
Scale 1791Scale 1791: Aerygyllian, Ian Ring Music TheoryAerygyllian
Scale 1779Scale 1779: Zynyllic, Ian Ring Music TheoryZynyllic
Scale 1783Scale 1783: Youlan Scale, Ian Ring Music TheoryYoulan Scale
Scale 1771Scale 1771: Kuwian, Ian Ring Music TheoryKuwian
Scale 1755Scale 1755: Octatonic, Ian Ring Music TheoryOctatonic
Scale 1723Scale 1723: JG Octatonic, Ian Ring Music TheoryJG Octatonic
Scale 1659Scale 1659: Maqam Shadd'araban, Ian Ring Music TheoryMaqam Shadd'araban
Scale 1915Scale 1915: Thydygic, Ian Ring Music TheoryThydygic
Scale 2043Scale 2043: Maqam Tarzanuyn, Ian Ring Music TheoryMaqam Tarzanuyn
Scale 1275Scale 1275: Stagyllic, Ian Ring Music TheoryStagyllic
Scale 1531Scale 1531: Styptygic, Ian Ring Music TheoryStyptygic
Scale 763Scale 763: Doryllic, Ian Ring Music TheoryDoryllic
Scale 2811Scale 2811: Barygic, Ian Ring Music TheoryBarygic
Scale 3835Scale 3835: Katodyllian, Ian Ring Music TheoryKatodyllian

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.