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Scale 729: "Stygimic"

Scale 729: Stygimic, 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
Stygimic
Dozenal
Ofrian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,3,4,6,7,9}

Forte Number

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

6-27

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: 873

Hemitonia

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

2 (dihemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

5

Prime Form

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

no
prime: 603

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.

[3, 1, 2, 1, 2, 3]

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.

<2, 2, 5, 2, 2, 2>

Interval Spectrum

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

p2m2n5s2d2t2

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

Spectra Variation

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

2.333

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.

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.

2.366

Polygon Perimeter

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

5.864

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.

(6, 21, 62)

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}231.57
Minor Triadscm{0,3,7}331.43
am{9,0,4}331.43
Diminished Triads{0,3,6}231.57
d♯°{3,6,9}231.71
f♯°{6,9,0}231.57
{9,0,3}231.57
Parsimonious Voice Leading Between Common Triads of Scale 729. Created by Ian Ring ©2019 cm cm c°->cm d#° d#° c°->d#° C C cm->C cm->a° am am C->am f#° f#° d#°->f#° f#°->am a°->am

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.

Diameter3
Radius3
Self-Centeredyes

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Major: {0, 4, 7}
Diminished: {3, 6, 9}

Modes

Modes are the rotational transformation of this scale. Scale 729 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 603
Scale 603: Aeolygimic, Ian Ring Music TheoryAeolygimicThis is the prime mode
3rd mode:
Scale 2349
Scale 2349: Raga Ghantana, Ian Ring Music TheoryRaga Ghantana
4th mode:
Scale 1611
Scale 1611: Dacrimic, Ian Ring Music TheoryDacrimic
5th mode:
Scale 2853
Scale 2853: Baptimic, Ian Ring Music TheoryBaptimic
6th mode:
Scale 1737
Scale 1737: Raga Madhukauns, Ian Ring Music TheoryRaga Madhukauns

Prime

The prime form of this scale is Scale 603

Scale 603Scale 603: Aeolygimic, Ian Ring Music TheoryAeolygimic

Complement

The hexatonic modal family [729, 603, 2349, 1611, 2853, 1737] (Forte: 6-27) is the complement of the hexatonic modal family [603, 729, 1611, 1737, 2349, 2853] (Forte: 6-27)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 729 is 873

Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 729 is chiral, and its enantiomorph is scale 873

Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic

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> 729       T0I <11,0> 873
T1 <1,1> 1458      T1I <11,1> 1746
T2 <1,2> 2916      T2I <11,2> 3492
T3 <1,3> 1737      T3I <11,3> 2889
T4 <1,4> 3474      T4I <11,4> 1683
T5 <1,5> 2853      T5I <11,5> 3366
T6 <1,6> 1611      T6I <11,6> 2637
T7 <1,7> 3222      T7I <11,7> 1179
T8 <1,8> 2349      T8I <11,8> 2358
T9 <1,9> 603      T9I <11,9> 621
T10 <1,10> 1206      T10I <11,10> 1242
T11 <1,11> 2412      T11I <11,11> 2484
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2889      T0MI <7,0> 603
T1M <5,1> 1683      T1MI <7,1> 1206
T2M <5,2> 3366      T2MI <7,2> 2412
T3M <5,3> 2637      T3MI <7,3> 729
T4M <5,4> 1179      T4MI <7,4> 1458
T5M <5,5> 2358      T5MI <7,5> 2916
T6M <5,6> 621      T6MI <7,6> 1737
T7M <5,7> 1242      T7MI <7,7> 3474
T8M <5,8> 2484      T8MI <7,8> 2853
T9M <5,9> 873      T9MI <7,9> 1611
T10M <5,10> 1746      T10MI <7,10> 3222
T11M <5,11> 3492      T11MI <7,11> 2349

The transformations that map this set to itself are: T0, T3MI

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 731Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating Heptamode
Scale 733Scale 733: Donian, Ian Ring Music TheoryDonian
Scale 721Scale 721: Raga Dhavalashri, Ian Ring Music TheoryRaga Dhavalashri
Scale 725Scale 725: Raga Yamuna Kalyani, Ian Ring Music TheoryRaga Yamuna Kalyani
Scale 713Scale 713: Thoptitonic, Ian Ring Music TheoryThoptitonic
Scale 745Scale 745: Kolimic, Ian Ring Music TheoryKolimic
Scale 761Scale 761: Ponian, Ian Ring Music TheoryPonian
Scale 665Scale 665: Raga Mohanangi, Ian Ring Music TheoryRaga Mohanangi
Scale 697Scale 697: Lagimic, Ian Ring Music TheoryLagimic
Scale 601Scale 601: Bycritonic, Ian Ring Music TheoryBycritonic
Scale 857Scale 857: Aeolydimic, Ian Ring Music TheoryAeolydimic
Scale 985Scale 985: Mela Sucaritra, Ian Ring Music TheoryMela Sucaritra
Scale 217Scale 217: Biwian, Ian Ring Music TheoryBiwian
Scale 473Scale 473: Aeralimic, Ian Ring Music TheoryAeralimic
Scale 1241Scale 1241: Pygimic, Ian Ring Music TheoryPygimic
Scale 1753Scale 1753: Hungarian Major, Ian Ring Music TheoryHungarian Major
Scale 2777Scale 2777: Aeolian Harmonic, Ian Ring Music TheoryAeolian Harmonic

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.