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Scale 3743: "Thadygic"

Scale 3743: Thadygic, 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
Thadygic
Dozenal
Xusian
Xutian

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

{0,1,2,3,4,7,9,10,11}

Forte Number

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

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: 3887

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 Structure

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

[1, 1, 1, 1, 3, 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.

<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{0,4,7}442.07
D♯{3,7,10}342.29
G{7,11,2}242.43
A{9,1,4}342.43
Minor Triadscm{0,3,7}342.14
em{4,7,11}342.14
gm{7,10,2}342.43
am{9,0,4}342.29
Augmented TriadsD♯+{3,7,11}442.07
Diminished Triadsc♯°{1,4,7}242.43
{4,7,10}242.5
{7,10,1}242.57
{9,0,3}242.5
a♯°{10,1,4}242.57
Parsimonious Voice Leading Between Common Triads of Scale 3743. Created by Ian Ring ©2019 cm cm C C cm->C D#+ D#+ cm->D#+ cm->a° c#° c#° C->c#° em em C->em am am C->am A A c#°->A D# D# D#->D#+ D#->e° gm gm D#->gm D#+->em Parsimonious Voice Leading Between Common Triads of Scale 3743. Created by Ian Ring ©2019 G D#+->G e°->em g°->gm a#° a#° g°->a#° gm->G a°->am am->A A->a#°

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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 3743 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3919
Scale 3919: Lynygic, Ian Ring Music TheoryLynygic
3rd mode:
Scale 4007
Scale 4007: Doptygic, Ian Ring Music TheoryDoptygic
4th mode:
Scale 4051
Scale 4051: Ionilygic, Ian Ring Music TheoryIonilygic
5th mode:
Scale 4073
Scale 4073: Sathygic, Ian Ring Music TheorySathygic
6th mode:
Scale 1021
Scale 1021: Ladygic, Ian Ring Music TheoryLadygic
7th mode:
Scale 1279
Scale 1279: Sarygic, Ian Ring Music TheorySarygic
8th mode:
Scale 2687
Scale 2687: Thacrygic, Ian Ring Music TheoryThacrygic
9th mode:
Scale 3391
Scale 3391: Aeolynygic, Ian Ring Music TheoryAeolynygic

Prime

The prime form of this scale is Scale 767

Scale 767Scale 767: Raptygic, Ian Ring Music TheoryRaptygic

Complement

The enneatonic modal family [3743, 3919, 4007, 4051, 4073, 1021, 1279, 2687, 3391] (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 3743 is 3887

Scale 3887Scale 3887: Phrathygic, Ian Ring Music TheoryPhrathygic

Enantiomorph

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

Scale 3887Scale 3887: Phrathygic, Ian Ring Music TheoryPhrathygic

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

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 3741Scale 3741: Zydyllic, Ian Ring Music TheoryZydyllic
Scale 3739Scale 3739: Epanyllic, Ian Ring Music TheoryEpanyllic
Scale 3735Scale 3735: Xupian, Ian Ring Music TheoryXupian
Scale 3727Scale 3727: Tholyllic, Ian Ring Music TheoryTholyllic
Scale 3759Scale 3759: Darygic, Ian Ring Music TheoryDarygic
Scale 3775Scale 3775: Loptyllian, Ian Ring Music TheoryLoptyllian
Scale 3807Scale 3807: Bagyllian, Ian Ring Music TheoryBagyllian
Scale 3615Scale 3615: Octatonic Chromatic 4, Ian Ring Music TheoryOctatonic Chromatic 4
Scale 3679Scale 3679: Rycrygic, Ian Ring Music TheoryRycrygic
Scale 3871Scale 3871: Nonatonic Chromatic 5, Ian Ring Music TheoryNonatonic Chromatic 5
Scale 3999Scale 3999: Decatonic Chromatic 6, Ian Ring Music TheoryDecatonic Chromatic 6
Scale 3231Scale 3231: Kataptyllic, Ian Ring Music TheoryKataptyllic
Scale 3487Scale 3487: Byptygic, Ian Ring Music TheoryByptygic
Scale 2719Scale 2719: Zocryllic, Ian Ring Music TheoryZocryllic
Scale 1695Scale 1695: Phrodyllic, Ian Ring Music TheoryPhrodyllic

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.