The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1145: "Zygimic"

Scale 1145: Zygimic, 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
Zygimic
Dozenal
Hakian

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

{0,3,4,5,6,10}

Forte Number

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

6-Z41

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: 965

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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.

5

Prime Form

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

no
prime: 335

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, 1, 1, 4, 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.

<3, 3, 2, 2, 3, 2>

Interval Spectrum

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

p3m2n2s3d3t2

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> = {1,2,3,4}
<2> = {2,4,5,6}
<3> = {3,5,6,7,9}
<4> = {6,7,8,10}
<5> = {8,9,10,11}

Spectra Variation

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

3.333

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.

2.116

Polygon Perimeter

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

5.699

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.

(20, 17, 64)

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
Minor Triadsd♯m{3,6,10}110.5
Diminished Triads{0,3,6}110.5

The following pitch classes are not present in any of the common triads: {4,5}

Parsimonious Voice Leading Between Common Triads of Scale 1145. Created by Ian Ring ©2019 d#m d#m c°->d#m

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1145 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 655
Scale 655: Kataptimic, Ian Ring Music TheoryKataptimic
3rd mode:
Scale 2375
Scale 2375: Aeolaptimic, Ian Ring Music TheoryAeolaptimic
4th mode:
Scale 3235
Scale 3235: Pothimic, Ian Ring Music TheoryPothimic
5th mode:
Scale 3665
Scale 3665: Stalimic, Ian Ring Music TheoryStalimic
6th mode:
Scale 485
Scale 485: Stoptimic, Ian Ring Music TheoryStoptimic

Prime

The prime form of this scale is Scale 335

Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic

Complement

The hexatonic modal family [1145, 655, 2375, 3235, 3665, 485] (Forte: 6-Z41) is the complement of the hexatonic modal family [215, 1475, 1805, 2155, 2785, 3125] (Forte: 6-Z12)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1145 is 965

Scale 965Scale 965: Ionothimic, Ian Ring Music TheoryIonothimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1145 is chiral, and its enantiomorph is scale 965

Scale 965Scale 965: Ionothimic, Ian Ring Music TheoryIonothimic

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> 1145       T0I <11,0> 965
T1 <1,1> 2290      T1I <11,1> 1930
T2 <1,2> 485      T2I <11,2> 3860
T3 <1,3> 970      T3I <11,3> 3625
T4 <1,4> 1940      T4I <11,4> 3155
T5 <1,5> 3880      T5I <11,5> 2215
T6 <1,6> 3665      T6I <11,6> 335
T7 <1,7> 3235      T7I <11,7> 670
T8 <1,8> 2375      T8I <11,8> 1340
T9 <1,9> 655      T9I <11,9> 2680
T10 <1,10> 1310      T10I <11,10> 1265
T11 <1,11> 2620      T11I <11,11> 2530
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 335      T0MI <7,0> 3665
T1M <5,1> 670      T1MI <7,1> 3235
T2M <5,2> 1340      T2MI <7,2> 2375
T3M <5,3> 2680      T3MI <7,3> 655
T4M <5,4> 1265      T4MI <7,4> 1310
T5M <5,5> 2530      T5MI <7,5> 2620
T6M <5,6> 965      T6MI <7,6> 1145
T7M <5,7> 1930      T7MI <7,7> 2290
T8M <5,8> 3860      T8MI <7,8> 485
T9M <5,9> 3625      T9MI <7,9> 970
T10M <5,10> 3155      T10MI <7,10> 1940
T11M <5,11> 2215      T11MI <7,11> 3880

The transformations that map this set to itself are: T0, 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 1147Scale 1147: Epynian, Ian Ring Music TheoryEpynian
Scale 1149Scale 1149: Bydian, Ian Ring Music TheoryBydian
Scale 1137Scale 1137: Stonitonic, Ian Ring Music TheoryStonitonic
Scale 1141Scale 1141: Rynimic, Ian Ring Music TheoryRynimic
Scale 1129Scale 1129: Raga Jayakauns, Ian Ring Music TheoryRaga Jayakauns
Scale 1113Scale 1113: Locrian Pentatonic 2, Ian Ring Music TheoryLocrian Pentatonic 2
Scale 1081Scale 1081: Goxian, Ian Ring Music TheoryGoxian
Scale 1209Scale 1209: Raga Bhanumanjari, Ian Ring Music TheoryRaga Bhanumanjari
Scale 1273Scale 1273: Ronian, Ian Ring Music TheoryRonian
Scale 1401Scale 1401: Pagian, Ian Ring Music TheoryPagian
Scale 1657Scale 1657: Ionothian, Ian Ring Music TheoryIonothian
Scale 121Scale 121: Asoian, Ian Ring Music TheoryAsoian
Scale 633Scale 633: Kydimic, Ian Ring Music TheoryKydimic
Scale 2169Scale 2169: Nefian, Ian Ring Music TheoryNefian
Scale 3193Scale 3193: Zathian, Ian Ring Music TheoryZathian

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.