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Scale 3157: "Zyptimic"

Scale 3157: Zyptimic, 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
Zyptimic
Dozenal
Gruian

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

{0,2,4,6,10,11}

Forte Number

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

6-22

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: 1351

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.

1 (uncohemitonic)

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: 343

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.

[2, 2, 2, 4, 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.

<2, 4, 1, 4, 2, 2>

Interval Spectrum

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

p2m4ns4d2t2

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

Spectra Variation

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

3

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.

2.232

Polygon Perimeter

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

5.767

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.

(10, 23, 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 Triadsbm{11,2,6}110.5
Augmented TriadsD+{2,6,10}110.5

The following pitch classes are not present in any of the common triads: {0,4}

Parsimonious Voice Leading Between Common Triads of Scale 3157. Created by Ian Ring ©2019 D+ D+ bm bm D+->bm

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 3157 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1813
Scale 1813: Katothimic, Ian Ring Music TheoryKatothimic
3rd mode:
Scale 1477
Scale 1477: Raga Jaganmohanam, Ian Ring Music TheoryRaga Jaganmohanam
4th mode:
Scale 1393
Scale 1393: Mycrimic, Ian Ring Music TheoryMycrimic
5th mode:
Scale 343
Scale 343: Ionorimic, Ian Ring Music TheoryIonorimicThis is the prime mode
6th mode:
Scale 2219
Scale 2219: Phrydimic, Ian Ring Music TheoryPhrydimic

Prime

The prime form of this scale is Scale 343

Scale 343Scale 343: Ionorimic, Ian Ring Music TheoryIonorimic

Complement

The hexatonic modal family [3157, 1813, 1477, 1393, 343, 2219] (Forte: 6-22) is the complement of the hexatonic modal family [343, 1393, 1477, 1813, 2219, 3157] (Forte: 6-22)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3157 is 1351

Scale 1351Scale 1351: Aeraptimic, Ian Ring Music TheoryAeraptimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3157 is chiral, and its enantiomorph is scale 1351

Scale 1351Scale 1351: Aeraptimic, Ian Ring Music TheoryAeraptimic

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> 3157       T0I <11,0> 1351
T1 <1,1> 2219      T1I <11,1> 2702
T2 <1,2> 343      T2I <11,2> 1309
T3 <1,3> 686      T3I <11,3> 2618
T4 <1,4> 1372      T4I <11,4> 1141
T5 <1,5> 2744      T5I <11,5> 2282
T6 <1,6> 1393      T6I <11,6> 469
T7 <1,7> 2786      T7I <11,7> 938
T8 <1,8> 1477      T8I <11,8> 1876
T9 <1,9> 2954      T9I <11,9> 3752
T10 <1,10> 1813      T10I <11,10> 3409
T11 <1,11> 3626      T11I <11,11> 2723
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1477      T0MI <7,0> 1141
T1M <5,1> 2954      T1MI <7,1> 2282
T2M <5,2> 1813      T2MI <7,2> 469
T3M <5,3> 3626      T3MI <7,3> 938
T4M <5,4> 3157       T4MI <7,4> 1876
T5M <5,5> 2219      T5MI <7,5> 3752
T6M <5,6> 343      T6MI <7,6> 3409
T7M <5,7> 686      T7MI <7,7> 2723
T8M <5,8> 1372      T8MI <7,8> 1351
T9M <5,9> 2744      T9MI <7,9> 2702
T10M <5,10> 1393      T10MI <7,10> 1309
T11M <5,11> 2786      T11MI <7,11> 2618

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

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 3159Scale 3159: Stocrian, Ian Ring Music TheoryStocrian
Scale 3153Scale 3153: Zathitonic, Ian Ring Music TheoryZathitonic
Scale 3155Scale 3155: Ladimic, Ian Ring Music TheoryLadimic
Scale 3161Scale 3161: Kodimic, Ian Ring Music TheoryKodimic
Scale 3165Scale 3165: Mylian, Ian Ring Music TheoryMylian
Scale 3141Scale 3141: Kanitonic, Ian Ring Music TheoryKanitonic
Scale 3149Scale 3149: Phrycrimic, Ian Ring Music TheoryPhrycrimic
Scale 3173Scale 3173: Zarimic, Ian Ring Music TheoryZarimic
Scale 3189Scale 3189: Aeolonian, Ian Ring Music TheoryAeolonian
Scale 3093Scale 3093: Buqian, Ian Ring Music TheoryBuqian
Scale 3125Scale 3125: Tonian, Ian Ring Music TheoryTonian
Scale 3221Scale 3221: Bycrimic, Ian Ring Music TheoryBycrimic
Scale 3285Scale 3285: Mela Citrambari, Ian Ring Music TheoryMela Citrambari
Scale 3413Scale 3413: Leading Whole-tone, Ian Ring Music TheoryLeading Whole-tone
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian
Scale 2133Scale 2133: Raga Kumurdaki, Ian Ring Music TheoryRaga Kumurdaki
Scale 2645Scale 2645: Raga Mruganandana, Ian Ring Music TheoryRaga Mruganandana
Scale 1109Scale 1109: Kataditonic, Ian Ring Music TheoryKataditonic

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.