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Scale 2851: "Katoptimic"

Scale 2851: Katoptimic, 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
Katoptimic
Dozenal
Sabian

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

{0,1,5,8,9,11}

Forte Number

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

6-15

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: 2203

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.

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: 311

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, 4, 3, 1, 2, 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.

<3, 2, 3, 4, 2, 1>

Interval Spectrum

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

p2m4n3s2d3t

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

(21, 18, 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
Major TriadsC♯{1,5,8}221.2
F{5,9,0}221.2
Minor Triadsfm{5,8,0}321
Augmented TriadsC♯+{1,5,9}231.4
Diminished Triads{5,8,11}131.6
Parsimonious Voice Leading Between Common Triads of Scale 2851. Created by Ian Ring ©2019 C# C# C#+ C#+ C#->C#+ fm fm C#->fm F F C#+->F f°->fm fm->F

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
Radius2
Self-Centeredno
Central VerticesC♯, fm, F
Peripheral VerticesC♯+, f°

Modes

Modes are the rotational transformation of this scale. Scale 2851 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3473
Scale 3473: Lathimic, Ian Ring Music TheoryLathimic
3rd mode:
Scale 473
Scale 473: Aeralimic, Ian Ring Music TheoryAeralimic
4th mode:
Scale 571
Scale 571: Kynimic, Ian Ring Music TheoryKynimic
5th mode:
Scale 2333
Scale 2333: Stynimic, Ian Ring Music TheoryStynimic
6th mode:
Scale 1607
Scale 1607: Epytimic, Ian Ring Music TheoryEpytimic

Prime

The prime form of this scale is Scale 311

Scale 311Scale 311: Stagimic, Ian Ring Music TheoryStagimic

Complement

The hexatonic modal family [2851, 3473, 473, 571, 2333, 1607] (Forte: 6-15) is the complement of the hexatonic modal family [311, 881, 1811, 2203, 2953, 3149] (Forte: 6-15)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2851 is 2203

Scale 2203Scale 2203: Dorimic, Ian Ring Music TheoryDorimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2851 is chiral, and its enantiomorph is scale 2203

Scale 2203Scale 2203: Dorimic, Ian Ring Music TheoryDorimic

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> 2851       T0I <11,0> 2203
T1 <1,1> 1607      T1I <11,1> 311
T2 <1,2> 3214      T2I <11,2> 622
T3 <1,3> 2333      T3I <11,3> 1244
T4 <1,4> 571      T4I <11,4> 2488
T5 <1,5> 1142      T5I <11,5> 881
T6 <1,6> 2284      T6I <11,6> 1762
T7 <1,7> 473      T7I <11,7> 3524
T8 <1,8> 946      T8I <11,8> 2953
T9 <1,9> 1892      T9I <11,9> 1811
T10 <1,10> 3784      T10I <11,10> 3622
T11 <1,11> 3473      T11I <11,11> 3149
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 691      T0MI <7,0> 2473
T1M <5,1> 1382      T1MI <7,1> 851
T2M <5,2> 2764      T2MI <7,2> 1702
T3M <5,3> 1433      T3MI <7,3> 3404
T4M <5,4> 2866      T4MI <7,4> 2713
T5M <5,5> 1637      T5MI <7,5> 1331
T6M <5,6> 3274      T6MI <7,6> 2662
T7M <5,7> 2453      T7MI <7,7> 1229
T8M <5,8> 811      T8MI <7,8> 2458
T9M <5,9> 1622      T9MI <7,9> 821
T10M <5,10> 3244      T10MI <7,10> 1642
T11M <5,11> 2393      T11MI <7,11> 3284

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 2849Scale 2849: Rubian, Ian Ring Music TheoryRubian
Scale 2853Scale 2853: Baptimic, Ian Ring Music TheoryBaptimic
Scale 2855Scale 2855: Epocrain, Ian Ring Music TheoryEpocrain
Scale 2859Scale 2859: Phrycrian, Ian Ring Music TheoryPhrycrian
Scale 2867Scale 2867: Socrian, Ian Ring Music TheorySocrian
Scale 2819Scale 2819: Rujian, Ian Ring Music TheoryRujian
Scale 2835Scale 2835: Ionygimic, Ian Ring Music TheoryIonygimic
Scale 2883Scale 2883: Savian, Ian Ring Music TheorySavian
Scale 2915Scale 2915: Aeolydian, Ian Ring Music TheoryAeolydian
Scale 2979Scale 2979: Gyptian, Ian Ring Music TheoryGyptian
Scale 2595Scale 2595: Rolitonic, Ian Ring Music TheoryRolitonic
Scale 2723Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika
Scale 2339Scale 2339: Raga Kshanika, Ian Ring Music TheoryRaga Kshanika
Scale 3363Scale 3363: Rogimic, Ian Ring Music TheoryRogimic
Scale 3875Scale 3875: Aeryptian, Ian Ring Music TheoryAeryptian
Scale 803Scale 803: Loritonic, Ian Ring Music TheoryLoritonic
Scale 1827Scale 1827: Katygimic, Ian Ring Music TheoryKatygimic

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.