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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

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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

Analysis

Cardinality6 (hexatonic)
Pitch Class Set{0,1,5,8,9,11}
Forte Number6-15
Rotational Symmetrynone
Reflection Axesnone
Palindromicno
Chiralityyes
enantiomorph: 2203
Hemitonia3 (trihemitonic)
Cohemitonia1 (uncohemitonic)
Imperfections4
Modes5
Prime?no
prime: 311
Deep Scaleno
Interval Vector323421
Interval Spectrump2m4n3s2d3t
Distribution Spectra<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 Variation3.333
Maximally Evenno
Maximal Area Setno
Interior Area2.116
Myhill Propertyno
Balancedno
Ridge Tonesnone
ProprietyImproper
Heliotonicno

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:

T0 2851  T0I 2203
T1 1607  T1I 311
T2 3214  T2I 622
T3 2333  T3I 1244
T4 571  T4I 2488
T5 1142  T5I 881
T6 2284  T6I 1762
T7 473  T7I 3524
T8 946  T8I 2953
T9 1892  T9I 1811
T10 3784  T10I 3622
T11 3473  T11I 3149

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, Ian Ring Music Theory
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, Ian Ring Music Theory
Scale 2835Scale 2835: Ionygimic, Ian Ring Music TheoryIonygimic
Scale 2883Scale 2883, Ian Ring Music Theory
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. The software used to generate this analysis is an open source project at GitHub. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. 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.