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Scale 3365: "Katolimic"

Scale 3365: Katolimic, 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

41161837294116105072918310504116183
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
Katolimic

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

{0,2,5,8,10,11}

Forte Number

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

6-Z45

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.

[5]

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.

no

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: 605

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

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, 3, 4, 2, 2, 2]

Interval Spectrum

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

p2m2n4s3d2t2

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

Spectra Variation

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

2.667

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.

2.366

Polygon Perimeter

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

5.864

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.

[10]

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

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 TriadsA♯{10,2,5}231.5
Minor Triadsfm{5,8,0}231.5
Diminished Triads{2,5,8}231.5
{5,8,11}231.5
g♯°{8,11,2}231.5
{11,2,5}231.5
Parsimonious Voice Leading Between Common Triads of Scale 3365. Created by Ian Ring ©2019 fm fm d°->fm A# A# d°->A# f°->fm g#° g#° f°->g#° g#°->b° A#->b°

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
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3365 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1865
Scale 1865: Thagimic, Ian Ring Music TheoryThagimic
3rd mode:
Scale 745
Scale 745: Kolimic, Ian Ring Music TheoryKolimic
4th mode:
Scale 605
Scale 605: Dycrimic, Ian Ring Music TheoryDycrimicThis is the prime mode
5th mode:
Scale 1175
Scale 1175: Epycrimic, Ian Ring Music TheoryEpycrimic
6th mode:
Scale 2635
Scale 2635: Gocrimic, Ian Ring Music TheoryGocrimic

Prime

The prime form of this scale is Scale 605

Scale 605Scale 605: Dycrimic, Ian Ring Music TheoryDycrimic

Complement

The hexatonic modal family [3365, 1865, 745, 605, 1175, 2635] (Forte: 6-Z45) is the complement of the hexatonic modal family [365, 1115, 1675, 1745, 2605, 2885] (Forte: 6-Z23)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3365 is 1175

Scale 1175Scale 1175: Epycrimic, Ian Ring Music TheoryEpycrimic

Transformations:

T0 3365  T0I 1175
T1 2635  T1I 2350
T2 1175  T2I 605
T3 2350  T3I 1210
T4 605  T4I 2420
T5 1210  T5I 745
T6 2420  T6I 1490
T7 745  T7I 2980
T8 1490  T8I 1865
T9 2980  T9I 3730
T10 1865  T10I 3365
T11 3730  T11I 2635

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 3367Scale 3367: Moptian, Ian Ring Music TheoryMoptian
Scale 3361Scale 3361, Ian Ring Music Theory
Scale 3363Scale 3363: Rogimic, Ian Ring Music TheoryRogimic
Scale 3369Scale 3369: Mixolimic, Ian Ring Music TheoryMixolimic
Scale 3373Scale 3373: Lodian, Ian Ring Music TheoryLodian
Scale 3381Scale 3381: Katanian, Ian Ring Music TheoryKatanian
Scale 3333Scale 3333, Ian Ring Music Theory
Scale 3349Scale 3349: Aeolocrimic, Ian Ring Music TheoryAeolocrimic
Scale 3397Scale 3397: Sydimic, Ian Ring Music TheorySydimic
Scale 3429Scale 3429: Marian, Ian Ring Music TheoryMarian
Scale 3493Scale 3493: Rathian, Ian Ring Music TheoryRathian
Scale 3109Scale 3109, Ian Ring Music Theory
Scale 3237Scale 3237: Raga Brindabani Sarang, Ian Ring Music TheoryRaga Brindabani Sarang
Scale 3621Scale 3621: Gylimic, Ian Ring Music TheoryGylimic
Scale 3877Scale 3877: Thanian, Ian Ring Music TheoryThanian
Scale 2341Scale 2341: Raga Priyadharshini, Ian Ring Music TheoryRaga Priyadharshini
Scale 2853Scale 2853: Baptimic, Ian Ring Music TheoryBaptimic
Scale 1317Scale 1317: Chaio, Ian Ring Music TheoryChaio

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.