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Scale 2045: "Katogyllian"

Scale 2045: Katogyllian, 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
Katogyllian

Analysis

Cardinality10 (decatonic)
Pitch Class Set{0,2,3,4,5,6,7,8,9,10}
Forte Number10-2
Rotational Symmetrynone
Reflection Axes0
Palindromicyes
Chiralityno
Hemitonia8 (multihemitonic)
Cohemitonia7 (multicohemitonic)
Imperfections2
Modes9
Prime?no
prime: 1535
Deep Scaleno
Interval Vector898884
Interval Spectrump8m8n8s9d8t4
Distribution Spectra<1> = {1,2}
<2> = {2,3,4}
<3> = {3,4,5}
<4> = {4,5,6}
<5> = {5,6,7}
<6> = {6,7,8}
<7> = {7,8,9}
<8> = {8,9,10}
<9> = {10,11}
Spectra Variation1.6
Maximally Evenno
Maximal Area Setyes
Interior Area2.866
Myhill Propertyno
Balancedno
Ridge Tones[0]
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{0,4,7}352.7
D{2,6,9}452.6
D♯{3,7,10}452.6
F{5,9,0}452.6
G♯{8,0,3}352.7
A♯{10,2,5}252.8
Minor Triadscm{0,3,7}452.6
dm{2,5,9}452.6
d♯m{3,6,10}452.6
fm{5,8,0}352.7
gm{7,10,2}252.8
am{9,0,4}352.7
Augmented TriadsC+{0,4,8}452.6
D+{2,6,10}452.6
Diminished Triads{0,3,6}252.8
{2,5,8}252.9
d♯°{3,6,9}252.8
{4,7,10}252.9
f♯°{6,9,0}252.8
{9,0,3}253
Parsimonious Voice Leading Between Common Triads of Scale 2045. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m C C cm->C D# D# cm->D# G# G# cm->G# C+ C+ C->C+ C->e° fm fm C+->fm C+->G# am am C+->am dm dm d°->dm d°->fm D D dm->D F F dm->F A# A# dm->A# D+ D+ D->D+ d#° d#° D->d#° f#° f#° D->f#° D+->d#m gm gm D+->gm D+->A# d#°->d#m d#m->D# D#->e° D#->gm fm->F F->f#° F->am G#->a° a°->am

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.

Diameter5
Radius5
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2045 can be rotated to make 9 other scales. The 1st mode is itself.

2nd mode:
Scale 1535
Scale 1535: Mixodyllian, Ian Ring Music TheoryMixodyllianThis is the prime mode
3rd mode:
Scale 2815
Scale 2815: Aeradyllian, Ian Ring Music TheoryAeradyllian
4th mode:
Scale 3455
Scale 3455: Ryptyllian, Ian Ring Music TheoryRyptyllian
5th mode:
Scale 3775
Scale 3775: Loptyllian, Ian Ring Music TheoryLoptyllian
6th mode:
Scale 3935
Scale 3935: Kataphyllian, Ian Ring Music TheoryKataphyllian
7th mode:
Scale 4015
Scale 4015: Phradyllian, Ian Ring Music TheoryPhradyllian
8th mode:
Scale 4055
Scale 4055: Dagyllian, Ian Ring Music TheoryDagyllian
9th mode:
Scale 4075
Scale 4075: Katyllian, Ian Ring Music TheoryKatyllian
10th mode:
Scale 4085
Scale 4085: Sydyllian, Ian Ring Music TheorySydyllian

Prime

The prime form of this scale is Scale 1535

Scale 1535Scale 1535: Mixodyllian, Ian Ring Music TheoryMixodyllian

Complement

The decatonic modal family [2045, 1535, 2815, 3455, 3775, 3935, 4015, 4055, 4075, 4085] (Forte: 10-2) is the complement of the modal family [5, 1025] (Forte: 2-2)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2045 is itself, because it is a palindromic scale!

Scale 2045Scale 2045: Katogyllian, Ian Ring Music TheoryKatogyllian

Transformations:

T0 2045  T0I 2045
T1 4090  T1I 4090
T2 4085  T2I 4085
T3 4075  T3I 4075
T4 4055  T4I 4055
T5 4015  T5I 4015
T6 3935  T6I 3935
T7 3775  T7I 3775
T8 3455  T8I 3455
T9 2815  T9I 2815
T10 1535  T10I 1535
T11 3070  T11I 3070

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 2047Scale 2047: Monatic, Ian Ring Music TheoryMonatic
Scale 2041Scale 2041: Aeolacrygic, Ian Ring Music TheoryAeolacrygic
Scale 2043Scale 2043: Maqam Tarzanuyn, Ian Ring Music TheoryMaqam Tarzanuyn
Scale 2037Scale 2037: Sythygic, Ian Ring Music TheorySythygic
Scale 2029Scale 2029: Kiourdi, Ian Ring Music TheoryKiourdi
Scale 2013Scale 2013: Mocrygic, Ian Ring Music TheoryMocrygic
Scale 1981Scale 1981: Houseini, Ian Ring Music TheoryHouseini
Scale 1917Scale 1917: Thydyllian, Ian Ring Music TheoryThydyllian
Scale 1789Scale 1789: Blues Enneatonic II, Ian Ring Music TheoryBlues Enneatonic II
Scale 1533Scale 1533: Katycrygic, Ian Ring Music TheoryKatycrygic
Scale 1021Scale 1021: Ladygic, Ian Ring Music TheoryLadygic
Scale 3069Scale 3069: Maqam Shawq Afza, Ian Ring Music TheoryMaqam Shawq Afza
Scale 4093Scale 4093: Aerycratic, Ian Ring Music TheoryAerycratic

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.