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Scale 2925: "Diminished"

Scale 2925: Diminished, Ian Ring Music Theory
This highly symmetrical scale is also called "Octatonic", but in modern times it is usually called "Diminished" to differentiate it from the "other" Octatonic scale, and its only mode, Scale 1755. Because it was associated with compositions by Dutch Composer Willem Pijper (1894-1947), it has been known as "Pijper's Scale", and notable composer/condustor Anthon van der Horst (1899-1965) called it "Modus Conjunctus".

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

Western Modern
Diminished
Octatonic
Whole-Half Step Scale
Auxiliary Diminished
Named After Composers
Modus Conjunctus
Pijper's Scale
Messiaen
Messiaen Mode 2 Inverse
Exoticisms
Arabian A
Zeitler
Epadyllic

Analysis

Cardinality8 (octatonic)
Pitch Class Set{0,2,3,5,6,8,9,11}
Forte Number8-28
Rotational Symmetry3, 6, 9 semitones
Reflection Axes1, 2.5, 4, 5.5
Palindromicno
Chiralityno
Hemitonia4 (multihemitonic)
Cohemitonia0 (ancohemitonic)
Imperfections4
Modes1
Prime?no
prime: 1755
Deep Scaleno
Interval Vector448444
Interval Spectrump4m4n8s4d4t4
Distribution Spectra<1> = {1,2}
<2> = {3}
<3> = {4,5}
<4> = {6}
<5> = {7,8}
<6> = {9}
<7> = {10,11}
Spectra Variation0.5
Maximally Evenyes
Maximal Area Setyes
Interior Area2.732
Myhill Propertyno
Balancedyes
Ridge Tones[2,5,8,11]
ProprietyStrictly Proper
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 TriadsD{2,6,9}442.13
F{5,9,0}442.13
G♯{8,0,3}442.13
B{11,3,6}442.13
Minor Triadsdm{2,5,9}442.13
fm{5,8,0}442.13
g♯m{8,11,3}442.13
bm{11,2,6}442.13
Diminished Triads{0,3,6}242.38
{2,5,8}242.38
d♯°{3,6,9}242.38
{5,8,11}242.38
f♯°{6,9,0}242.38
g♯°{8,11,2}242.38
{9,0,3}242.38
{11,2,5}242.38
Parsimonious Voice Leading Between Common Triads of Scale 2925. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B dm dm d°->dm fm fm d°->fm D D dm->D F F dm->F dm->b° d#° d#° D->d#° f#° f#° D->f#° bm bm D->bm d#°->B f°->fm g#m g#m f°->g#m fm->F fm->G# F->f#° F->a° g#° g#° g#°->g#m g#°->bm g#m->G# g#m->B G#->a° b°->bm bm->B

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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2925 can be rotated to make 1 other scales. The 1st mode is itself.

2nd mode:
Scale 1755
Scale 1755: Octatonic, Ian Ring Music TheoryOctatonicThis is the prime mode

Prime

The prime form of this scale is Scale 1755

Scale 1755Scale 1755: Octatonic, Ian Ring Music TheoryOctatonic

Complement

The octatonic modal family [2925, 1755] (Forte: 8-28) is the complement of the tetratonic modal family [585] (Forte: 4-28)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2925 is 1755

Scale 1755Scale 1755: Octatonic, Ian Ring Music TheoryOctatonic

Transformations:

T0 2925  T0I 1755
T1 1755  T1I 3510
T2 3510  T2I 2925
T3 2925  T3I 1755
T4 1755  T4I 3510
T5 3510  T5I 2925
T6 2925  T6I 1755
T7 1755  T7I 3510
T8 3510  T8I 2925
T9 2925  T9I 1755
T10 1755  T10I 3510
T11 3510  T11I 2925

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 2927Scale 2927: Rodygic, Ian Ring Music TheoryRodygic
Scale 2921Scale 2921: Pogian, Ian Ring Music TheoryPogian
Scale 2923Scale 2923: Baryllic, Ian Ring Music TheoryBaryllic
Scale 2917Scale 2917: Nohkan Flute Scale, Ian Ring Music TheoryNohkan Flute Scale
Scale 2933Scale 2933, Ian Ring Music Theory
Scale 2941Scale 2941: Laptygic, Ian Ring Music TheoryLaptygic
Scale 2893Scale 2893: Lylian, Ian Ring Music TheoryLylian
Scale 2909Scale 2909: Mocryllic, Ian Ring Music TheoryMocryllic
Scale 2861Scale 2861: Katothian, Ian Ring Music TheoryKatothian
Scale 2989Scale 2989: Bebop Minor, Ian Ring Music TheoryBebop Minor
Scale 3053Scale 3053: Zycrygic, Ian Ring Music TheoryZycrygic
Scale 2669Scale 2669: Jeths' Mode, Ian Ring Music TheoryJeths' Mode
Scale 2797Scale 2797: Stalyllic, Ian Ring Music TheoryStalyllic
Scale 2413Scale 2413: Locrian Natural 2, Ian Ring Music TheoryLocrian Natural 2
Scale 3437Scale 3437, Ian Ring Music Theory
Scale 3949Scale 3949: Koptygic, Ian Ring Music TheoryKoptygic
Scale 877Scale 877: Moravian Pistalkova, Ian Ring Music TheoryMoravian Pistalkova
Scale 1901Scale 1901: Ionidyllic, Ian Ring Music TheoryIonidyllic

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.