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Scale 1995: "Sideways Scale"

Scale 1995: Sideways Scale, 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

12Tone
Sideways Scale
Dozenal
Mecian
Western Modern
Dimygian
Zeitler
Aeolacryllic

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

8 (octatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,3,6,7,8,9,10}

Forte Number

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

8-13

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: 2685

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

5 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

3 (tricohemitonic)

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.

3

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.

7

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 735

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, 2, 3, 1, 1, 1, 1, 2]

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.

<5, 5, 6, 4, 5, 3>

Interval Spectrum

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

p5m4n6s5d5t3

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

Spectra Variation

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

2.5

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.

2.616

Polygon Perimeter

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

6.002

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.

(36, 72, 151)

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♯{3,7,10}341.91
F♯{6,10,1}341.91
G♯{8,0,3}242.27
Minor Triadscm{0,3,7}342
d♯m{3,6,10}441.82
f♯m{6,9,1}342
Diminished Triads{0,3,6}242.09
d♯°{3,6,9}242.09
f♯°{6,9,0}242.27
{7,10,1}242.18
{9,0,3}242.36
Parsimonious Voice Leading Between Common Triads of Scale 1995. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m D# D# cm->D# G# G# cm->G# d#° d#° d#°->d#m f#m f#m d#°->f#m d#m->D# F# F# d#m->F# D#->g° f#° f#° f#°->f#m f#°->a° f#m->F# F#->g° G#->a°

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 1995 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3045
Scale 3045: Raptyllic, Ian Ring Music TheoryRaptyllic
3rd mode:
Scale 1785
Scale 1785: Tharyllic, Ian Ring Music TheoryTharyllic
4th mode:
Scale 735
Scale 735: Sylyllic, Ian Ring Music TheorySylyllicThis is the prime mode
5th mode:
Scale 2415
Scale 2415: Lothyllic, Ian Ring Music TheoryLothyllic
6th mode:
Scale 3255
Scale 3255: Daryllic, Ian Ring Music TheoryDaryllic
7th mode:
Scale 3675
Scale 3675: Monyllic, Ian Ring Music TheoryMonyllic
8th mode:
Scale 3885
Scale 3885: Styryllic, Ian Ring Music TheoryStyryllic

Prime

The prime form of this scale is Scale 735

Scale 735Scale 735: Sylyllic, Ian Ring Music TheorySylyllic

Complement

The octatonic modal family [1995, 3045, 1785, 735, 2415, 3255, 3675, 3885] (Forte: 8-13) is the complement of the tetratonic modal family [75, 705, 1545, 2085] (Forte: 4-13)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1995 is 2685

Scale 2685Scale 2685: Ionoryllic, Ian Ring Music TheoryIonoryllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1995 is chiral, and its enantiomorph is scale 2685

Scale 2685Scale 2685: Ionoryllic, Ian Ring Music TheoryIonoryllic

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> 1995       T0I <11,0> 2685
T1 <1,1> 3990      T1I <11,1> 1275
T2 <1,2> 3885      T2I <11,2> 2550
T3 <1,3> 3675      T3I <11,3> 1005
T4 <1,4> 3255      T4I <11,4> 2010
T5 <1,5> 2415      T5I <11,5> 4020
T6 <1,6> 735      T6I <11,6> 3945
T7 <1,7> 1470      T7I <11,7> 3795
T8 <1,8> 2940      T8I <11,8> 3495
T9 <1,9> 1785      T9I <11,9> 2895
T10 <1,10> 3570      T10I <11,10> 1695
T11 <1,11> 3045      T11I <11,11> 3390
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2685      T0MI <7,0> 1995
T1M <5,1> 1275      T1MI <7,1> 3990
T2M <5,2> 2550      T2MI <7,2> 3885
T3M <5,3> 1005      T3MI <7,3> 3675
T4M <5,4> 2010      T4MI <7,4> 3255
T5M <5,5> 4020      T5MI <7,5> 2415
T6M <5,6> 3945      T6MI <7,6> 735
T7M <5,7> 3795      T7MI <7,7> 1470
T8M <5,8> 3495      T8MI <7,8> 2940
T9M <5,9> 2895      T9MI <7,9> 1785
T10M <5,10> 1695      T10MI <7,10> 3570
T11M <5,11> 3390      T11MI <7,11> 3045

The transformations that map this set to itself are: T0, T0MI

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 1993Scale 1993: Katoptian, Ian Ring Music TheoryKatoptian
Scale 1997Scale 1997: Raga Cintamani, Ian Ring Music TheoryRaga Cintamani
Scale 1999Scale 1999: Zacrygic, Ian Ring Music TheoryZacrygic
Scale 1987Scale 1987: Mexian, Ian Ring Music TheoryMexian
Scale 1991Scale 1991: Phryptyllic, Ian Ring Music TheoryPhryptyllic
Scale 2003Scale 2003: Podyllic, Ian Ring Music TheoryPodyllic
Scale 2011Scale 2011: Raphygic, Ian Ring Music TheoryRaphygic
Scale 2027Scale 2027: Boptygic, Ian Ring Music TheoryBoptygic
Scale 1931Scale 1931: Stogian, Ian Ring Music TheoryStogian
Scale 1963Scale 1963: Epocryllic, Ian Ring Music TheoryEpocryllic
Scale 1867Scale 1867: Solian, Ian Ring Music TheorySolian
Scale 1739Scale 1739: Mela Sadvidhamargini, Ian Ring Music TheoryMela Sadvidhamargini
Scale 1483Scale 1483: Mela Bhavapriya, Ian Ring Music TheoryMela Bhavapriya
Scale 971Scale 971: Mela Gavambodhi, Ian Ring Music TheoryMela Gavambodhi
Scale 3019Scale 3019: Subian, Ian Ring Music TheorySubian
Scale 4043Scale 4043: Phrocrygic, Ian Ring Music TheoryPhrocrygic

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.