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Scale 1155

Scale 1155, 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).

Analysis

Cardinality

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

4 (tetratonic)

Pitch Class Set

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

{0,1,7,10}

Forte Number

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

4-13

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: 2085

Hemitonia

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

1 (unhemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

3

Prime Form

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

no
prime: 75

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, 6, 3, 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.

<1, 1, 2, 0, 1, 1>

Interval Spectrum

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

pn2sdt

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> = {1,2,3,6}
<2> = {3,5,7,9}
<3> = {6,9,10,11}

Spectra Variation

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

4

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.

1.183

Polygon Perimeter

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

4.932

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.

(4, 3, 18)

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
Diminished Triads{7,10,1}000

The following pitch classes are not present in any of the common triads: {0}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 1155 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 2625
Scale 2625, Ian Ring Music Theory
3rd mode:
Scale 105
Scale 105, Ian Ring Music Theory
4th mode:
Scale 525
Scale 525, Ian Ring Music Theory

Prime

The prime form of this scale is Scale 75

Scale 75Scale 75: Iloian, Ian Ring Music TheoryIloian

Complement

The tetratonic modal family [1155, 2625, 105, 525] (Forte: 4-13) is the complement of the octatonic modal family [735, 1785, 1995, 2415, 3045, 3255, 3675, 3885] (Forte: 8-13)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1155 is 2085

Scale 2085Scale 2085: Mogian, Ian Ring Music TheoryMogian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1155 is chiral, and its enantiomorph is scale 2085

Scale 2085Scale 2085: Mogian, Ian Ring Music TheoryMogian

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> 1155       T0I <11,0> 2085
T1 <1,1> 2310      T1I <11,1> 75
T2 <1,2> 525      T2I <11,2> 150
T3 <1,3> 1050      T3I <11,3> 300
T4 <1,4> 2100      T4I <11,4> 600
T5 <1,5> 105      T5I <11,5> 1200
T6 <1,6> 210      T6I <11,6> 2400
T7 <1,7> 420      T7I <11,7> 705
T8 <1,8> 840      T8I <11,8> 1410
T9 <1,9> 1680      T9I <11,9> 2820
T10 <1,10> 3360      T10I <11,10> 1545
T11 <1,11> 2625      T11I <11,11> 3090
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2085      T0MI <7,0> 1155
T1M <5,1> 75      T1MI <7,1> 2310
T2M <5,2> 150      T2MI <7,2> 525
T3M <5,3> 300      T3MI <7,3> 1050
T4M <5,4> 600      T4MI <7,4> 2100
T5M <5,5> 1200      T5MI <7,5> 105
T6M <5,6> 2400      T6MI <7,6> 210
T7M <5,7> 705      T7MI <7,7> 420
T8M <5,8> 1410      T8MI <7,8> 840
T9M <5,9> 2820      T9MI <7,9> 1680
T10M <5,10> 1545      T10MI <7,10> 3360
T11M <5,11> 3090      T11MI <7,11> 2625

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 1153Scale 1153: Choian, Ian Ring Music TheoryChoian
Scale 1157Scale 1157: Alkian, Ian Ring Music TheoryAlkian
Scale 1159Scale 1159: Hasian, Ian Ring Music TheoryHasian
Scale 1163Scale 1163: Raga Rukmangi, Ian Ring Music TheoryRaga Rukmangi
Scale 1171Scale 1171: Raga Manaranjani I, Ian Ring Music TheoryRaga Manaranjani I
Scale 1187Scale 1187: Kokin-joshi, Ian Ring Music TheoryKokin-joshi
Scale 1219Scale 1219: Hidian, Ian Ring Music TheoryHidian
Scale 1027Scale 1027: Geqian, Ian Ring Music TheoryGeqian
Scale 1091Scale 1091: Pedian, Ian Ring Music TheoryPedian
Scale 1283Scale 1283: Hurian, Ian Ring Music TheoryHurian
Scale 1411Scale 1411: Iroian, Ian Ring Music TheoryIroian
Scale 1667Scale 1667: Kekian, Ian Ring Music TheoryKekian
Scale 131Scale 131: Atoian, Ian Ring Music TheoryAtoian
Scale 643Scale 643: Duxian, Ian Ring Music TheoryDuxian
Scale 2179Scale 2179, Ian Ring Music Theory
Scale 3203Scale 3203: Etrian, Ian Ring Music TheoryEtrian

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.