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Scale 3155: "Ladimic"

Scale 3155: Ladimic, 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

Zeitler
Ladimic

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

{0,1,4,6,10,11}

Forte Number

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

6-Z41

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: 2375

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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.

5

Prime Form

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

no
prime: 335

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 3, 2, 4, 1, 1]

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.

<3, 3, 2, 2, 3, 2>

Interval Spectrum

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

p3m2n2s3d3t2

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> = {1,2,3,4}
<2> = {2,4,5,6}
<3> = {3,5,6,7,9}
<4> = {6,7,8,10}
<5> = {8,9,10,11}

Spectra Variation

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

3.333

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.

2.116

Polygon Perimeter

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

5.699

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.

(20, 17, 64)

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 TriadsF♯{6,10,1}110.5
Diminished Triadsa♯°{10,1,4}110.5

The following pitch classes are not present in any of the common triads: {0,11}

Parsimonious Voice Leading Between Common Triads of Scale 3155. Created by Ian Ring ©2019 F# F# a#° a#° F#->a#°

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3155 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3625
Scale 3625: Podimic, Ian Ring Music TheoryPodimic
3rd mode:
Scale 965
Scale 965: Ionothimic, Ian Ring Music TheoryIonothimic
4th mode:
Scale 1265
Scale 1265: Pynimic, Ian Ring Music TheoryPynimic
5th mode:
Scale 335
Scale 335: Zanimic, Ian Ring Music TheoryZanimicThis is the prime mode
6th mode:
Scale 2215
Scale 2215: Ranimic, Ian Ring Music TheoryRanimic

Prime

The prime form of this scale is Scale 335

Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic

Complement

The hexatonic modal family [3155, 3625, 965, 1265, 335, 2215] (Forte: 6-Z41) is the complement of the hexatonic modal family [215, 1475, 1805, 2155, 2785, 3125] (Forte: 6-Z12)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3155 is 2375

Scale 2375Scale 2375: Aeolaptimic, Ian Ring Music TheoryAeolaptimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3155 is chiral, and its enantiomorph is scale 2375

Scale 2375Scale 2375: Aeolaptimic, Ian Ring Music TheoryAeolaptimic

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> 3155       T0I <11,0> 2375
T1 <1,1> 2215      T1I <11,1> 655
T2 <1,2> 335      T2I <11,2> 1310
T3 <1,3> 670      T3I <11,3> 2620
T4 <1,4> 1340      T4I <11,4> 1145
T5 <1,5> 2680      T5I <11,5> 2290
T6 <1,6> 1265      T6I <11,6> 485
T7 <1,7> 2530      T7I <11,7> 970
T8 <1,8> 965      T8I <11,8> 1940
T9 <1,9> 1930      T9I <11,9> 3880
T10 <1,10> 3860      T10I <11,10> 3665
T11 <1,11> 3625      T11I <11,11> 3235
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 485      T0MI <7,0> 1265
T1M <5,1> 970      T1MI <7,1> 2530
T2M <5,2> 1940      T2MI <7,2> 965
T3M <5,3> 3880      T3MI <7,3> 1930
T4M <5,4> 3665      T4MI <7,4> 3860
T5M <5,5> 3235      T5MI <7,5> 3625
T6M <5,6> 2375      T6MI <7,6> 3155
T7M <5,7> 655      T7MI <7,7> 2215
T8M <5,8> 1310      T8MI <7,8> 335
T9M <5,9> 2620      T9MI <7,9> 670
T10M <5,10> 1145      T10MI <7,10> 1340
T11M <5,11> 2290      T11MI <7,11> 2680

The transformations that map this set to itself are: T0, T6MI

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 3153Scale 3153: Zathitonic, Ian Ring Music TheoryZathitonic
Scale 3157Scale 3157: Zyptimic, Ian Ring Music TheoryZyptimic
Scale 3159Scale 3159: Stocrian, Ian Ring Music TheoryStocrian
Scale 3163Scale 3163: Rogian, Ian Ring Music TheoryRogian
Scale 3139Scale 3139, Ian Ring Music Theory
Scale 3147Scale 3147: Ryrimic, Ian Ring Music TheoryRyrimic
Scale 3171Scale 3171: Zythimic, Ian Ring Music TheoryZythimic
Scale 3187Scale 3187: Koptian, Ian Ring Music TheoryKoptian
Scale 3091Scale 3091, Ian Ring Music Theory
Scale 3123Scale 3123, Ian Ring Music Theory
Scale 3219Scale 3219: Ionaphimic, Ian Ring Music TheoryIonaphimic
Scale 3283Scale 3283: Mela Visvambhari, Ian Ring Music TheoryMela Visvambhari
Scale 3411Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
Scale 3667Scale 3667: Kaptian, Ian Ring Music TheoryKaptian
Scale 2131Scale 2131, Ian Ring Music Theory
Scale 2643Scale 2643: Raga Hamsanandi, Ian Ring Music TheoryRaga Hamsanandi
Scale 1107Scale 1107: Mogitonic, Ian Ring Music TheoryMogitonic

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.