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# Scale 1231: "Logian" ### 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).

Zeitler
Logian
Dozenal
Hilian

## Analysis

#### Cardinality

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

7 (heptatonic)

#### Pitch Class Set

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

{0,1,2,3,6,7,10}

#### Forte Number

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

7-Z18

#### 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: 3685

#### Hemitonia

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

4 (multihemitonic)

#### 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.

6

#### Prime Form

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

no
prime: 755

#### 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, 1, 1, 3, 1, 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.

<4, 3, 4, 4, 4, 2>

#### Interval Spectrum

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

p4m4n4s3d4t2

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

#### Spectra Variation

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

2.571

#### 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.

2.433

#### Polygon Perimeter

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

5.899

#### 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.

(23, 34, 100)

## Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony. 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

F♯{6,10,1}242
d♯m{3,6,10}331.5
gm{7,10,2}331.5
{7,10,1}242

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.

Diameter 4 3 no D+, d♯m, D♯, gm c°, cm, F♯, g°

## Modes

Modes are the rotational transformation of this scale. Scale 1231 can be rotated to make 6 other scales. The 1st mode is itself.

 2nd mode:Scale 2663 Lalian 3rd mode:Scale 3379 Verdi's Scala Enigmatica Descending 4th mode:Scale 3737 Phrocrian 5th mode:Scale 979 Mela Dhavalambari 6th mode:Scale 2537 Laptian 7th mode:Scale 829 Lygian

## Prime

The prime form of this scale is Scale 755

 Scale 755 Phrythian

## Complement

The heptatonic modal family [1231, 2663, 3379, 3737, 979, 2537, 829] (Forte: 7-Z18) is the complement of the pentatonic modal family [179, 779, 1633, 2137, 2437] (Forte: 5-Z18)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1231 is 3685

 Scale 3685 Kodian

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1231 is chiral, and its enantiomorph is scale 3685

 Scale 3685 Kodian

## 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> 1231       T0I <11,0> 3685
T1 <1,1> 2462      T1I <11,1> 3275
T2 <1,2> 829      T2I <11,2> 2455
T3 <1,3> 1658      T3I <11,3> 815
T4 <1,4> 3316      T4I <11,4> 1630
T5 <1,5> 2537      T5I <11,5> 3260
T6 <1,6> 979      T6I <11,6> 2425
T7 <1,7> 1958      T7I <11,7> 755
T8 <1,8> 3916      T8I <11,8> 1510
T9 <1,9> 3737      T9I <11,9> 3020
T10 <1,10> 3379      T10I <11,10> 1945
T11 <1,11> 2663      T11I <11,11> 3890
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3181      T0MI <7,0> 1735
T1M <5,1> 2267      T1MI <7,1> 3470
T2M <5,2> 439      T2MI <7,2> 2845
T3M <5,3> 878      T3MI <7,3> 1595
T4M <5,4> 1756      T4MI <7,4> 3190
T5M <5,5> 3512      T5MI <7,5> 2285
T6M <5,6> 2929      T6MI <7,6> 475
T7M <5,7> 1763      T7MI <7,7> 950
T8M <5,8> 3526      T8MI <7,8> 1900
T9M <5,9> 2957      T9MI <7,9> 3800
T10M <5,10> 1819      T10MI <7,10> 3505
T11M <5,11> 3638      T11MI <7,11> 2915

The transformations that map this set to itself are: T0

## 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 1229 Raga Simharava Scale 1227 Thacrimic Scale 1223 Phryptimic Scale 1239 Epaptian Scale 1247 Aeodyllic Scale 1263 Stynyllic Scale 1167 Aerodimic Scale 1199 Magian Scale 1103 Lynimic Scale 1359 Aerygian Scale 1487 Mothyllic Scale 1743 Epigyllic Scale 207 Beqian Scale 719 Kanian Scale 2255 Dylian Scale 3279 Pythyllic

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.