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# Scale 1245: "Lathian" ### 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
Lathian
Dozenal
Bupian

## 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

{0,2,3,4,6,7,10}

#### Forte Number

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

7-26

#### 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: 1893

#### 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.

1 (uncohemitonic)

#### 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.

4

#### 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: 699

#### 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.

[2, 1, 1, 2, 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.

<3, 4, 4, 5, 3, 2>

#### Interval Spectrum

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

p3m5n4s4d3t2

#### 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> = {4,5,6,7}
<4> = {5,6,7,8}
<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.

2.286

#### 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.

2.549

#### Polygon Perimeter

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

5.967

#### 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.

(13, 34, 98)

## 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

D♯{3,7,10}421.25
d♯m{3,6,10}331.5
gm{7,10,2}231.75
{4,7,10}231.75

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 2 no D♯ C, D+

## Modes

Modes are the rotational transformation of this scale. Scale 1245 can be rotated to make 6 other scales. The 1st mode is itself.

 2nd mode:Scale 1335 Elephant Scale 3rd mode:Scale 2715 Kynian 4th mode:Scale 3405 Stynian 5th mode:Scale 1875 Persichetti Scale 6th mode:Scale 2985 Epacrian 7th mode:Scale 885 Sathian

## Prime

The prime form of this scale is Scale 699

 Scale 699 Aerothian

## Complement

The heptatonic modal family [1245, 1335, 2715, 3405, 1875, 2985, 885] (Forte: 7-26) is the complement of the pentatonic modal family [309, 849, 1101, 1299, 2697] (Forte: 5-26)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1245 is 1893

 Scale 1893 Ionylian

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1245 is chiral, and its enantiomorph is scale 1893

 Scale 1893 Ionylian

## 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> 1245       T0I <11,0> 1893
T1 <1,1> 2490      T1I <11,1> 3786
T2 <1,2> 885      T2I <11,2> 3477
T3 <1,3> 1770      T3I <11,3> 2859
T4 <1,4> 3540      T4I <11,4> 1623
T5 <1,5> 2985      T5I <11,5> 3246
T6 <1,6> 1875      T6I <11,6> 2397
T7 <1,7> 3750      T7I <11,7> 699
T8 <1,8> 3405      T8I <11,8> 1398
T9 <1,9> 2715      T9I <11,9> 2796
T10 <1,10> 1335      T10I <11,10> 1497
T11 <1,11> 2670      T11I <11,11> 2994
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3405      T0MI <7,0> 1623
T1M <5,1> 2715      T1MI <7,1> 3246
T2M <5,2> 1335      T2MI <7,2> 2397
T3M <5,3> 2670      T3MI <7,3> 699
T4M <5,4> 1245       T4MI <7,4> 1398
T5M <5,5> 2490      T5MI <7,5> 2796
T6M <5,6> 885      T6MI <7,6> 1497
T7M <5,7> 1770      T7MI <7,7> 2994
T8M <5,8> 3540      T8MI <7,8> 1893
T9M <5,9> 2985      T9MI <7,9> 3786
T10M <5,10> 1875      T10MI <7,10> 3477
T11M <5,11> 3750      T11MI <7,11> 2859

The transformations that map this set to itself are: T0, T4M

## 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 1247 Aeodyllic Scale 1241 Pygimic Scale 1243 Epylian Scale 1237 Salimic Scale 1229 Raga Simharava Scale 1261 Modified Blues Scale 1277 Zadyllic Scale 1181 Katagimic Scale 1213 Gyrian Scale 1117 Raptimic Scale 1373 Storian Scale 1501 Stygyllic Scale 1757 Kunian Scale 221 Biyian Scale 733 Donian Scale 2269 Pygian Scale 3293 Saryllic

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.