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Scale 1357: "Takemitsu Linea Mode 2"

Scale 1357: Takemitsu Linea Mode 2, 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

Named After Composers
Takemitsu Linea Mode 2
Takemitsu Tree Line Mode 2
Zeitler
Katonimic
Dozenal
Ihoian

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

{0,2,3,6,8,10}

Forte Number

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

6-34

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: 1621

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.

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.

5

Prime Form

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

no
prime: 683

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

Interval Spectrum

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

p2m4n2s4dt2

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

Spectra Variation

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

1.667

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.

2.482

Polygon Perimeter

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

5.932

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".

Proper

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.

(0, 4, 45)

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 TriadsG♯{8,0,3}131.5
Minor Triadsd♯m{3,6,10}221
Augmented TriadsD+{2,6,10}131.5
Diminished Triads{0,3,6}221
Parsimonious Voice Leading Between Common Triads of Scale 1357. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# D+ D+ D+->d#m

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.

Diameter3
Radius2
Self-Centeredno
Central Verticesc°, d♯m
Peripheral VerticesD+, G♯

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Augmented: {2, 6, 10}
Major: {8, 0, 3}

Modes

Modes are the rotational transformation of this scale. Scale 1357 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1363
Scale 1363: Gygimic, Ian Ring Music TheoryGygimic
3rd mode:
Scale 2729
Scale 2729: Aeragimic, Ian Ring Music TheoryAeragimic
4th mode:
Scale 853
Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
5th mode:
Scale 1237
Scale 1237: Salimic, Ian Ring Music TheorySalimic
6th mode:
Scale 1333
Scale 1333: Lyptimic, Ian Ring Music TheoryLyptimic

Prime

The prime form of this scale is Scale 683

Scale 683Scale 683: Stogimic, Ian Ring Music TheoryStogimic

Complement

The hexatonic modal family [1357, 1363, 2729, 853, 1237, 1333] (Forte: 6-34) is the complement of the hexatonic modal family [683, 1369, 1381, 1429, 1621, 2389] (Forte: 6-34)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1357 is 1621

Scale 1621Scale 1621: Scriabin's Prometheus, Ian Ring Music TheoryScriabin's Prometheus

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1357 is chiral, and its enantiomorph is scale 1621

Scale 1621Scale 1621: Scriabin's Prometheus, Ian Ring Music TheoryScriabin's Prometheus

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> 1357       T0I <11,0> 1621
T1 <1,1> 2714      T1I <11,1> 3242
T2 <1,2> 1333      T2I <11,2> 2389
T3 <1,3> 2666      T3I <11,3> 683
T4 <1,4> 1237      T4I <11,4> 1366
T5 <1,5> 2474      T5I <11,5> 2732
T6 <1,6> 853      T6I <11,6> 1369
T7 <1,7> 1706      T7I <11,7> 2738
T8 <1,8> 3412      T8I <11,8> 1381
T9 <1,9> 2729      T9I <11,9> 2762
T10 <1,10> 1363      T10I <11,10> 1429
T11 <1,11> 2726      T11I <11,11> 2858
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1117      T0MI <7,0> 1861
T1M <5,1> 2234      T1MI <7,1> 3722
T2M <5,2> 373      T2MI <7,2> 3349
T3M <5,3> 746      T3MI <7,3> 2603
T4M <5,4> 1492      T4MI <7,4> 1111
T5M <5,5> 2984      T5MI <7,5> 2222
T6M <5,6> 1873      T6MI <7,6> 349
T7M <5,7> 3746      T7MI <7,7> 698
T8M <5,8> 3397      T8MI <7,8> 1396
T9M <5,9> 2699      T9MI <7,9> 2792
T10M <5,10> 1303      T10MI <7,10> 1489
T11M <5,11> 2606      T11MI <7,11> 2978

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 1359Scale 1359: Aerygian, Ian Ring Music TheoryAerygian
Scale 1353Scale 1353: Raga Harikauns, Ian Ring Music TheoryRaga Harikauns
Scale 1355Scale 1355: Raga Bhavani, Ian Ring Music TheoryRaga Bhavani
Scale 1349Scale 1349: Tholitonic, Ian Ring Music TheoryTholitonic
Scale 1365Scale 1365: Whole Tone, Ian Ring Music TheoryWhole Tone
Scale 1373Scale 1373: Storian, Ian Ring Music TheoryStorian
Scale 1389Scale 1389: Minor Locrian, Ian Ring Music TheoryMinor Locrian
Scale 1293Scale 1293: Huxian, Ian Ring Music TheoryHuxian
Scale 1325Scale 1325: Phradimic, Ian Ring Music TheoryPhradimic
Scale 1421Scale 1421: Raga Trimurti, Ian Ring Music TheoryRaga Trimurti
Scale 1485Scale 1485: Minor Romani, Ian Ring Music TheoryMinor Romani
Scale 1101Scale 1101: Stothitonic, Ian Ring Music TheoryStothitonic
Scale 1229Scale 1229: Raga Simharava, Ian Ring Music TheoryRaga Simharava
Scale 1613Scale 1613: Thylimic, Ian Ring Music TheoryThylimic
Scale 1869Scale 1869: Katyrian, Ian Ring Music TheoryKatyrian
Scale 333Scale 333: Bogitonic, Ian Ring Music TheoryBogitonic
Scale 845Scale 845: Raga Neelangi, Ian Ring Music TheoryRaga Neelangi
Scale 2381Scale 2381: Takemitsu Linea Mode 1, Ian Ring Music TheoryTakemitsu Linea Mode 1
Scale 3405Scale 3405: Stynian, Ian Ring Music TheoryStynian

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.