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Scale 549: "Lahuzu 4 Tone Type 4"

Scale 549: Lahuzu 4 Tone Type 4, 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

Southeast Asia
Lahuzu 4 Tone Type 4
Zeitler
Rothic
Dozenal
DIRian

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

{0,2,5,9}

Forte Number

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

4-26

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.

[1]

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.

no

Hemitonia

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

0 (anhemitonic)

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.

2

Modes

Modes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

4

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 297

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, an indicator of maximum hierarchization.

no

Interval Structure

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

[2, 3, 4, 3]

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.

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

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0, 0.333, 0.5, 0.333, 0.667, 0>

Interval Spectrum

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

p2mn2s

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

Spectra Variation

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

1.5

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.

1.866

Polygon Perimeter

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

5.56

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

[2]

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".

Strictly 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, 0, 14)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

1

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.222

Generator

This scale has no generator.

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{5,9,0}110.5
Minor Triadsdm{2,5,9}110.5
Parsimonious Voice Leading Between Common Triads of Scale 549. Created by Ian Ring ©2019 dm dm F F dm->F

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 549 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 1161
Scale 1161: Bi Yu, Ian Ring Music TheoryBi Yu
3rd mode:
Scale 657
Scale 657: Lahuzu 4 Tone Type 3, Ian Ring Music TheoryLahuzu 4 Tone Type 3
4th mode:
Scale 297
Scale 297: Karen 4 Tone Type 3, Ian Ring Music TheoryKaren 4 Tone Type 3This is the prime mode

Prime

The prime form of this scale is Scale 297

Scale 297Scale 297: Karen 4 Tone Type 3, Ian Ring Music TheoryKaren 4 Tone Type 3

Complement

The tetratonic modal family [549, 1161, 657, 297] (Forte: 4-26) is the complement of the octatonic modal family [1467, 1719, 1773, 1899, 2781, 2907, 2997, 3501] (Forte: 8-26)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 549 is 1161

Scale 1161Scale 1161: Bi Yu, Ian Ring Music TheoryBi Yu

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> 549       T0I <11,0> 1161
T1 <1,1> 1098      T1I <11,1> 2322
T2 <1,2> 2196      T2I <11,2> 549
T3 <1,3> 297      T3I <11,3> 1098
T4 <1,4> 594      T4I <11,4> 2196
T5 <1,5> 1188      T5I <11,5> 297
T6 <1,6> 2376      T6I <11,6> 594
T7 <1,7> 657      T7I <11,7> 1188
T8 <1,8> 1314      T8I <11,8> 2376
T9 <1,9> 2628      T9I <11,9> 657
T10 <1,10> 1161      T10I <11,10> 1314
T11 <1,11> 2322      T11I <11,11> 2628
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1539      T0MI <7,0> 2061
T1M <5,1> 3078      T1MI <7,1> 27
T2M <5,2> 2061      T2MI <7,2> 54
T3M <5,3> 27      T3MI <7,3> 108
T4M <5,4> 54      T4MI <7,4> 216
T5M <5,5> 108      T5MI <7,5> 432
T6M <5,6> 216      T6MI <7,6> 864
T7M <5,7> 432      T7MI <7,7> 1728
T8M <5,8> 864      T8MI <7,8> 3456
T9M <5,9> 1728      T9MI <7,9> 2817
T10M <5,10> 3456      T10MI <7,10> 1539
T11M <5,11> 2817      T11MI <7,11> 3078

The transformations that map this set to itself are: T0, T2I

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 551Scale 551: Aeoloditonic, Ian Ring Music TheoryAeoloditonic
Scale 545Scale 545: Karen 3 Tone Type 4, Ian Ring Music TheoryKaren 3 Tone Type 4
Scale 547Scale 547: Pyrric, Ian Ring Music TheoryPyrric
Scale 553Scale 553: Phradic, Ian Ring Music TheoryPhradic
Scale 557Scale 557: Raga Abhogi, Ian Ring Music TheoryRaga Abhogi
Scale 565Scale 565: Aeolyphritonic, Ian Ring Music TheoryAeolyphritonic
Scale 517Scale 517: ALUian, Ian Ring Music TheoryALUian
Scale 533Scale 533: DEHian, Ian Ring Music TheoryDEHian
Scale 581Scale 581: Bolic, Ian Ring Music TheoryBolic
Scale 613Scale 613: Phralitonic, Ian Ring Music TheoryPhralitonic
Scale 677Scale 677: Scottish Pentatonic, Ian Ring Music TheoryScottish Pentatonic
Scale 805Scale 805: Rothitonic, Ian Ring Music TheoryRothitonic
Scale 37Scale 37: Akha 3 Tone, Ian Ring Music TheoryAkha 3 Tone
Scale 293Scale 293: Raga Haripriya, Ian Ring Music TheoryRaga Haripriya
Scale 1061Scale 1061: Karen 4 Tone Type 4, Ian Ring Music TheoryKaren 4 Tone Type 4
Scale 1573Scale 1573: Raga Guhamanohari, Ian Ring Music TheoryRaga Guhamanohari
Scale 2597Scale 2597: Raga Rasranjani, Ian Ring Music TheoryRaga Rasranjani

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.