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Scale 1829: "Pathimic"

Scale 1829: Pathimic, 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
Pathimic
Dozenal
LEFian

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

 {0,2,5,8,9,10}

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-Z46

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: 1181

Hemitonia

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

 2 (dihemitonic)

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.

 3

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.

 6

Prime Form

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

 no
prime: 599

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, 3, 1, 1, 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.

 <2, 3, 3, 3, 3, 1>

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.

<0.4, 0.5, 0.6, 0.25, 0.6, 0.333>

Interval Spectrum

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

 p3m3n3s3d2t

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

Spectra Variation

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

 2.667

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.

2.366

Polygon Perimeter

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

5.864

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.

 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.

(12, 17, 65)

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.

0.554

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.

0.133

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}221.2
A♯{10,2,5}131.6
Minor Triadsdm{2,5,9}321
fm{5,8,0}231.4
Diminished Triadsd°{2,5,8}221.2
Parsimonious Voice Leading Between Common Triads of Scale 1829. Created by Ian Ring ©2019 d° d° dm dm d°->dm fm fm d°->fm F F dm->F A# A# dm->A# fm->F

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.

Diameter3
Radius2
Self-Centeredno
Central Verticesd°, dm, F
Peripheral Verticesfm, A♯

Modes

Modes are the rotational transformation of this scale. Scale 1829 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1481		Scale 1481: Zagimic, Ian Ring Music TheoryZagimic
3rd mode:
Scale 697		Scale 697: Lagimic, Ian Ring Music TheoryLagimic
4th mode:
Scale 599		Scale 599: Thyrimic, Ian Ring Music TheoryThyrimicThis is the prime mode
5th mode:
Scale 2347		Scale 2347: Raga Viyogavarali, Ian Ring Music TheoryRaga Viyogavarali
6th mode:
Scale 3221		Scale 3221: Bycrimic, Ian Ring Music TheoryBycrimic

Prime

The prime form of this scale is Scale 599

Scale 599Scale 599: Thyrimic, Ian Ring Music TheoryThyrimic

Complement

The hexatonic modal family [1829, 1481, 697, 599, 2347, 3221] (Forte: 6-Z46) is the complement of the hexatonic modal family [347, 1457, 1579, 1733, 2221, 2837] (Forte: 6-Z24)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1829 is 1181

Scale 1181Scale 1181: Katagimic, Ian Ring Music TheoryKatagimic

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
111010010011101829k = 1h = 1
211010010011101829k = 2h = 1
311010010011101829k = 3h = 1
42(10)(10)0(10)011(10)1829k = 4h = 2
52(10)(10)0(10)011(10)1829k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1829 is chiral, and its enantiomorph is scale 1181

Scale 1181Scale 1181: Katagimic, Ian Ring Music TheoryKatagimic

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 1829       T0I <11,0> 1181
T1 <1,1> 3658      T1I <11,1> 2362
T2 <1,2> 3221      T2I <11,2> 629
T3 <1,3> 2347      T3I <11,3> 1258
T4 <1,4> 599      T4I <11,4> 2516
T5 <1,5> 1198      T5I <11,5> 937
T6 <1,6> 2396      T6I <11,6> 1874
T7 <1,7> 697      T7I <11,7> 3748
T8 <1,8> 1394      T8I <11,8> 3401
T9 <1,9> 2788      T9I <11,9> 2707
T10 <1,10> 1481      T10I <11,10> 1319
T11 <1,11> 2962      T11I <11,11> 2638
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1559      T0MI <7,0> 3341
T1M <5,1> 3118      T1MI <7,1> 2587
T2M <5,2> 2141      T2MI <7,2> 1079
T3M <5,3> 187      T3MI <7,3> 2158
T4M <5,4> 374      T4MI <7,4> 221
T5M <5,5> 748      T5MI <7,5> 442
T6M <5,6> 1496      T6MI <7,6> 884
T7M <5,7> 2992      T7MI <7,7> 1768
T8M <5,8> 1889      T8MI <7,8> 3536
T9M <5,9> 3778      T9MI <7,9> 2977
T10M <5,10> 3461      T10MI <7,10> 1859
T11M <5,11> 2827      T11MI <7,11> 3718

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 1831Scale 1831: Pothian, Ian Ring Music TheoryPothian
Scale 1825Scale 1825: LECian, Ian Ring Music TheoryLECian
Scale 1827Scale 1827: Katygimic, Ian Ring Music TheoryKatygimic
Scale 1833Scale 1833: Ionacrimic, Ian Ring Music TheoryIonacrimic
Scale 1837Scale 1837: Dalian, Ian Ring Music TheoryDalian
Scale 1845Scale 1845: Mixolydian Augmented, Ian Ring Music TheoryMixolydian Augmented
Scale 1797Scale 1797: LALian, Ian Ring Music TheoryLALian
Scale 1813Scale 1813: Katothimic, Ian Ring Music TheoryKatothimic
Scale 1861Scale 1861: Phrygimic, Ian Ring Music TheoryPhrygimic
Scale 1893Scale 1893: Ionylian, Ian Ring Music TheoryIonylian
Scale 1957Scale 1957: Pyrian, Ian Ring Music TheoryPyrian
Scale 1573Scale 1573: Raga Guhamanohari, Ian Ring Music TheoryRaga Guhamanohari
Scale 1701Scale 1701: Mixolydian Hexatonic, Ian Ring Music TheoryMixolydian Hexatonic
Scale 1317Scale 1317: Chaio, Ian Ring Music TheoryChaio
Scale 805Scale 805: Rothitonic, Ian Ring Music TheoryRothitonic
Scale 2853Scale 2853: Baptimic, Ian Ring Music TheoryBaptimic
Scale 3877Scale 3877: Thanian, Ian Ring Music TheoryThanian

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.