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Scale 1729: "Kowian"

Scale 1729: Kowian, 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




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

5 (pentatonic)

Pitch Class Set

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


Forte Number

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


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


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.



A palindromic scale has the same pattern of intervals both ascending and descending.



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

enantiomorph: 109


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

2 (dihemitonic)


A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)


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.



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.


Prime Form

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

prime: 91


Indicates if the scale can be constructed using a generator, and an origin.


Deep Scale

A deep scale is one where the interval vector has 6 different digits.


Interval Structure

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

[6, 1, 2, 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, 2, 3, 1, 1, 1>

Interval Spectrum

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


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> = {1,2,6}
<2> = {3,7,8}
<3> = {4,5,9}
<4> = {6,10,11}

Spectra Variation

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


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


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.


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.


Polygon Perimeter

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


Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.



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.


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.



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".


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.

(14, 1, 30)

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
Diminished Triadsf♯°{6,9,0}000

The following pitch classes are not present in any of the common triads: {7,10}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.


Modes are the rotational transformation of this scale. Scale 1729 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 91
Scale 91: Anoian, Ian Ring Music TheoryAnoianThis is the prime mode
3rd mode:
Scale 2093
Scale 2093: Mulian, Ian Ring Music TheoryMulian
4th mode:
Scale 1547
Scale 1547: Jopian, Ian Ring Music TheoryJopian
5th mode:
Scale 2821
Scale 2821: Rukian, Ian Ring Music TheoryRukian


The prime form of this scale is Scale 91

Scale 91Scale 91: Anoian, Ian Ring Music TheoryAnoian


The pentatonic modal family [1729, 91, 2093, 1547, 2821] (Forte: 5-10) is the complement of the heptatonic modal family [607, 761, 1993, 2351, 3223, 3659, 3877] (Forte: 7-10)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1729 is 109

Scale 109Scale 109: Amsian, Ian Ring Music TheoryAmsian


Only scales that are chiral will have an enantiomorph. Scale 1729 is chiral, and its enantiomorph is scale 109

Scale 109Scale 109: Amsian, Ian Ring Music TheoryAmsian


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> 1729       T0I <11,0> 109
T1 <1,1> 3458      T1I <11,1> 218
T2 <1,2> 2821      T2I <11,2> 436
T3 <1,3> 1547      T3I <11,3> 872
T4 <1,4> 3094      T4I <11,4> 1744
T5 <1,5> 2093      T5I <11,5> 3488
T6 <1,6> 91      T6I <11,6> 2881
T7 <1,7> 182      T7I <11,7> 1667
T8 <1,8> 364      T8I <11,8> 3334
T9 <1,9> 728      T9I <11,9> 2573
T10 <1,10> 1456      T10I <11,10> 1051
T11 <1,11> 2912      T11I <11,11> 2102
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2629      T0MI <7,0> 1099
T1M <5,1> 1163      T1MI <7,1> 2198
T2M <5,2> 2326      T2MI <7,2> 301
T3M <5,3> 557      T3MI <7,3> 602
T4M <5,4> 1114      T4MI <7,4> 1204
T5M <5,5> 2228      T5MI <7,5> 2408
T6M <5,6> 361      T6MI <7,6> 721
T7M <5,7> 722      T7MI <7,7> 1442
T8M <5,8> 1444      T8MI <7,8> 2884
T9M <5,9> 2888      T9MI <7,9> 1673
T10M <5,10> 1681      T10MI <7,10> 3346
T11M <5,11> 3362      T11MI <7,11> 2597

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 1731Scale 1731: Koxian, Ian Ring Music TheoryKoxian
Scale 1733Scale 1733: Raga Sarasvati, Ian Ring Music TheoryRaga Sarasvati
Scale 1737Scale 1737: Raga Madhukauns, Ian Ring Music TheoryRaga Madhukauns
Scale 1745Scale 1745: Raga Vutari, Ian Ring Music TheoryRaga Vutari
Scale 1761Scale 1761: Kuqian, Ian Ring Music TheoryKuqian
Scale 1665Scale 1665: Kejian, Ian Ring Music TheoryKejian
Scale 1697Scale 1697: Raga Kuntvarali, Ian Ring Music TheoryRaga Kuntvarali
Scale 1601Scale 1601: Juwian, Ian Ring Music TheoryJuwian
Scale 1857Scale 1857: Liwian, Ian Ring Music TheoryLiwian
Scale 1985Scale 1985: Mewian, Ian Ring Music TheoryMewian
Scale 1217Scale 1217: Hician, Ian Ring Music TheoryHician
Scale 1473Scale 1473: Javian, Ian Ring Music TheoryJavian
Scale 705Scale 705: Edrian, Ian Ring Music TheoryEdrian
Scale 2753Scale 2753: Ritian, Ian Ring Music TheoryRitian
Scale 3777Scale 3777: Yarian, Ian Ring Music TheoryYarian

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 ( 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.