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Scale 1249: "Howian"

Scale 1249: Howian, 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: 229


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.

1 (uncohemitonic)


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: 167


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.

[5, 1, 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.

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

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.

(9, 9, 38)

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.


Modes are the rotational transformation of this scale. Scale 1249 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 167
Scale 167: Barian, Ian Ring Music TheoryBarianThis is the prime mode
3rd mode:
Scale 2131
Scale 2131: Nahian, Ian Ring Music TheoryNahian
4th mode:
Scale 3113
Scale 3113: Tigian, Ian Ring Music TheoryTigian
5th mode:
Scale 901
Scale 901: Bofian, Ian Ring Music TheoryBofian


The prime form of this scale is Scale 167

Scale 167Scale 167: Barian, Ian Ring Music TheoryBarian


The pentatonic modal family [1249, 167, 2131, 3113, 901] (Forte: 5-14) is the complement of the heptatonic modal family [431, 1507, 1933, 2263, 2801, 3179, 3637] (Forte: 7-14)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1249 is 229

Scale 229Scale 229: Bidian, Ian Ring Music TheoryBidian


Only scales that are chiral will have an enantiomorph. Scale 1249 is chiral, and its enantiomorph is scale 229

Scale 229Scale 229: Bidian, Ian Ring Music TheoryBidian


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> 1249       T0I <11,0> 229
T1 <1,1> 2498      T1I <11,1> 458
T2 <1,2> 901      T2I <11,2> 916
T3 <1,3> 1802      T3I <11,3> 1832
T4 <1,4> 3604      T4I <11,4> 3664
T5 <1,5> 3113      T5I <11,5> 3233
T6 <1,6> 2131      T6I <11,6> 2371
T7 <1,7> 167      T7I <11,7> 647
T8 <1,8> 334      T8I <11,8> 1294
T9 <1,9> 668      T9I <11,9> 2588
T10 <1,10> 1336      T10I <11,10> 1081
T11 <1,11> 2672      T11I <11,11> 2162
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2119      T0MI <7,0> 3139
T1M <5,1> 143      T1MI <7,1> 2183
T2M <5,2> 286      T2MI <7,2> 271
T3M <5,3> 572      T3MI <7,3> 542
T4M <5,4> 1144      T4MI <7,4> 1084
T5M <5,5> 2288      T5MI <7,5> 2168
T6M <5,6> 481      T6MI <7,6> 241
T7M <5,7> 962      T7MI <7,7> 482
T8M <5,8> 1924      T8MI <7,8> 964
T9M <5,9> 3848      T9MI <7,9> 1928
T10M <5,10> 3601      T10MI <7,10> 3856
T11M <5,11> 3107      T11MI <7,11> 3617

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 1251Scale 1251: Sylimic, Ian Ring Music TheorySylimic
Scale 1253Scale 1253: Zolimic, Ian Ring Music TheoryZolimic
Scale 1257Scale 1257: Blues Scale, Ian Ring Music TheoryBlues Scale
Scale 1265Scale 1265: Pynimic, Ian Ring Music TheoryPynimic
Scale 1217Scale 1217: Hician, Ian Ring Music TheoryHician
Scale 1233Scale 1233: Ionoditonic, Ian Ring Music TheoryIonoditonic
Scale 1185Scale 1185: Genus Primum Inverse, Ian Ring Music TheoryGenus Primum Inverse
Scale 1121Scale 1121: Guwian, Ian Ring Music TheoryGuwian
Scale 1377Scale 1377: Insian, Ian Ring Music TheoryInsian
Scale 1505Scale 1505: Jepian, Ian Ring Music TheoryJepian
Scale 1761Scale 1761: Kuqian, Ian Ring Music TheoryKuqian
Scale 225Scale 225: Bibian, Ian Ring Music TheoryBibian
Scale 737Scale 737: Truian, Ian Ring Music TheoryTruian
Scale 2273Scale 2273: Nurian, Ian Ring Music TheoryNurian
Scale 3297Scale 3297: Ullian, Ian Ring Music TheoryUllian

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.