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Scale 3297: "Ullian"

Scale 3297: Ullian, 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

Dozenal
Ullian

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

{0,5,6,7,10,11}

Forte Number

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

6-Z6

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.

[2.5]

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.

4 (multihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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 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: 231

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.

[5, 1, 1, 3, 1, 1]

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.

<4, 2, 1, 2, 4, 2>

Interval Spectrum

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

p4m2ns2d4t2

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

Spectra Variation

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

3

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.

1.75

Polygon Perimeter

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

5.417

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.

[5]

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.

(24, 10, 51)

Common Triads

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

Modes

Modes are the rotational transformation of this scale. Scale 3297 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 231
Scale 231: Bifian, Ian Ring Music TheoryBifianThis is the prime mode
3rd mode:
Scale 2163
Scale 2163: Nebian, Ian Ring Music TheoryNebian
4th mode:
Scale 3129
Scale 3129: Toqian, Ian Ring Music TheoryToqian
5th mode:
Scale 903
Scale 903: Fosian, Ian Ring Music TheoryFosian
6th mode:
Scale 2499
Scale 2499: Pirian, Ian Ring Music TheoryPirian

Prime

The prime form of this scale is Scale 231

Scale 231Scale 231: Bifian, Ian Ring Music TheoryBifian

Complement

The hexatonic modal family [3297, 231, 2163, 3129, 903, 2499] (Forte: 6-Z6) is the complement of the hexatonic modal family [399, 483, 2247, 2289, 3171, 3633] (Forte: 6-Z38)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3297 is 231

Scale 231Scale 231: Bifian, Ian Ring Music TheoryBifian

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> 3297       T0I <11,0> 231
T1 <1,1> 2499      T1I <11,1> 462
T2 <1,2> 903      T2I <11,2> 924
T3 <1,3> 1806      T3I <11,3> 1848
T4 <1,4> 3612      T4I <11,4> 3696
T5 <1,5> 3129      T5I <11,5> 3297
T6 <1,6> 2163      T6I <11,6> 2499
T7 <1,7> 231      T7I <11,7> 903
T8 <1,8> 462      T8I <11,8> 1806
T9 <1,9> 924      T9I <11,9> 3612
T10 <1,10> 1848      T10I <11,10> 3129
T11 <1,11> 3696      T11I <11,11> 2163
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2247      T0MI <7,0> 3171
T1M <5,1> 399      T1MI <7,1> 2247
T2M <5,2> 798      T2MI <7,2> 399
T3M <5,3> 1596      T3MI <7,3> 798
T4M <5,4> 3192      T4MI <7,4> 1596
T5M <5,5> 2289      T5MI <7,5> 3192
T6M <5,6> 483      T6MI <7,6> 2289
T7M <5,7> 966      T7MI <7,7> 483
T8M <5,8> 1932      T8MI <7,8> 966
T9M <5,9> 3864      T9MI <7,9> 1932
T10M <5,10> 3633      T10MI <7,10> 3864
T11M <5,11> 3171      T11MI <7,11> 3633

The transformations that map this set to itself are: T0, T5I

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 3299Scale 3299: Syptian, Ian Ring Music TheorySyptian
Scale 3301Scale 3301: Chromatic Mixolydian Inverse, Ian Ring Music TheoryChromatic Mixolydian Inverse
Scale 3305Scale 3305: Chromatic Hypophrygian, Ian Ring Music TheoryChromatic Hypophrygian
Scale 3313Scale 3313: Aeolacrian, Ian Ring Music TheoryAeolacrian
Scale 3265Scale 3265: Urrian, Ian Ring Music TheoryUrrian
Scale 3281Scale 3281: Raga Vijayavasanta, Ian Ring Music TheoryRaga Vijayavasanta
Scale 3233Scale 3233: Unbian, Ian Ring Music TheoryUnbian
Scale 3169Scale 3169: Tupian, Ian Ring Music TheoryTupian
Scale 3425Scale 3425: Vihian, Ian Ring Music TheoryVihian
Scale 3553Scale 3553: Wehian, Ian Ring Music TheoryWehian
Scale 3809Scale 3809: Yelian, Ian Ring Music TheoryYelian
Scale 2273Scale 2273: Nurian, Ian Ring Music TheoryNurian
Scale 2785Scale 2785: Ronian, Ian Ring Music TheoryRonian
Scale 1249Scale 1249: Howian, Ian Ring Music TheoryHowian

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.