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Scale 3085: "Tepian"

Scale 3085: Tepian, 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
Tepian

Analysis

Cardinality

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

{0,2,3,10,11}

Forte Number

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

5-3

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: 1543

Hemitonia

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

3 (trihemitonic)

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.

4

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.

4

Prime Form

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

no
prime: 55

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.

[2, 1, 7, 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.

<3, 2, 2, 2, 1, 0>

Interval Spectrum

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

pm2n2s2d3

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

Spectra Variation

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

4.8

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.

0.933

Polygon Perimeter

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

4.485

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.

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.

(15, 2, 30)

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 3085 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1795
Scale 1795: Lakian, Ian Ring Music TheoryLakian
3rd mode:
Scale 2945
Scale 2945: Sihian, Ian Ring Music TheorySihian
4th mode:
Scale 55
Scale 55: Aspian, Ian Ring Music TheoryAspianThis is the prime mode
5th mode:
Scale 2075
Scale 2075: Mozian, Ian Ring Music TheoryMozian

Prime

The prime form of this scale is Scale 55

Scale 55Scale 55: Aspian, Ian Ring Music TheoryAspian

Complement

The pentatonic modal family [3085, 1795, 2945, 55, 2075] (Forte: 5-3) is the complement of the heptatonic modal family [319, 1009, 2207, 3151, 3623, 3859, 3977] (Forte: 7-3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3085 is 1543

Scale 1543Scale 1543: Jomian, Ian Ring Music TheoryJomian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3085 is chiral, and its enantiomorph is scale 1543

Scale 1543Scale 1543: Jomian, Ian Ring Music TheoryJomian

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> 3085       T0I <11,0> 1543
T1 <1,1> 2075      T1I <11,1> 3086
T2 <1,2> 55      T2I <11,2> 2077
T3 <1,3> 110      T3I <11,3> 59
T4 <1,4> 220      T4I <11,4> 118
T5 <1,5> 440      T5I <11,5> 236
T6 <1,6> 880      T6I <11,6> 472
T7 <1,7> 1760      T7I <11,7> 944
T8 <1,8> 3520      T8I <11,8> 1888
T9 <1,9> 2945      T9I <11,9> 3776
T10 <1,10> 1795      T10I <11,10> 3457
T11 <1,11> 3590      T11I <11,11> 2819
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1165      T0MI <7,0> 1573
T1M <5,1> 2330      T1MI <7,1> 3146
T2M <5,2> 565      T2MI <7,2> 2197
T3M <5,3> 1130      T3MI <7,3> 299
T4M <5,4> 2260      T4MI <7,4> 598
T5M <5,5> 425      T5MI <7,5> 1196
T6M <5,6> 850      T6MI <7,6> 2392
T7M <5,7> 1700      T7MI <7,7> 689
T8M <5,8> 3400      T8MI <7,8> 1378
T9M <5,9> 2705      T9MI <7,9> 2756
T10M <5,10> 1315      T10MI <7,10> 1417
T11M <5,11> 2630      T11MI <7,11> 2834

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 3087Scale 3087: Hexatonic Chromatic 3, Ian Ring Music TheoryHexatonic Chromatic 3
Scale 3081Scale 3081: Temian, Ian Ring Music TheoryTemian
Scale 3083Scale 3083: Rehian, Ian Ring Music TheoryRehian
Scale 3077Scale 3077: Tekian, Ian Ring Music TheoryTekian
Scale 3093Scale 3093: Buqian, Ian Ring Music TheoryBuqian
Scale 3101Scale 3101: Tiyian, Ian Ring Music TheoryTiyian
Scale 3117Scale 3117: Tijian, Ian Ring Music TheoryTijian
Scale 3149Scale 3149: Phrycrimic, Ian Ring Music TheoryPhrycrimic
Scale 3213Scale 3213: Eponimic, Ian Ring Music TheoryEponimic
Scale 3341Scale 3341: Vahian, Ian Ring Music TheoryVahian
Scale 3597Scale 3597: Wijian, Ian Ring Music TheoryWijian
Scale 2061Scale 2061: Morian, Ian Ring Music TheoryMorian
Scale 2573Scale 2573: Pulian, Ian Ring Music TheoryPulian
Scale 1037Scale 1037: Warao Tetratonic, Ian Ring Music TheoryWarao Tetratonic

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.