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Scale 1865: "Thagimic"

Scale 1865: Thagimic, 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
Thagimic

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

{0,3,6,8,9,10}

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-Z45

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.

[3]

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.

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.

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.

5

Prime Form

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

no
prime: 605

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 Formula

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

[3, 3, 2, 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, 4, 2, 2, 2>

Interval Spectrum

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

p2m2n4s3d2t2

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

[6]

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.

(16, 17, 62)

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 TriadsG♯{8,0,3}231.5
Minor Triadsd♯m{3,6,10}231.5
Diminished Triads{0,3,6}231.5
d♯°{3,6,9}231.5
f♯°{6,9,0}231.5
{9,0,3}231.5
Parsimonious Voice Leading Between Common Triads of Scale 1865. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# d#° d#° d#°->d#m f#° f#° d#°->f#° f#°->a° G#->a°

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
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1865 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 745
Scale 745: Kolimic, Ian Ring Music TheoryKolimic
3rd mode:
Scale 605
Scale 605: Dycrimic, Ian Ring Music TheoryDycrimicThis is the prime mode
4th mode:
Scale 1175
Scale 1175: Epycrimic, Ian Ring Music TheoryEpycrimic
5th mode:
Scale 2635
Scale 2635: Gocrimic, Ian Ring Music TheoryGocrimic
6th mode:
Scale 3365
Scale 3365: Katolimic, Ian Ring Music TheoryKatolimic

Prime

The prime form of this scale is Scale 605

Scale 605Scale 605: Dycrimic, Ian Ring Music TheoryDycrimic

Complement

The hexatonic modal family [1865, 745, 605, 1175, 2635, 3365] (Forte: 6-Z45) is the complement of the hexatonic modal family [365, 1115, 1675, 1745, 2605, 2885] (Forte: 6-Z23)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1865 is 605

Scale 605Scale 605: Dycrimic, Ian Ring Music TheoryDycrimic

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> 1865       T0I <11,0> 605
T1 <1,1> 3730      T1I <11,1> 1210
T2 <1,2> 3365      T2I <11,2> 2420
T3 <1,3> 2635      T3I <11,3> 745
T4 <1,4> 1175      T4I <11,4> 1490
T5 <1,5> 2350      T5I <11,5> 2980
T6 <1,6> 605      T6I <11,6> 1865
T7 <1,7> 1210      T7I <11,7> 3730
T8 <1,8> 2420      T8I <11,8> 3365
T9 <1,9> 745      T9I <11,9> 2635
T10 <1,10> 1490      T10I <11,10> 1175
T11 <1,11> 2980      T11I <11,11> 2350
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 605      T0MI <7,0> 1865
T1M <5,1> 1210      T1MI <7,1> 3730
T2M <5,2> 2420      T2MI <7,2> 3365
T3M <5,3> 745      T3MI <7,3> 2635
T4M <5,4> 1490      T4MI <7,4> 1175
T5M <5,5> 2980      T5MI <7,5> 2350
T6M <5,6> 1865       T6MI <7,6> 605
T7M <5,7> 3730      T7MI <7,7> 1210
T8M <5,8> 3365      T8MI <7,8> 2420
T9M <5,9> 2635      T9MI <7,9> 745
T10M <5,10> 1175      T10MI <7,10> 1490
T11M <5,11> 2350      T11MI <7,11> 2980

The transformations that map this set to itself are: T0, T6I, T6M, T0MI

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 1867Scale 1867: Solian, Ian Ring Music TheorySolian
Scale 1869Scale 1869: Katyrian, Ian Ring Music TheoryKatyrian
Scale 1857Scale 1857, Ian Ring Music Theory
Scale 1861Scale 1861: Phrygimic, Ian Ring Music TheoryPhrygimic
Scale 1873Scale 1873: Dathimic, Ian Ring Music TheoryDathimic
Scale 1881Scale 1881: Katorian, Ian Ring Music TheoryKatorian
Scale 1897Scale 1897: Ionopian, Ian Ring Music TheoryIonopian
Scale 1801Scale 1801, Ian Ring Music Theory
Scale 1833Scale 1833: Ionacrimic, Ian Ring Music TheoryIonacrimic
Scale 1929Scale 1929: Aeolycrimic, Ian Ring Music TheoryAeolycrimic
Scale 1993Scale 1993: Katoptian, Ian Ring Music TheoryKatoptian
Scale 1609Scale 1609: Thyritonic, Ian Ring Music TheoryThyritonic
Scale 1737Scale 1737: Raga Madhukauns, Ian Ring Music TheoryRaga Madhukauns
Scale 1353Scale 1353: Raga Harikauns, Ian Ring Music TheoryRaga Harikauns
Scale 841Scale 841: Phrothitonic, Ian Ring Music TheoryPhrothitonic
Scale 2889Scale 2889: Thoptimic, Ian Ring Music TheoryThoptimic
Scale 3913Scale 3913: Bonian, Ian Ring Music TheoryBonian

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.