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Scale 1595: "Dacrian"

Scale 1595: Dacrian, 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
Dacrian
Dozenal
Agrian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,1,3,4,5,9,10}

Forte Number

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

7-Z38

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: 2957

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.

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.

3

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.

6

Prime Form

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

no
prime: 439

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.

[1, 2, 1, 1, 4, 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.

<4, 3, 4, 4, 4, 2>

Interval Spectrum

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

p4m4n4s3d4t2

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

Spectra Variation

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

2.571

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.

2.299

Polygon Perimeter

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

5.803

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.

(27, 24, 90)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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 TriadsF{5,9,0}231.57
A{9,1,4}321.29
Minor Triadsam{9,0,4}331.43
a♯m{10,1,5}241.86
Augmented TriadsC♯+{1,5,9}331.43
Diminished Triads{9,0,3}142.14
a♯°{10,1,4}231.71
Parsimonious Voice Leading Between Common Triads of Scale 1595. Created by Ian Ring ©2019 C#+ C#+ F F C#+->F A A C#+->A a#m a#m C#+->a#m am am F->am a°->am am->A a#° a#° A->a#° a#°->a#m

view full size

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.

Diameter4
Radius2
Self-Centeredno
Central VerticesA
Peripheral Verticesa°, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 1595 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2845
Scale 2845: Baptian, Ian Ring Music TheoryBaptian
3rd mode:
Scale 1735
Scale 1735: Mela Navanitam, Ian Ring Music TheoryMela Navanitam
4th mode:
Scale 2915
Scale 2915: Aeolydian, Ian Ring Music TheoryAeolydian
5th mode:
Scale 3505
Scale 3505: Stygian, Ian Ring Music TheoryStygian
6th mode:
Scale 475
Scale 475: Aeolygian, Ian Ring Music TheoryAeolygian
7th mode:
Scale 2285
Scale 2285: Aerogian, Ian Ring Music TheoryAerogian

Prime

The prime form of this scale is Scale 439

Scale 439Scale 439: Bythian, Ian Ring Music TheoryBythian

Complement

The heptatonic modal family [1595, 2845, 1735, 2915, 3505, 475, 2285] (Forte: 7-Z38) is the complement of the pentatonic modal family [295, 625, 905, 2195, 3145] (Forte: 5-Z38)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1595 is 2957

Scale 2957Scale 2957: Thygian, Ian Ring Music TheoryThygian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1595 is chiral, and its enantiomorph is scale 2957

Scale 2957Scale 2957: Thygian, Ian Ring Music TheoryThygian

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> 1595       T0I <11,0> 2957
T1 <1,1> 3190      T1I <11,1> 1819
T2 <1,2> 2285      T2I <11,2> 3638
T3 <1,3> 475      T3I <11,3> 3181
T4 <1,4> 950      T4I <11,4> 2267
T5 <1,5> 1900      T5I <11,5> 439
T6 <1,6> 3800      T6I <11,6> 878
T7 <1,7> 3505      T7I <11,7> 1756
T8 <1,8> 2915      T8I <11,8> 3512
T9 <1,9> 1735      T9I <11,9> 2929
T10 <1,10> 3470      T10I <11,10> 1763
T11 <1,11> 2845      T11I <11,11> 3526
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 815      T0MI <7,0> 3737
T1M <5,1> 1630      T1MI <7,1> 3379
T2M <5,2> 3260      T2MI <7,2> 2663
T3M <5,3> 2425      T3MI <7,3> 1231
T4M <5,4> 755      T4MI <7,4> 2462
T5M <5,5> 1510      T5MI <7,5> 829
T6M <5,6> 3020      T6MI <7,6> 1658
T7M <5,7> 1945      T7MI <7,7> 3316
T8M <5,8> 3890      T8MI <7,8> 2537
T9M <5,9> 3685      T9MI <7,9> 979
T10M <5,10> 3275      T10MI <7,10> 1958
T11M <5,11> 2455      T11MI <7,11> 3916

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 1593Scale 1593: Zogimic, Ian Ring Music TheoryZogimic
Scale 1597Scale 1597: Aeolodian, Ian Ring Music TheoryAeolodian
Scale 1599Scale 1599: Pocryllic, Ian Ring Music TheoryPocryllic
Scale 1587Scale 1587: Raga Rudra Pancama, Ian Ring Music TheoryRaga Rudra Pancama
Scale 1591Scale 1591: Rodian, Ian Ring Music TheoryRodian
Scale 1579Scale 1579: Sagimic, Ian Ring Music TheorySagimic
Scale 1563Scale 1563: Joyian, Ian Ring Music TheoryJoyian
Scale 1627Scale 1627: Zyptian, Ian Ring Music TheoryZyptian
Scale 1659Scale 1659: Maqam Shadd'araban, Ian Ring Music TheoryMaqam Shadd'araban
Scale 1723Scale 1723: JG Octatonic, Ian Ring Music TheoryJG Octatonic
Scale 1851Scale 1851: Zacryllic, Ian Ring Music TheoryZacryllic
Scale 1083Scale 1083: Goyian, Ian Ring Music TheoryGoyian
Scale 1339Scale 1339: Kycrian, Ian Ring Music TheoryKycrian
Scale 571Scale 571: Kynimic, Ian Ring Music TheoryKynimic
Scale 2619Scale 2619: Ionyrian, Ian Ring Music TheoryIonyrian
Scale 3643Scale 3643: Kydyllic, Ian Ring Music TheoryKydyllic

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.