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Scale 1977: "Dagyllic"

Scale 1977: Dagyllic, 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
Dagyllic
Dozenal
Marian

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,3,4,5,7,8,9,10}

Forte Number

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

8-14

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: 957

Hemitonia

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

5 (multihemitonic)

Cohemitonia

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

3 (tricohemitonic)

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.

7

Prime Form

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

no
prime: 759

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.

[3, 1, 1, 2, 1, 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.

<5, 5, 5, 5, 6, 2>

Interval Spectrum

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

p6m5n5s5d5t2

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

Spectra Variation

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

2.25

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.

2.616

Polygon Perimeter

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

6.002

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.

(34, 56, 136)

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 TriadsC{0,4,7}331.7
D♯{3,7,10}252.5
F{5,9,0}252.5
G♯{8,0,3}331.7
Minor Triadscm{0,3,7}341.9
fm{5,8,0}242.1
am{9,0,4}341.9
Augmented TriadsC+{0,4,8}431.5
Diminished Triads{4,7,10}242.3
{9,0,3}242.1
Parsimonious Voice Leading Between Common Triads of Scale 1977. Created by Ian Ring ©2019 cm cm C C cm->C D# D# cm->D# G# G# cm->G# C+ C+ C->C+ C->e° fm fm C+->fm C+->G# am am C+->am D#->e° F F fm->F F->am G#->a° a°->am

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.

Diameter5
Radius3
Self-Centeredno
Central VerticesC, C+, G♯
Peripheral VerticesD♯, F

Modes

Modes are the rotational transformation of this scale. Scale 1977 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 759
Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllicThis is the prime mode
3rd mode:
Scale 2427
Scale 2427: Katoryllic, Ian Ring Music TheoryKatoryllic
4th mode:
Scale 3261
Scale 3261: Dodyllic, Ian Ring Music TheoryDodyllic
5th mode:
Scale 1839
Scale 1839: Zogyllic, Ian Ring Music TheoryZogyllic
6th mode:
Scale 2967
Scale 2967: Madyllic, Ian Ring Music TheoryMadyllic
7th mode:
Scale 3531
Scale 3531: Neveseri, Ian Ring Music TheoryNeveseri
8th mode:
Scale 3813
Scale 3813: Aeologyllic, Ian Ring Music TheoryAeologyllic

Prime

The prime form of this scale is Scale 759

Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic

Complement

The octatonic modal family [1977, 759, 2427, 3261, 1839, 2967, 3531, 3813] (Forte: 8-14) is the complement of the tetratonic modal family [141, 417, 1059, 2577] (Forte: 4-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1977 is 957

Scale 957Scale 957: Phronyllic, Ian Ring Music TheoryPhronyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1977 is chiral, and its enantiomorph is scale 957

Scale 957Scale 957: Phronyllic, Ian Ring Music TheoryPhronyllic

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> 1977       T0I <11,0> 957
T1 <1,1> 3954      T1I <11,1> 1914
T2 <1,2> 3813      T2I <11,2> 3828
T3 <1,3> 3531      T3I <11,3> 3561
T4 <1,4> 2967      T4I <11,4> 3027
T5 <1,5> 1839      T5I <11,5> 1959
T6 <1,6> 3678      T6I <11,6> 3918
T7 <1,7> 3261      T7I <11,7> 3741
T8 <1,8> 2427      T8I <11,8> 3387
T9 <1,9> 759      T9I <11,9> 2679
T10 <1,10> 1518      T10I <11,10> 1263
T11 <1,11> 3036      T11I <11,11> 2526
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2847      T0MI <7,0> 3867
T1M <5,1> 1599      T1MI <7,1> 3639
T2M <5,2> 3198      T2MI <7,2> 3183
T3M <5,3> 2301      T3MI <7,3> 2271
T4M <5,4> 507      T4MI <7,4> 447
T5M <5,5> 1014      T5MI <7,5> 894
T6M <5,6> 2028      T6MI <7,6> 1788
T7M <5,7> 4056      T7MI <7,7> 3576
T8M <5,8> 4017      T8MI <7,8> 3057
T9M <5,9> 3939      T9MI <7,9> 2019
T10M <5,10> 3783      T10MI <7,10> 4038
T11M <5,11> 3471      T11MI <7,11> 3981

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 1979Scale 1979: Aeradygic, Ian Ring Music TheoryAeradygic
Scale 1981Scale 1981: Houseini, Ian Ring Music TheoryHouseini
Scale 1969Scale 1969: Stylian, Ian Ring Music TheoryStylian
Scale 1973Scale 1973: Zyryllic, Ian Ring Music TheoryZyryllic
Scale 1961Scale 1961: Soptian, Ian Ring Music TheorySoptian
Scale 1945Scale 1945: Zarian, Ian Ring Music TheoryZarian
Scale 2009Scale 2009: Stacryllic, Ian Ring Music TheoryStacryllic
Scale 2041Scale 2041: Aeolacrygic, Ian Ring Music TheoryAeolacrygic
Scale 1849Scale 1849: Chromatic Hypodorian Inverse, Ian Ring Music TheoryChromatic Hypodorian Inverse
Scale 1913Scale 1913: Lofian, Ian Ring Music TheoryLofian
Scale 1721Scale 1721: Mela Vagadhisvari, Ian Ring Music TheoryMela Vagadhisvari
Scale 1465Scale 1465: Mela Ragavardhani, Ian Ring Music TheoryMela Ragavardhani
Scale 953Scale 953: Mela Yagapriya, Ian Ring Music TheoryMela Yagapriya
Scale 3001Scale 3001: Lonyllic, Ian Ring Music TheoryLonyllic
Scale 4025Scale 4025: Kalygic, Ian Ring Music TheoryKalygic

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.