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Scale 3419: "Magen Abot 1"

Scale 3419: Magen Abot 1, 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

Jewish
Magen Abot 1
Zeitler
Danyllic
Dozenal
Vidian

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

{0,1,3,4,6,8,10,11}

Forte Number

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

8-22

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: 2903

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.

7

Prime Form

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

no
prime: 1391

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

Interval Spectrum

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

p6m5n5s6d4t2

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

Spectra Variation

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

1.75

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.

yes

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.

2.732

Polygon Perimeter

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

6.071

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.

(10, 59, 137)

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 TriadsE{4,8,11}242.1
F♯{6,10,1}242.3
G♯{8,0,3}341.9
B{11,3,6}341.9
Minor Triadsc♯m{1,4,8}242.1
d♯m{3,6,10}242.1
g♯m{8,11,3}341.9
Augmented TriadsC+{0,4,8}341.9
Diminished Triads{0,3,6}242.1
a♯°{10,1,4}242.3
Parsimonious Voice Leading Between Common Triads of Scale 3419. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B C+ C+ c#m c#m C+->c#m E E C+->E C+->G# a#° a#° c#m->a#° d#m d#m F# F# d#m->F# d#m->B g#m g#m E->g#m F#->a#° g#m->G# g#m->B

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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3419 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3757
Scale 3757: Raga Mian Ki Malhar, Ian Ring Music TheoryRaga Mian Ki Malhar
3rd mode:
Scale 1963
Scale 1963: Epocryllic, Ian Ring Music TheoryEpocryllic
4th mode:
Scale 3029
Scale 3029: Ionocryllic, Ian Ring Music TheoryIonocryllic
5th mode:
Scale 1781
Scale 1781: Gocryllic, Ian Ring Music TheoryGocryllic
6th mode:
Scale 1469
Scale 1469: Epiryllic, Ian Ring Music TheoryEpiryllic
7th mode:
Scale 1391
Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllicThis is the prime mode
8th mode:
Scale 2743
Scale 2743: Staptyllic, Ian Ring Music TheoryStaptyllic

Prime

The prime form of this scale is Scale 1391

Scale 1391Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllic

Complement

The octatonic modal family [3419, 3757, 1963, 3029, 1781, 1469, 1391, 2743] (Forte: 8-22) is the complement of the tetratonic modal family [149, 673, 1061, 1289] (Forte: 4-22)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3419 is 2903

Scale 2903Scale 2903: Gothyllic, Ian Ring Music TheoryGothyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3419 is chiral, and its enantiomorph is scale 2903

Scale 2903Scale 2903: Gothyllic, Ian Ring Music TheoryGothyllic

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> 3419       T0I <11,0> 2903
T1 <1,1> 2743      T1I <11,1> 1711
T2 <1,2> 1391      T2I <11,2> 3422
T3 <1,3> 2782      T3I <11,3> 2749
T4 <1,4> 1469      T4I <11,4> 1403
T5 <1,5> 2938      T5I <11,5> 2806
T6 <1,6> 1781      T6I <11,6> 1517
T7 <1,7> 3562      T7I <11,7> 3034
T8 <1,8> 3029      T8I <11,8> 1973
T9 <1,9> 1963      T9I <11,9> 3946
T10 <1,10> 3926      T10I <11,10> 3797
T11 <1,11> 3757      T11I <11,11> 3499
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 509      T0MI <7,0> 2033
T1M <5,1> 1018      T1MI <7,1> 4066
T2M <5,2> 2036      T2MI <7,2> 4037
T3M <5,3> 4072      T3MI <7,3> 3979
T4M <5,4> 4049      T4MI <7,4> 3863
T5M <5,5> 4003      T5MI <7,5> 3631
T6M <5,6> 3911      T6MI <7,6> 3167
T7M <5,7> 3727      T7MI <7,7> 2239
T8M <5,8> 3359      T8MI <7,8> 383
T9M <5,9> 2623      T9MI <7,9> 766
T10M <5,10> 1151      T10MI <7,10> 1532
T11M <5,11> 2302      T11MI <7,11> 3064

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 3417Scale 3417: Golian, Ian Ring Music TheoryGolian
Scale 3421Scale 3421: Aerothyllic, Ian Ring Music TheoryAerothyllic
Scale 3423Scale 3423: Lothygic, Ian Ring Music TheoryLothygic
Scale 3411Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
Scale 3415Scale 3415: Ionaptyllic, Ian Ring Music TheoryIonaptyllic
Scale 3403Scale 3403: Bylian, Ian Ring Music TheoryBylian
Scale 3435Scale 3435: Prokofiev, Ian Ring Music TheoryProkofiev
Scale 3451Scale 3451: Garygic, Ian Ring Music TheoryGarygic
Scale 3355Scale 3355: Bagian, Ian Ring Music TheoryBagian
Scale 3387Scale 3387: Aeryptyllic, Ian Ring Music TheoryAeryptyllic
Scale 3483Scale 3483: Mixotharyllic, Ian Ring Music TheoryMixotharyllic
Scale 3547Scale 3547: Sadygic, Ian Ring Music TheorySadygic
Scale 3163Scale 3163: Rogian, Ian Ring Music TheoryRogian
Scale 3291Scale 3291: Lygyllic, Ian Ring Music TheoryLygyllic
Scale 3675Scale 3675: Monyllic, Ian Ring Music TheoryMonyllic
Scale 3931Scale 3931: Aerygic, Ian Ring Music TheoryAerygic
Scale 2395Scale 2395: Zoptian, Ian Ring Music TheoryZoptian
Scale 2907Scale 2907: Magen Abot 2, Ian Ring Music TheoryMagen Abot 2
Scale 1371Scale 1371: Superlocrian, Ian Ring Music TheorySuperlocrian

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.