The Exciting Universe Of Music Theory

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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 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).

## CardinalityCardinality is the count of how many pitches are in the scale. |
5 (pentatonic) |

## Pitch Class SetThe tones in this scale, expressed as numbers from 0 to 11 |
{0,4,5,10,11} |

## Forte NumberA code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations. |
5-7 |

## Rotational SymmetrySome scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity. |
none |

## Reflection AxesIf 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 |

## PalindromicityA palindromic scale has the same pattern of intervals both ascending and descending. |
no |

## ChiralityA chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph. |
yes enantiomorph: 391 |

## HemitoniaA hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist. |
3 (trihemitonic) |

## CohemitoniaA cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist. |
1 (uncohemitonic) |

## ImperfectionsAn 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 |

## ModesModes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes. |
5 |

## Prime FormDescribes if this scale is in prime form, using the Starr/Rahn algorithm. |
no prime: 199 |

## GeneratorIndicates if the scale can be constructed using a generator, and an origin. |
none |

## Deep ScaleA deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization. |
no |

## Interval StructureDefines the scale as the sequence of intervals between one tone and the next. |
[4, 1, 5, 1, 1] |

## Interval VectorDescribes 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, 1, 0, 1, 3, 2> |

## Proportional Saturation VectorFirst described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible. | <0.75, 0.25, 0, 0, 0.75, 1> |

## Interval SpectrumThe same as the Interval Vector, but expressed in a syntax used by Howard Hanson. |
p^{3}msd^{3}t^{2} |

## Distribution SpectraDescribes 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> = {1,4,5} <2> = {2,5,6} <3> = {6,7,10} <4> = {7,8,11} |

## Spectra VariationDetermined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality. |
3.2 |

## Maximally EvenA scale is maximally even if the tones are optimally spaced apart from each other. |
no |

## Maximal Area SetA 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 AreaArea 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. |
1.433 |

## Polygon PerimeterPerimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle. |
5.217 |

## Myhill PropertyA scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval. |
no |

## Centre of Gravity DistanceWhen tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero. |
0.2 |

## Ridge TonesRidge 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 |

## ProprietyAlso 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 ProfileDefined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where | (4, 8, 30) |

## Coherence QuotientThe Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale. | 0.52 |

## Sameness QuotientThe Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity. | 0.25 |

This scale has no generator.

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes are the rotational transformation of this scale. Scale 3121 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode: Scale 451 | Raga Saugandhini | ||||

3rd mode: Scale 2273 | NURian | ||||

4th mode: Scale 199 | Raga Nabhomani | This is the prime mode | |||

5th mode: Scale 2147 | NARian |

The prime form of this scale is Scale 199

Scale 199 | Raga Nabhomani |

The pentatonic modal family [3121, 451, 2273, 199, 2147] (Forte: 5-7) is the complement of the heptatonic modal family [463, 967, 2279, 2531, 3187, 3313, 3641] (Forte: 7-7)

The inverse of a scale is a reflection using the root as its axis. The inverse of 3121 is 391

Scale 391 | CIYian |

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

## Contradictions (4) | ## Ambiguities(8) |

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable *k* is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

k | Hierarchizability | Breakdown Pattern | Diagram |
---|---|---|---|

1 | 1 | 100011000011 | |

2 | 3 | 1([0][0][0]11)0([0][0][0]11) | |

3 | 3 | 1([0][0][0]11)0([0][0][0]11) | |

4 | 3 | 1([0][0][0]11)0([0][0][0]11) | |

5 | 3 | 1([0][0][0]11)0([0][0][0]11) |

Only scales that are chiral will have an enantiomorph. Scale 3121 is chiral, and its enantiomorph is scale 391

Scale 391 | CIYian |

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position with origin in the center | (0, -0.2) |
---|---|

Distance from Center | 0.2 |

Angle in degrees measured clockwise starting from the root. | 0 |

Angle in cents 100 cents = 1 semitone. | 0 |

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`. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev | Operation | Result | Abbrev | Operation | Result | |||
---|---|---|---|---|---|---|---|---|

T_{0} | <1,0> | 3121 | T_{0}I | <11,0> | 391 | |||

T_{1} | <1,1> | 2147 | T_{1}I | <11,1> | 782 | |||

T_{2} | <1,2> | 199 | T_{2}I | <11,2> | 1564 | |||

T_{3} | <1,3> | 398 | T_{3}I | <11,3> | 3128 | |||

T_{4} | <1,4> | 796 | T_{4}I | <11,4> | 2161 | |||

T_{5} | <1,5> | 1592 | T_{5}I | <11,5> | 227 | |||

T_{6} | <1,6> | 3184 | T_{6}I | <11,6> | 454 | |||

T_{7} | <1,7> | 2273 | T_{7}I | <11,7> | 908 | |||

T_{8} | <1,8> | 451 | T_{8}I | <11,8> | 1816 | |||

T_{9} | <1,9> | 902 | T_{9}I | <11,9> | 3632 | |||

T_{10} | <1,10> | 1804 | T_{10}I | <11,10> | 3169 | |||

T_{11} | <1,11> | 3608 | T_{11}I | <11,11> | 2243 | |||

Abbrev | Operation | Result | Abbrev | Operation | Result | |||

T_{0}M | <5,0> | 391 | T_{0}MI | <7,0> | 3121 | |||

T_{1}M | <5,1> | 782 | T_{1}MI | <7,1> | 2147 | |||

T_{2}M | <5,2> | 1564 | T_{2}MI | <7,2> | 199 | |||

T_{3}M | <5,3> | 3128 | T_{3}MI | <7,3> | 398 | |||

T_{4}M | <5,4> | 2161 | T_{4}MI | <7,4> | 796 | |||

T_{5}M | <5,5> | 227 | T_{5}MI | <7,5> | 1592 | |||

T_{6}M | <5,6> | 454 | T_{6}MI | <7,6> | 3184 | |||

T_{7}M | <5,7> | 908 | T_{7}MI | <7,7> | 2273 | |||

T_{8}M | <5,8> | 1816 | T_{8}MI | <7,8> | 451 | |||

T_{9}M | <5,9> | 3632 | T_{9}MI | <7,9> | 902 | |||

T_{10}M | <5,10> | 3169 | T_{10}MI | <7,10> | 1804 | |||

T_{11}M | <5,11> | 2243 | T_{11}MI | <7,11> | 3608 |

The transformations that map this set to itself are: T_{0}, T_{0}MI

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 3123 | TOMian | |||

Scale 3125 | TONian | |||

Scale 3129 | TOQian | |||

Scale 3105 | TIBian | |||

Scale 3113 | TIGian | |||

Scale 3089 | TIRian | |||

Scale 3153 | Zathitonic | |||

Scale 3185 | Messiaen Mode 5 Rotation 1 | |||

Scale 3249 | Raga Tilang | |||

Scale 3377 | Phralimic | |||

Scale 3633 | Daptimic | |||

Scale 2097 | MUNian | |||

Scale 2609 | Raga Bhinna Shadja | |||

Scale 1073 | GOSian |

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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, and ©2023 Robert Bedwell, used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

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 naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Thanks to u/howaboot for inventing the Center of Gravity metrics.