A Study of Scales

Where we discuss every possible combination of notes

This exploration of scales is based on work by William Zeitler1. In fact much of the material on this page repeats Zeitler's findings, presented here along with additional observations and/or PHP code used to generate the scales. This exploration also owes a debt to Andrew Duncan's work on combinatorial music theory2. Most of the code in this treatise is the result of continuous exploration and development of PHPMusicTools, an open-source project at GitHub. The section related to scale-chord relationships is indebted to Enrico Dell'Aquila, and his video about Mode Colours

Assumptions

This exploration of scales is based in a musical universe founded on two assumptions:

  • Octave Equivalence
    We assume that for the purpose of defining a scale, a pitch is functionally equivalent to another pitch separated by an octave. So it follows that if you're playing a scale in one octave, if you continue the pattern into the next octave you will play pitches with the same name.

  • 12 tone equal temperament
    We're using the 12 tones of an equally-tempered tuning system, as you'd find on a piano. Equal temperament asserts that the perceptual (or functional) relationship between two pitches is the same as the relationship between two other pitches with the same chromatic interval distance. Other tuning systems have the notion of scales, but they're not being considered in this study.

Representing a scale

When I began piano lessons as a child, I learned that a scale was made up of whole- and half-steps. The scale was essentially a stack of intervals piled up, that happened to add up to an octave. While this view of the scale is useful for understanding diatonic melodies, it's not a convenient paradigm for analysis. Instead, we should regard a scale as being a set of 12 possibilities, and each one is either on or off.

The major scale, in lights.

What we have in the 12-tone system is a binary "word" made of 12 bits. We can assign one bit to each degree of the chromatic scale, and use the power of binary arithmetic and logic to do some pretty awesome analysis with them. When represented as bits it reads from right to left - the lowest bit is the root, and each bit going from right to left ascends by one semitone.

The total number of possible combinations of on and off bits is called the "power set". The number of sets in a power set of size n is (2^n). Using a word of 12 bits, the power set (2^12) is equal to 4096. The fun thing about binary power sets is that we can produce every possible combination, by merely invoking the integers from 0 (no tones) to 4095 (all 12 tones).

This means that every possible combination of tones in the 12-tone set can be represented by a number between 0 and 4095. We don't need to remember the fancy names like "phrygian", we can just call it scale number 1451. Convenient!

$allscales = range(0, 4095);
decimalbinary
0 000000000000 no notes in the scale
1 000000000001 just the root tone
1365 010101010101 whole tone scale
2741 101010110101 major scale
4095 111111111111 chromatic scale

An important concept here is that any set of tones can be represented by a number. This number is not "ordinal" - it's not merely describing the position of the set in an indexed scheme; it's also not "cardinal" because it's not describing an amount of something. This is a nominal number because the number *is* the scale. You can do binary arithmetic with it, and you are adding and subtracting scales with no need to convert the scale into some other representation.


Scale Finder

Not sure what scale you've got? Check the boxes below tones that appear in your scale. Then follow the link to learn more about it!


Binary:
Decimal:

Because scales are cyclical - they repeat and continue beyond a single octave - it is intuitive to represent them in "bracelet" notation. This visual representation is useful because it helps conceptualize modes and rotation (which we'll get into further below).

Here is the major scale, represented as a bracelet. Black beads are on bits, white beads are off bits.

If you imagine the bracelet with clock numbers, the topmost bead - at 12 o'clock - is the root, or 1st degree of the scale. Beware that in some of the math, this is the "0" position, as if we're counting semitones above the root. To play a scale ascending, start from the top and move around it clockwise. To play the scale descending, start at 12 o'clock and go around counterclockwise.

Interval Pattern

Another popular way of representing a scale is by its interval pattern. When I was learning the major scale, I was taught to say aloud: "tone, tone, semitone, tone, tone, tone, semitone". Many music theorists like to represent a scale this way because it's accurate and easy to understand: "TTSTTTS". Having a scale's interval pattern has merit as an intermediary step can make some kinds of analysis simpler. Expressed numerically - which is more convenient for computation - the major scale is [2,2,1,2,2,2,1].

Pitch Class Sets

Yet another way to represent a scale is as a "pitch class set", where the tones are assigned numbers 0 to 11 (sometimes using "T" and "E" for 10 and 11), and the set enumerates the ones present in the scale. A pitch class set for the major scale is notated like this: {0,2,4,5,7,9,11}. The "scales" we'll study here are a subset of Pitch Classes (ie those that have a root, and obey Zeitler's Rules3) and we can use many of the same mathematical tricks to manipulate them.

What is a scale?

Or more importantly, what is *not* a scale?

Now that we have the superset of all possible sets of tones, we can whittle it down to exclude ones that we don't consider to be a legitimate "scale". We can do this with just two rules.

  • A scale starts on the root tone.

    This means any set of notes that doesn't have that first bit turned on is not eligible to be called a scale. This cuts our power set in exactly half, leaving 2048 sets.

    In binary, it's easy to see that the first bit is on or off. In decimal, you can easily tell this by whether the number is odd or even. All even numbers have the first bit off; therefore all scales are represented by an odd number.

    We could have cut some corners in our discussion of scales by omitting the root tone (always assumed to be on) to work with 11 bits instead of 12, but there are compelling reasons for keeping the 12-bit number for our scales, such as simplifying the analysis of symmetry, simplifying the calculation of modes, and performing analysis of sonorities that do not include the root, where an even number is appropriate.

    function hasRootTone($scale) { // returns true if the first bit is not a zero return (1 & $scale) != 0; } $allscales = array_filter($allscales, 'hasRootTone');

    scales remaining: 2048

  • A scale does not have any leaps greater than n semitones.

    For the purposes of this exercise we are saying n = 4, a.k.a. a major third. Any collection of tones that has an interval greater than a major third is not considered a "scale". This configuration is consistent with Zeitler's constant used to generate his comprehensive list of scales.

    function doesNotHaveFourConsecutiveOffBits($scale) { $c = 0; for ($i=0; $i<12; $i++) { if (!($scale & (1 << ($i)))) { $c++; if ($c >= 4) { return false; } } else { $c = 0; } } return true; } $allscales = array_filter($allscales, 'doesNotHaveFourConsecutiveOffBits');

    scales remaining: 1490

Now that we've whittled our set of tones to only the ones we'd call a "scale", let's count how many there are with each number of tones.

function countOnBits($scale) { $c = 0; for ($i=0; $i<12; $i++) { if ($scale & (1 << ($i))) { $c++; } } return $c; }
number of tones how many scales
10
20
31
431
5155
6336
7413
8322
9165
1055
1111
121

Heptatonics

The equal temperament system of 12 tones has a special affinity for heptatonic scales. Of the 1490 scales, 413 of them are heptatonic, comprising 28% of the total. The predominant scales used in popular music are heptatonic, including those derived most closely from the circle of fifths.

Some heptatonics are diatonic. To be diatonic means that it includes only notes in the prevailing key; so the set of diatonic scales are very limited to just the major scale and its 6 modes - and only when used in a context where the modes are offset in respect to the tonic key.

Hepatonia prima, secunda, tertia

(coming soon)

Symmetry

There are two kinds of symmetry of interest to scale analysis. They are rotational symmetry and reflective symmetry.

Rotational Symmetry

Rotational symmetry is the symmetry that occurs by transposing a scale up or down by an interval, and observing whether the transposition and the original have an identical pattern of steps.

The set of 12 tones has 5 axes of symmetry. The twelve can be divided by 1, 2, 3, 4, and 6.

Any scale containing this kind of symmetry can reproduce its own tones by transposition, and is also called a "mode of limited transposition" (Messaien)

function symmetries($scale) { $rotateme = $scale; $symmetries = array(); for ($i = 0; $i < 12; $i++) { $rotateme = rotate_bitmask($rotateme); if ($rotateme == $scale) { $symmetries[] = $i; } } return $symmetries; }

Below are all the scales that have rotational symmetry.

axes of symmetryinterval of repetitionscales
1,2,3,4,5,6,7,8,9,10,11 semitone
2,4,6,8,10 whole tone
3,6,9 minor thirds

585 
4,8 major thirds

273 

819 
6 tritones

325 

455 

715 

845 

975 

number of notes in scale Placement of rotational symmetries
1234567891011
300010001000
400100300100
500000000000
6010301003010
700000000000
8002001000200
900030003000
1000000500000
1100000000000
1211111111111

A curious numeric pattern

You'll notice that in the list of scales exhibiting rotational symmetry, many have a decimal representation that ends with the digit 5. A closer look shows that the pattern exists for all scales that have symmetry on axes of 6, 4, and 2 - where all the notes in the scale are in pairs a tritone apart. In the binary schema, each pair of notes that are a tritone apart add up to a multiple of 10, except for the root pair that adds up to 65. Therefore, all scales with rotational symmetry on the tritone will end with the digit 5.


1 + 64
= 65

2 + 128
= 130

4 + 256
= 260

8 + 512
= 520

16 + 1024
= 1040

32 + 2048
= 2080

Messiaen's Modes - and their truncations

The French composer Messiaen studied scales with rotational symmetry, because they are useful in creating sonorities with an ambiguous tonic. He famously named seven such modes, and asserted that they are the only ones that may exist; all other symmetrical scales are truncations of those. We can prove that Messiaen was correct, by putting every symmetrical scale into a chart of their truncation relationships.

In all of Messiaen's modes, the modes are regarded as identical, which makes sense because the whole point was to create ambiguous tonality. Therefore when we speak of one of the Messiaenic modes, we're talking about a whole modal family of scales, like this one which he labeled Mode 7:

Truncation is the process of removing notes from a scale, without affecting its symmetry. The concept is best explained by example. Let's take the mode which Messiaen called the Second Mode:

That scale is symmetrical along the axis of a 3-semitone interval, i.e. by transposing it up a minor third. When we remove tones to create a truncation, we must preserve that symmetry, by removing four notes from the scale, like this:


585 

In addition to the 3-semitone symmetry, that scale is also symmetrical along the axis of a 6-semitone interval. We can create two different truncations of 1755  that preserve that symmetry:

Technically, all of Messiaen's modes are truncated forms of 4095 , the 12-tone scale, which is symmetrical at all intervals.

Hierarchy of truncations

This diagram illustrates that all symmetrical scales are truncations of the Messaienic modes (labeled "M" plus their number as assigned by Messaien). M1, M2, M3, and M7 are all truncations of the chromatic scale 4095; M2, M4, and M6 are truncations of M7; M1 and M5 are truncaions of M6. M1 is also a truncation of M3, and M5 is also a truncation of M4. All the other rotationally symmetrical scales are labeled "T#" and are shown where they belong in the hierarchy of truncations.

Modal FamilyScalesis truncation of
*  
Messiaen's Modes of Limited Transposition
M1 *, M3, M6
M2 *, M7
M3 *
M4 M7
M5 M6, M4
M6 M7
M7 *
Truncations
T1

585 
M2, T5, T4
T2 M3
T3

273 
M1, T2
T4 M2, M6, M4
T5 M2, M6, M4
T6 T4, T5, M5, M1

Going Further

  • In this study we are ignoring sonorities that aren't like a "scale" by disallowing large interval leaps (see the first section). If we ignore those rules, what other truncated sets exist? (spoiler hint: not very many!)
  • Does the naming of certain scales by Messiaen seem haphazard? Why did he name M5, which is merely a truncation of both M6 and M4, which are in turn truncations of M7?

Reflective Symmetry

A scale can be said to have reflective symmetry if it has the same interval pattern whether ascending or descending. A reflectively symmetrical scale, symmetrical on the axis of the root tone, is called a palindromic scale.

function isPalindromic($scale) { for ($i=1; $i<=5; $i++) { if ( (bool)($scale & (1 << $i)) !== (bool)($scale & (1 << (12 - $i))) ) { return false; } } return true; }

Here are all the scales that are palindromic:


273 

337 

433 

497 

585 

681 

745 

793 

857 

953 

Chirality

An object is chiral if it is distinguishable from its mirror image, and can't be transformed into its mirror image by rotation. Chirality is an important concept in knot theory, and also has applications in molecular chemistry.

The concept of chirality is apparent when you consider "handedness" of a shape, such as a glove. A pair of gloves has a left hand and a right hand, and you can't transform a left glove into a right one by flipping it over or rotating it. The glove, therefore, is a chiral object. Conversely, a sock can be worn by the right or left foot; the mirror image of a sock would still be the same sock. Therefore a sock is not a chiral object.

Palindromic scales are achiral. But not all non-palindromic scales are chiral. For example, consider the scales 1105  and 325  (below). One is the mirror image (reflection) of the other. They are not palindromic scales. They are also achiral, because one can transform into the other by rotation.

The major scale is achiral. If you reflect the major scale, you get the phrygian scale; these two scales are modes (rotations) of each other. Therefore neither of them are chiral. Similarly, none of the other ecclesiastical major modes are chiral.

What makes chirality interesting? For one thing chirality indicates that a scale does not have any symmetry along any axis. If you examine all achiral scales, they will all have some axis of reflective symmetry - it's just not necessarily the root tone (which would make it palindromic).

Some achiral scales, and their axes of symmetry
function isChiral($scale) { $reflected = reflect_bitmask($scale); for ($i = 0; $i < 12; $i++) { $reflected = rotate_bitmask($reflected, 1, 1); if ($reflected == $scale) { return false; } } return true; }

A chiral object and its mirror image are called enantiomorphs. (source) function enantiomorph($scale) { $scale = reflect_bitmask($scale); $scale = rotate_bitmask($scale, -1, 1); return $scale; }
ScaleNameChirality / Enantiomorph

273 
augmented triadachiral

585 
diminished seventhachiral

661 
pentatonic majorachiral

859 
ultralocrian
pentatonic minorachiral
blues

741 
whole toneachiral
alteredachiral
locrianachiral
half diminishedachiral
major locrianachiral
phrygianachiral
aeolianachiral
phrygian dominant
lydian diminished
lydian minorachiral
bebop locrian
prometheus
locrian natural 6
dorianachiral
mixolydianachiral
bebop dorianachiral
ukranian dorian
acousticachiral
hungarian major

877 
octatonicachiral
hirajoshi

355 
fifth mode of limited transpositionachiral
augmentedachiral
minor neapolitan
harmonic minor
enigmaticachiral
hungarian minorachiral
fourth mode of limited transpositionachiral
major neapolitanachiral
heptatonia secondaachiral
majorachiral
lydianachiral
lydian #2

875 
ionian augmented
lydian augmentedachiral
diminishedachiral
bebop minor
bebop majorachiral
seventh mode of limited transpositionachiral
enigmatic
sixth mode of limited transpositionachiral
third mode of limited transpositionachiral
bebop dominantachiral
chromaticachiral


Going Further

  • Reflective symmetry can occur with an axis on other notes of the scale. Are those interesting?
  • The reflection axis can be on a tone, or between two tones. Is that interesting?
  • Is there a more optimal or elegant way to find the reflective symmetry axes of a scale?
  • Are there chiral enantiomorph pairs that are both named scales?

Combined Symmetry

Using similar methods to those above, we can identify all the scales that are both palindromic and have rotational symmetry. Here they are:


273 

585 

Interval Spectrum / Richness / Interval Vector

Howard Hanson, in the book "Harmonic Materials"4, posits that the character of a sonority is defined by the intervals that are within it. This definition is also called an "Interval Vector"5. Furthermore, in his system an interval is equivalent to its inverse, i.e. the sonority of a perfect 5th is the same as the perfect 4th. Hanson categorizes all intervals as being one of six classes, and gives each a letter: p m n s d t, ordered from most consonant (p) to most dissonant (t). When an interval appears more than once in a sonority, it is superscripted with a number, like p2.


P - the Perfects (5 or 7)

This is the interval of a perfect 5th, or perfect 4th.


M - The Major Third (4 or 8)

This is the interval of a major 3rd, or minor 6th


N - The Minor Third (3 or 9)

This is the interval of a minor 3rd, or a major 6th


S - the second (2 or 10)

This is the interval of a major 2nd, or minor 7th


D - the Diminished (1 or 11)

Intervals of a minor 2nd, or a major 7th


T - the Tritone (6 semitones)

For example, in Hanson's analysis, a C major triad has the sonority pmn, because it contains one perfect 5th between the C and G (p), a major third between C and E (m), and a minor third between the E and G (n). Interesting to note that a minor triad of A-C-E is also a pmn sonority, just with the arrangement of the minor and major intervals swapped. The diminished seventh chord 585  has the sonority n4t2 because it contains four different minor thirds, and two tritones.

We can count the appearances of an interval using a method called "cyclic autocorrelation". To find out how many of interval X appear in a scale, we rotate a copy of the scale by X degrees, and then count the number of positions where an "on" bit appears in both the original and copy.

function find_spectrum($scale) { $spectrum = array(); $rotateme = $scale; for ($i=0; $i<6; $i++) { $rotateme = rotate_bitmask($rotateme, $direction = 1, $amount = 1); $spectrum[$i] = countOnBits($scale & $rotateme); } // special rule: if there is a tritone in the sonority, it will show up twice, so we divide by 2 $spectrum[5] = $spectrum[5] / 2; return $spectrum; }

All members of the same modal family will have the same interval spectrum, because the spectrum doesn't care about rotation, and modes are merely rotations of each other. And there will be many examples of different families that have the same interval spectrum - for example, 281  and 275  both have the spectrum "pm3nd", but they are not modes of each other.

Below is a table of some of the named scales, and their interval spectrum. You'll notice that modal families will all have the same spectrum; hence every named mode of the diatonic scale (lydian, dorian, phrygian etc.) has the same spectrum of p6m3n4s5d2t

273 (augmented triad) m3
585 (diminished seventh) n4t2
661 (pentatonic major) p4mn2s3
859 (ultralocrian) p4m4n5s3d3t2
1193 (pentatonic minor) p4mn2s3
1257 (blues) p4m2n3s3d2t
1365 (whole tone) m6s6t3
1371 (altered) p4m4n4s5d2t2
1387 (locrian) p6m3n4s5d2t
1389 (half diminished) p4m4n4s5d2t2
1397 (major locrian) p2m6n2s6d2t3
1451 (phrygian) p6m3n4s5d2t
1453 (aeolian) p6m3n4s5d2t
1459 (phrygian dominant) p4m4n5s3d3t2
1485 (lydian diminished) p4m5n3s4d3t2
1493 (lydian minor) p2m6n2s6d2t3
1499 (bebop locrian) p5m5n6s5d4t3
1621 (prometheus) p2m4n2s4dt2
1643 (locrian natural 6) p4m4n5s3d3t2
1709 (dorian) p6m3n4s5d2t
1717 (mixolydian) p6m3n4s5d2t
1725 (bebop dorian) p7m4n5s6d4t2
1741 (ukranian dorian) p4m4n5s3d3t2
1749 (acoustic) p4m4n4s5d2t2
1753 (hungarian major) p3m3n6s3d3t3
1755 (octatonic) p4m4n8s4d4t4
2257 (hirajoshi) p3m2nsd2t
2275 (fifth mode of limited transposition) p4m2s2d4t3
2457 (augmented) p3m6n3d3
2475 (minor neapolitan) p4m5n3s4d3t2
2477 (harmonic minor) p4m4n5s3d3t2
2483 (enigmatic) p4m5n4s2d4t2
2509 (hungarian minor) p4m5n4s2d4t2
2535 (fourth mode of limited transposition) p6m4n4s4d6t4
2731 (major neapolitan) p2m6n2s6d2t3
2733 (heptatonia seconda) p4m4n4s5d2t2
2741 (major) p6m3n4s5d2t
2773 (lydian) p6m3n4s5d2t
2777 (lydian #2) p4m4n5s3d3t2
2869 (ionian augmented) p4m4n5s3d3t2
2901 (lydian augmented) p4m4n4s5d2t2
2925 (diminished) p4m4n8s4d4t4
2989 (bebop minor) p5m5n6s5d4t3
2997 (bebop major) p6m5n6s5d4t2
3055 (seventh mode of limited transposition) p8m8n8s8d8t5
3411 (enigmatic) p4m4n3s5d3t2
3445 (sixth mode of limited transposition) p4m6n4s6d4t4
3549 (third mode of limited transposition) p6m9n6s6d6t3
3765 (bebop dominant) p7m4n5s6d4t2
4095 (chromatic) p12m12n12s12d12t6

Going Further

  • Is there an optimal or elegant way to find all scales with a given spectrum?
  • What patterns appear in interval distribution?
  • Which are the most common, and least common spectra?

Hemitonia and Tritonia

One interesting approach to understanding scales is to look at the distribution of consonant and dissonant intervals in the scale. We did this already in the "spectrum analysis" above, but two intervals in particular are commonly regarded in more depth: tritones and semitones - which are deemed the most dissonant of intervals, and those that give a scale its spice and flavour.

A scale that contains semitones is called "hemitonic", and we will also refer to a hemitonic as the scale member that has the upper semitone neighbour (for example, mi-fa and ti-do in a major scale, the hemitones are mi and fa).

A scale that contains tritones is called "tritonic", and we will also refer to a tritonic as the scale member that has a tritone above it. This concept seems very close to the notion of "imperfection" (explained below)

Since hemitonia and tritonia are based on the interval spectrum, all the modes of a scale will have the same hemitonia. For example, the major scale is dihemitonic and ancohemitonic; thus so are dorian, phrygian, lydian, etc.

Number of tones# of Hemitonic Scales# of Tritonic Scales
300
41224
5140150
6335335
7413413
8322322
9165165
105555
111111
1211

It's well to know that so many scales are hemitonic and tritonic (having more than zero semitones or tritones), but obviously some scales are more hemitonic than others. A scale could have no hemitonics ("ahemitonic"), one ("unihemitonic"), two ("dihemitonic"), three ("trihemitonic") and so on. Let's look at the count of hemitones in our scales.

Number of hemitones found in all scales
Number of hemitones
tones in scale 0123456789101112
31000000000000
4191200000000000
51580600000000000
61301501401500000000
70021140210420000000
800007016884000000
9000000847290000
10000000004510000
1100000000001100
120000000000001

Fun fact: there are no scales with 11 hemitones. Do you understand why?

Number of tritones found in all scales
Number of tritones
tones in scale 0123456789101112
31000000000000
471680000000000
55407530500000000
61121021466960000000
70014112196847000000
800006216884800000
9000000847290000
10000000004510000
1100000000001100
120000000000001

Cohemitonia

Cohemitonia is the presense of two semitones consecutively in scale order. For instance, if a scale has three cromatically consecutive notes, then it is cohemitonic. We will also refer to the cehemitonic as the tone that has two semitone neighbours above it.

Modes

There is a lot of confusion about what is a "mode", chiefly because the word is used slightly differently in various contexts.

When we say "C major", the word "major" refers to a specific pattern of whole- and half-steps. The "C" tells us to begin that pattern on the root tone of "C".

Modes are created when you use the same patterns of whole- and half-steps, but you begin on a different step. For instance, the "D Dorian" mode uses all the same notes as C major (the white keys on a piano), but it begins with D. Compared with the Major (also known as "Ionian" mode), the Dorian sounds different, because relative to the root note D, it has a minor third and a minor seventh.

The best way to understand modes is to think of a toy piano where the black keys are just painted on - all you have are the white keys: C D E F G A B. Can you play a song that sounds like it's in a minor key? You can't play a song in C minor, because that would require three flats. So instead you play the song with A as the root (A B C D E F G). That scale is a mode of the major scale, called the Aeolian Mode.

When you play that minor scale, you're not playing "C minor", you're playing the relative minor of C, which is "A minor". Modes are relatives of each other if they have the same pattern of steps, starting on different steps.

To compute a mode of the current scale, we "rotate" all the notes down one semitone. then if the rotated notes have an on bit in the root, then it is a mode of the original scale.

			101010110101 = 2741 - major scale, "ionian" mode
			110101011010 = 3418 - rotated down 1 semitone - not a scale
			011010101101 = 1709 - rotated down 2 semitones - "dorian"
			101101010110 = 2902 - rotated down 3 semitones - not a scale
			010110101011 = 1451 - rotated down 4 semitones - "phrygian"
			101011010101 = 2773 - rotated down 5 semitones - "lydian"
			110101101010 = 3434 - rotated down 6 semitones - not a scale
			011010110101 = 1717 - rotated down 7 semitones - "mixolydian"
			101101011010 = 2906 - rotated down 8 semitones - not a scale
			010110101101 = 1453 - rotated down 9 semitones - "aeolian"
			101011010110 = 2774 - rotated down 10 semitones - not a scale
			010101101011 = 1387 - rotated down 11 semitones - "locrian"
			
function modes($scale) { $rotateme = $scale; $modes = array(); for ($i = 0; $i < 12; $i++) { $rotateme = rotate_bitmask($rotateme); if (($rotateme & 1) == 0) { continue; } $modes[] = $rotateme; } return $modes; } function rotate_bitmask($bits, $direction = 1, $amount = 1) { for ($i = 0; $i < $amount; $i++) { if ($direction == 1) { $firstbit = $bits & 1; $bits = $bits >> 1; $bits = $bits | ($firstbit << 11); } else { $firstbit = $bits & (1 << 11); $bits = $bits << 1; $bits = $bits & ~(1 << 12); $bits = $bits | ($firstbit >> 11); } } return $bits; }

When we do this to every scale, we see modal relationships between scales, and we also discover symmetries when a scale is a mode of itself on another degree.

A complete list of all modal families

Modal families with 3 tones

m3273 (augmented triad) 

Modal families with 4 tones

pm3nd281 401 547 2321 
p2m2nd291 393 561 2193 
pm3nd275 305 785 2185 
m3s2t277 337 1093 1297 
pmn2st329 553 581 1169 
p2mn2s297 549 657 1161 
m2s2t2325 1105 
pmn2st293 593 649 1097 
n4t2585 (diminished seventh) 

Modal families with 5 tones

pm3ns2d2t285 465 1095 2595 3345 
p2m2n2sd2t457 569 583 2339 3217 
p2m3n2sd2313 551 913 2323 3209 
p2m2s2d2t2327 453 1137 2211 3153 
p2m2n2sd2t295 625 905 2195 3145 
pm3ns2d2t279 369 1809 2187 3141 
p2m3n2sd2283 433 1571 2189 2833 
p3m2n2s2d425 565 1165 1315 2705 
pm3n2s2dt309 849 1101 1299 2697 
p3mn2s2dt421 (japenese (b)) 653 (kumoi) 1129 1187 (insen) 2641 
pmn4sdt2589 617 841 1171 2633 
p2mn3s2dt361 557 1163 1681 2629 
p3m2nsd2t397 (hirajoshi) 419 (japanese (a)) 1123 (iwato) 2257 (hirajoshi) 2609 
pm2n2s2dt2333 837 1107 1233 2601 
p2mn3s2dt301 721 1099 1673 2597 
p2m4n2d2409 563 803 2329 2449 
p2m4n2d2307 787 817 2201 2441 
p2m3ns2dt405 675 1125 1305 2385 
p2m2n3sdt613 659 809 1177 2377 
pm2n2s2dt2357 651 1113 1617 2373 
p2m3n2d2t403 611 793 2249 2353 
p2m2n3sdt595 665 805 1225 2345 
pmn4sdt2587 601 713 1609 2341 
pm3n2s2dt345 555 1425 1605 2325 
p3m2nsd2t355 395 (balinese) 1585 2225 2245 
p2m3ns2dt339 789 1221 1329 2217 
p3mn2s2dt331 709 1201 1577 2213 
p3m2n2s2d299 689 1417 1573 2197 
m4s4t2341 1109 1301 1349 1361 
p2m2n2s3t597 681 1173 1317 1353 
p4mn2s3661 (pentatonic major) 677 1189 (egyptian) 1193 (pentatonic minor) 1321 

Modal families with 6 tones

p2m3n2s3d4t287 497 2191 3143 3619 3857 
p3m2n3s3d3t489 573 1167 2631 3363 3729 
p2m3n3s3d3t317 977 1103 2599 3347 3721 
p3m2n2s3d3t2485 655 1145 2375 3235 3665 
p2m2n4s2d3t2591 633 969 2343 3219 3657 
p2m3n3s3d3t377 559 1937 2327 3211 3653 
p4m2ns2d4t2399 483 2247 2289 3171 3633 
p3m2n2s3d3t2335 965 1265 2215 3155 3625 
p3m2n3s3d3t303 753 1929 2199 3147 3621 
p2m4n3s2d3t473 571 1607 2333 2851 3473 
p3m4n3s2d3315 945 1575 2205 2835 3465 
p2m4ns4d2t2469 1141 1309 1351 2723 3409 
p3m3n3s3d2t629 937 1181 1319 2707 3401 
pm4n2s4d2t2373 1117 1303 1873 2699 3397 
p3m3n2s2d3t2467 797 1223 2281 2659 3377 
p4m2n3s3d2t669 933 1191 1257 (blues) 2643 3369 
p2m2n4s3d2t2605 745 1175 1865 2635 3365 
p3m4n2s2d3t413 931 1127 2513 2611 3353 
pm4n2s4d2t2349 1111 1489 1861 2603 3349 
p3m3n2s2d3t2461 839 1139 2467 2617 3281 
p3m4n3sd3t627 807 921 2361 2451 3273 
p3m4n2s2d3t371 791 1841 2233 2443 3269 
p4m2n2s2d3t2459 711 1593 2277 2403 3249 
p4m3n2s3d2t679 917 1253 1337 2387 3241 
p4m2n3s3d2t663 741 1209 1833 2379 3237 
p3m4n3sd3t615 825 915 2355 2505 3225 
p3m3n3s3d2t599 697 1481 1829 2347 3221 
p3m4n3s2d3441 567 1827 2331 2961 3213 
p4m2s2d4t3455 2275 (fifth mode of limited transposition) 3185 
p4m2n2s2d3t2423 909 1251 2259 2673 3177 
p3m3n2s2d3t2407 739 1817 2251 2417 3173 
p3m3n2s2d3t2359 907 1649 2227 2501 3161 
p2m4ns4d2t2343 1393 1477 1813 2219 3157 
p2m4n3s2d3t311 881 1811 2203 2953 3149 
p3m3n3s3d2t437 1133 1307 1699 2701 2897 
p2m2n5s2d2t2621 873 1179 1683 2637 2889 
p2m2n4s3d2t2365 1115 1675 1745 2605 2885 
p3m4n3sd3t435 795 1635 2265 2445 2865 
p2m3n4s2d2t2667 869 1241 1619 (prometheus neopolitan) 2381 2857 
p2m2n5s2d2t2603 729 1611 1737 2349 2853 
p3m4n3sd3t411 867 1587 2253 2481 2841 
p3m3n3s3d2t347 1457 1579 1733 2221 2837 
p4m2n3s3d2t429 1131 1443 1677 2613 2769 
p2m3n4s2d2t2619 857 1427 1613 2357 2761 
p4m2n3s3d2t363 1419 1581 1713 2229 2757 
p4m3n2s3d2t427 1379 1421 1589 2261 2737 
p2m4n2s4dt2853 1237 1333 1357 1363 2729 
p4m2n3s4dt725 1205 1325 1355 1705 2725 
p3m4n3s2d2t821 851 1229 1331 2473 2713 
p5m2n3s4d693 1197 1323 1449 1701 (dominant 7th) 2709 
p2m2n4s2d2t3845 1235 (tritone scale) 2665 
p3m2n4s2d2t2723 813 1227 1689 2409 2661 
p3m2n4s2d2t2717 843 1203 1641 2469 2649 
p4m2n3s4dt685 1195 1385 1445 1685 2645 
p3m6n3d3819 (augmented inverse) 2457 (augmented) 
p3m4n3s2d2t691 811 1433 1637 2393 2453 
p2m2n4s2d2t3715 1625 2405 
p2m4n2s4dt2683 1369 1381 1429 1621 (prometheus) 2389 
m6s6t31365 (whole tone) 

Modal families with 7 tones

p3m4n4s4d5t505 575 2335 3215 3655 3875 3985 
p3m4n4s4d5t319 1009 2207 3151 3623 3859 3977 
p3m4n3s5d4t2501 1149 1311 2703 3399 3747 3921 
p3m3n5s4d4t2637 1001 1183 2639 3367 3731 3913 
p2m4n4s5d4t2381 1119 2001 2607 3351 3723 3909 
p4m4n3s3d5t2499 799 2297 2447 3271 (mela raghupriya (42)) 3683 3889 
p4m3n4s4d4t2671 997 1273 2383 3239 (mela tanarupi (6)) 3667 3881 
p3m3n5s4d4t2607 761 1993 2351 3223 3659 3877 
p4m4n3s3d5t2415 995 2255 2545 3175 3635 3865 
p3m4n3s5d4t2351 1521 1989 2223 3159 3627 3861 
p4m3n4s4d4t2493 1147 1679 2621 2887 3491 3793 
p3m4n5s3d4t2635 985 (mela sucharitra (67)) 1615 2365 2855 3475 3785 
p4m4n4s4d4t379 1583 1969 2237 2839 3467 3781 
p5m3n3s4d4t2491 1423 1597 2293 2759 (mela pavani (41)) 3427 3761 
p4m4n3s5d3t2981 (mela kantamani (61)) 1269 1341 1359 2727 (mela manavati (5)) 3411 (enigmatic) 3753 
p5m3n4s5d3t757 1213 1327 1961 2711 3403 3749 
p4m4n4s3d4t2829 979 (mela dhavalambari (49)) 1231 2537 2663 3379 3737 
p5m3n4s5d3t701 1199 1513 1957 2647 3371 3733 
p4m4n4s4d4t445 1135 1955 2615 3025 3355 3725 
p5m3n2s3d5t3487 911 2291 2503 (mela jhalavarali (39)) 3193 3299 3697 
p4m3n4s3d4t3847 973 (mela syamalangi (55)) 1267 2471 (mela ganamurti (3)) 2681 3283 (mela visvambari (54)) 3689 
p4m4n4s3d4t2755 815 1945 2425 2455 3275 (mela divyamani (48)) 3685 
p4m3n4s3d4t3719 971 (mela gavambodhi (43)) 1657 2407 2533 3251 (mela hatakambari (18)) 3673 
p4m4n3s5d3t2687 1401 1509 1941 2391 3243 (mela rupavati (12)) 3669 
p3m4n5s3d4t2623 889 1939 2359 3017 3227 3661 
p5m3n2s3d5t3463 967 (mela salagam (37)) 2279 2531 3187 3313 3641 
p5m3n3s4d4t2431 1507 1933 2263 2801 3179 3637 
p4m3n4s4d4t2367 1777 1931 2231 3013 3163 3629 
p3m5n3s4d4t2477 1143 1863 2619 2979 3357 3537 
p4m5n4s3d4t631 953 (mela yagapriya (31)) 1831 2363 2963 3229 3529 
p3m5n3s4d4t2375 1815 1905 2235 2955 3165 3525 
p4m4n4s3d4t2475 1595 1735 (mela navanitam (40)) 2285 2845 2915 3505 
p5m4n4s4d3t949 (mela mararanjani (25)) 1261 1339 1703 (mela vanaspati (4)) 2717 2899 3497 
p4m3n5s4d3t2749 1211 1687 1897 2653 2891 3493 
p4m6n4s2d4t827 947 (mela gayakapriya (13)) 1639 2461 2521 (mela dhatuvardhani (69)) 2867 3481 
p3m5n4s4d3t2699 1497 (mela jyotisvarupini (68)) 1623 1893 2397 2859 3477 
p4m5n4s3d4t443 1591 1891 2269 2843 2993 3469 
p4m4n2s4d4t3471 1479 (mela jalarnavam (38)) 1821 2283 2787 3189 3441 
p5m3n4s4d3t2941 (mela jhankaradhvani (19)) 1259 1447 (mela ratnangi (2)) 1693 2677 2771 (marva theta) 3433 
p3m4n4s4d3t3747 1431 1629 1881 2421 2763 (mela suvarnangi (47)) 3429 
p4m5n3s4d3t2939 (mela senavati (7)) 1383 1437 1653 2517 (mela latangi (63)) 2739 (mela suryakantam (17)) 3417 
p2m6n2s6d2t31367 1373 1397 (major locrian) 1493 (lydian minor) 1877 2731 (major neapolitan) 3413 (leading whole tone) 
p3m5n4s4d3t2885 1245 1335 1875 2715 2985 3405 
p5m4n3s3d4t2925 935 (mela kanakangi (1)) 1255 2515 (mela kamavarardhani (51)) 2675 3305 3385 
p3m4n4s4d3t3861 1239 1491 (mela namanarayani (50)) 1869 2667 2793 3381 
p4m3n5s4d3t2733 1207 1769 1867 2651 2981 3373 
p5m4n3s3d4t2743 919 1849 2419 (persian) 2507 (mela subhapantuvarali (45)) 3257 (mela chalanata (36)) 3301 
p4m5n4s2d4t2871 923 1651 (oriental (b)) 2483 (enigmatic) 2509 (hungarian minor) 2873 3289 (mela rasikapriya (72)) 
p4m5n3s4d3t2855 1395 (oriental (a)) 1485 (lydian diminished) 1845 2475 (minor neapolitan) 2745 (mela sulini (35)) 3285 (mela chitrambari (66)) 
p4m6n4s2d4t823 883 1843 2459 2489 (mela gangeyabhusani (33)) 2969 3277 (mela nitimati (60)) 
p5m3n4s4d3t2727 1483 (mela bhavapriya (44)) 1721 (mela vagadhisvari (34)) 1837 2411 2789 3253 (mela naganandini (30)) 
p5m4n4s4d3t695 1465 (mela ragavardhani (32)) 1765 1835 2395 2965 3245 (mela varunapriya (24)) 
p4m4n4s3d4t2439 1763 1819 2267 2929 2957 3181 
p3m3n6s3d3t3877 1243 1691 1747 (mela ramapriya (52)) 2669 2893 2921 
p3m3n6s3d3t3731 1627 1739 (mela sadvidhamargini (46)) 1753 (hungarian major) 2413 2861 2917 
p4m4n5s3d3t2875 1435 1645 1715 (mela chakravakam (16)) 2485 (mela sarasangi (27)) 2765 (mela dharmavati (59)) 2905 
p4m4n4s5d2t21371 (altered) 1389 (half diminished) 1461 (hindu) 1707 (javanese (pelog)) 1749 (acoustic) 2733 (heptatonia seconda) 2901 (lydian augmented) 
p4m4n5s3d3t2859 (ultralocrian) 1459 (phrygian dominant) 1643 (locrian natural 6) 1741 (ukranian dorian) 2477 (harmonic minor) 2777 (lydian #2) 2869 (ionian augmented) 
p6m3n4s5d2t1387 (locrian) 1451 (phrygian) 1453 (aeolian) 1709 (dorian) 1717 (mixolydian) 2741 (major) 2773 (lydian) 

Modal families with 8 tones

p4m5n5s6d6t2509 1151 2623 3359 3727 3911 4003 4049 
p4m5n6s5d6t2639 1017 2367 3231 3663 3879 3987 4041 
p4m5n5s6d6t2383 2033 2239 3167 3631 3863 3979 4037 
p5m5n5s5d6t2507 1599 2301 2847 3471 3783 3939 4017 
p5m5n5s6d5t21013 1277 1343 2719 3407 3751 3923 4009 
p5m4n6s6d5t2765 1215 2025 2655 3375 3735 3915 4005 
p5m6n5s4d6t2831 1011 2463 2553 3279 3687 3891 3993 
p5m5n5s6d5t2703 1529 2021 2399 3247 3671 3883 3989 
p5m5n5s5d6t2447 2019 2271 3057 3183 3639 3867 3981 
p5m5n4s5d6t3503 1823 2299 2959 3197 3527 3811 3953 
p5m4n6s5d5t31005 1275 1695 2685 2895 3495 3795 3945 
p4m5n6s5d5t3763 1631 2009 2429 2863 3479 3787 3941 
p5m5n5s5d5t31003 1439 1661 2549 2767 3431 3763 3929 
p4m6n4s7d4t31375 1405 1525 2005 2735 3415 3755 3925 
p4m5n6s5d5t3893 1247 2003 2671 3049 3383 3739 3917 
p6m5n4s4d6t3927 999 2511 2547 3303 3321 3699 3897 
p5m5n5s5d5t3863 1523 1997 2479 2809 3287 3691 3893 
p5m4n6s5d5t3735 1785 1995 2415 3045 3255 3675 3885 
p5m5n4s5d6t3479 1991 2287 3043 3191 3569 3643 3869 
p6m4n4s5d6t3495 1935 2295 3015 3195 3555 3645 3825 
p5m5n5s5d5t3989 1271 1871 2683 2983 3389 3539 3817 
p6m5n5s5d5t2759 1839 1977 2427 2967 3261 3531 3813 
p5m5n6s4d5t3987 1659 1743 2541 (algerian) 2877 2919 3507 3801 
p6m5n5s6d4t21403 (eight tone spanish) 1517 1711 (jewish (adonai malakh)) 1973 2749 2903 3499 3797 
p5m6n6s4d5t2891 1647 1971 2493 2871 3033 3483 3789 
p6m5n4s5d5t3983 1487 1853 2539 2791 3317 3443 3769 
p7m4n5s6d4t21455 1515 1725 (bebop dorian) 1965 2775 2805 (japanese (ichikosucho)) 3435 3765 (bebop dominant) 
p6m5n5s6d4t21391 1469 1781 1963 2743 3029 3419 (jewish (magan abot)) 3757 
p6m5n5s5d5t2957 1263 1959 2679 3027 3387 3561 3741 
p6m4n4s4d6t4975 2535 (fourth mode of limited transposition) 3315 3705 
p6m5n4s5d5t3943 1511 1949 2519 2803 3307 3449 3701 
p5m5n6s4d5t3879 1779 1947 2487 2937 3021 3291 3693 
p5m5n5s5d5t3751 1913 1943 2423 3019 3259 3557 3677 
p5m7n5s4d5t2955 1655 1895 2525 2875 2995 3485 3545 
p4m7n4s6d4t31399 1501 1879 1909 2747 2987 3421 3541 
p5m7n5s4d5t2887 1847 1907 2491 2971 3001 3293 3533 
p6m6n5s4d5t2951 1767 1851 2523 2931 2973 3309 3513 
p5m5n6s5d4t31499 (bebop locrian) 1723 1751 1901 2797 2909 2923 3509 
p6m5n6s5d4t21467 1719 1773 1899 2781 2907 2997 (bebop major) 3501 
p4m6n4s6d4t41495 1885 2795 3445 (sixth mode of limited transposition) 
p5m5n6s5d4t31463 1757 1771 1883 2779 2933 2989 (bebop minor) 3437 
p4m4n8s4d4t41755 (octatonic) 2925 (diminished) 

Modal families with 9 tones

p6m6n6s7d8t3511 2303 3199 3647 3871 3983 4039 4067 4081 
p6m6n7s7d7t31021 1279 2687 3391 3743 3919 4007 4051 4073 
p6m6n7s7d7t3767 2041 2431 3263 3679 3887 3991 4043 4069 
p6m7n7s6d7t31019 1663 2557 2879 3487 3791 3943 4019 4057 
p6m7n6s8d6t31407 1533 2037 2751 3423 3759 3927 4011 4053 
p6m7n7s6d7t3895 2035 2495 3065 3295 3695 3895 3995 4045 
p7m7n6s6d7t31015 1855 2555 2975 3325 3535 3815 3955 4025 
p7m6n7s7d6t31531 1727 2029 2813 2911 3503 3799 3947 4021 
p7m6n7s7d6t31471 1789 2027 2783 3061 3439 3767 3931 4013 
p7m7n6s6d7t3959 2023 2527 3059 3311 3577 3703 3899 3997 
p7m6n6s6d7t41007 1951 2551 3023 3323 3559 3709 3827 3961 
p6m7n6s7d6t41527 1887 2013 2811 2991 3453 3543 3819 3957 
p6m6n8s6d6t41759 1787 2011 2927 2941 3053 3511 3803 3949 
p6m7n6s7d6t41503 1917 2007 2799 3051 3447 3573 3771 3933 
p7m6n6s6d7t4991 1999 2543 3047 3319 3571 3707 3833 3901 
p8m6n6s7d6t31519 1967 1981 2807 3031 3451 3563 3773 3829 (japanese (taishikicho)) 
p7m7n7s6d6t31783 1903 1979 2939 2999 3037 (nine tone scale) 3517 3547 3821 
p7m7n7s6d6t31775 1915 1975 2935 3005 3035 3515 3565 3805 
p6m9n6s6d6t31911 3003 3549 (third mode of limited transposition) 

Modal families with 10 tones

p8m8n8s8d9t41023 2559 3327 3711 3903 3999 4047 4071 4083 4089 
p8m8n8s9d8t41535 2045 2815 3455 3775 3935 4015 4055 4075 4085 
p8m8n9s8d8t41791 2043 2943 3069 3519 3807 3951 4023 4059 4077 
p8m9n8s8d8t41919 2039 3007 3067 3551 3581 3823 3959 4027 4061 
p9m8n8s8d8t41983 2031 3039 3063 3567 3579 3831 3837 3963 4029 
p8m8n8s8d8t52015 3055 (seventh mode of limited transposition) 3575 3835 3965 

Modal families with 11 tones

p10m10n10s10d10t52047 3071 3583 3839 3967 4031 4063 4079 4087 4091 4093 

Modal families with 12 tones

p12m12n12s12d12t64095 (chromatic) 

Proximity

We would consider two scales to be "near" each other, if they bear many similarities; sharing many of the same notes, such that it would take few transformations to turn one into the other.

This distance measured by transformation is almost the same thing as a "Levenshtein Distance". The normal definition of a Levenshtein Distance on a string of characters allows three kinds of transformation: insertion, deletion, and substitution. Our scale transformations are different from a string transformation, in that scales can't do substitution, instead we move tones up or down with rules for handling collision. This difference means we can't employ all the elegant formulas bequeathed to us by Vladimir Levenshtein.

We can transform a scale in three ways:

  • Move a tone up or down by a semitone
  • Remove a tone
  • Add a tone

It is simple to generate all the scales at a distance of 1, just by performing all possible transformations to every interval above the root.

function findNearby($scale) { $near = array(); for ($i=1; $i<12; $i++) { if ($scale & (1 << ($i))) { $copy = $scale; $off = $copy ^ 1 << ($i); $near[] = $off; $copy = $off | 1 << ($i - 1); $near[] = $copy; if ($i != 11) { $copy = $off | 1 << ($i + 1); $near[] = $copy; } } else { $copy = $scale; $copy = $copy | 1 << ($i); $near[] = $copy; } } return $near; }

Example

Here are all the scales that are a distance of 1 from the major scale, aka 2741 , shown here as a simple C major scale:

add a tone at C#2743 
lower the D to D♭2739 (mela suryakantam (17)) 
raise the D to D#2745 (mela sulini (35)) 
delete the D2737 
add a tone at D#2749 
lower the E to E♭2733 (heptatonia seconda) 
raise the E to Fsame as deleting E
delete the E2725 
lower the F to Esame as deleting F
raise the F to F#2773 (lydian) 
delete the F2709 
add a tone at F#2805 (japanese (ichikosucho)) 
lower the G to G♭2677 
raise the G to G#2869 (ionian augmented) 
delete the G2613 
add a tone at G#2997 (bebop major) 
lower the A to A♭2485 (mela sarasangi (27)) 
raise the A to A#3253 (mela naganandini (30)) 
delete the A2229 
add a tone at A#3765 (bebop dominant) 
lower the B to B♭1717 (mixolydian) 
raise the B to Csame as deleting B
delete the B693 

Evidently since every scale will have a collection of neighbours similar to this one, a complete graph of scale proximity is large, but not unfathomably so.

We can also easily calculate the Levenshtein distance between two scales. Apply an XOR operation to expose changed bits between two scales. Adjacent on bits in the XOR where the bits were different in the original scale identify a "move" operation. Any bits remaining are either an addition or deletion.

function levenshtein_scale($scale1, $scale2) { $distance = 0; $d = $scale1 ^ $scale2; for ($i=0; $i<12; $i++) { if ( ($d & (1 << ($i))) && ($d &(1 << ($i+1))) && ($scale1 & (1 << ($i))) != ($scale1 & (1 << ($i+1))) ) { $distance++; $d = $d & ( ~ (1 << ($i))); $d = $d & ( ~ (1 << ($i+1))); } } for ($i=0; $i<12; $i++) { if (($d & (1 << ($i)))) { $distance++; } } return $distance; }

Imperfection

Imperfection is a concept invented (so far as I can tell) by William Zeitler, to describe the presence or absense of perfect fifths in the scale tones. Any tone in the scale that does not have the perfect fifth above it represented in the scale is an "imperfect" tone. The number of imperfections is a metric that plausibly correlates with the perception of dissonance in a sonority.

The only scale that has no imperfections is the 12-tone chromatic scale.

This table differs from Zeitler's6, because this script does not de-duplicate modes. If an imperfection exists in a scale, it will also exist in all the sibling modes of that scale. Hence the single imperfect tone in the 11-tone scale is found 11 times (in 11 scales that are all modally related), whereas Zeitler only counts it as one.

number of notes in scale # of Imperfections
0123456
10000000
20000000
30001000
400816700
50530754050
60669146102121
70784196112140
808841686200
9097284000
10010450000
1101100000
121000000

Going Further

  • Looking at the table, there are 5 scales of 5 tones with one imperfection. There are 6 scales of 6 tones with only one imperfection. The same patten continues right up to 11 tones. What causes this pattern?
  • The only 7-note scale with only one imperfection is the major diatonic scale (plus all its modes). Is it a coincidence that we find this scale sonically pleasing?

Negative

One peculiar way we can manipulate a scale is to "flip its bits" -- so that every bit that is on becomes off, and all that were of are turned on. If you flip a scale with a root tone, you will get a non-scale without a root tone; so it's not so useful to speak of negating a scale, instead we negate an entire modal family to find the modal family that is its negative.

For example, one that's easy to conceptualize is the major scale, which (in C) occupies all the white keys on a piano. The negative of the major scale is all the notes that aren't in the major scale - just the black keys, which interestingly have the pattern of a major pentatonic (with F# as the root). In pitch class set theory, the negative of a set is called its "complement", and Dr Forte named complementary pairs with matching numbers.

Chord Relationships

In the case of heptatonic chords, we can use the stacked-third members of a scale to create a chord. More on this later.

Glossary

achiral
Not having chirality, i.e. the mirror image can be achieved by rotation.
ancohemitonic
A scale that is not cohemitonic. This either means it contain no semitones (and thus is anhemitonic), or contain semitones (being hemitonic) where none of the semitones appear consecutively in scale order.
anhemitonic
A scale that does not include any semitones
atritonic
containing no tritones
chiral
The quality of being different from ones own mirror-image, in a way that can not be achieved by rotation.
cohemitonic
Cohemitonic scales contain two or more semitones (making them hemitonic) such that two or more of the semitones appear consecutively in scale order. Example: the Hungarian minor scale
dicohemitonic
A scale that contains exactly two semitones consecutively in scale order
dihemitonic
A scale that contains exactly two semitones
enantiomorph
the result of a transformation by reflection, i.e. with its interval pattern reversed
hemitonic
A scale that includes semitones
hepatonic
A scale with seven tones. For example, the major scale is heptatonic.
imperfection
A scale member where the perfect fifth above it is not in the scale
interval pattern
The sequence of semitones, tones, and larger intervals, that describe a scale. For example, a major scale is "T T S T T T S". Expressed numerically, a major scale has the interval pattern [2,2,1,2,2,2]; the final interval is implied.
interval spectrum
A signature invented by Howard Hanson, describing all the intervals that can be found in a sonority
octatonic
A scale with eight tones.
palindromic
A scale that has the same interval pattern forward and backward.
pentatonic
A scale with five tones
proximity
The number of transformations required to change one scale into another
root
The lowest tone of the scale, signifying the tone upon which all others are measured as an interval above
scale
A set of tones starting on a root, contained within one octave, having no more than a major third leap
sonority
The whole of a sound, comprised of all component tones
symmetry
Having the ability to transform into itself by reflection or rotation
tone
A single entity having a pitch, as in one member of a scale
trihemitonic
A scale that contains exactly three semitones
tritonic
containing one or more tritones
truncation
A scale produced by removing tones from another scale
unhemitonic
A scale that contains only one semitone

Citations

1 - William Zeitler, "All The Scales", http://allthescales.org

2 - Andrew Duncan, "Combinatorial Music Theory", Journal of the Audio Engineering Society, vol. 39, pp. 427-448. (1991 June), retrieved from http://andrewduncan.net/cmt/

3 - http://allthescales.org/

4 - Howard Hansen, "Harmonic Materials Of Modern Music", ISBN 978-0891972075

5 - Paul Nelson, "Pitch Class Sets" http://composertools.com/Theory/PCSets.pdf

6 - Zeitler, William, table of imperfections counted in scales. https://allthescales.org/intro.html#Perfection


This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. Scale notation generated by VexFlow, and MIDI playback by MIDI.js