20180705

All The Intervals

The so-called 'all interval tetrachord' is not exactly news to musicians, although that this news is only dozens, and not hundreds, of years old is possibly surprising. Its (or more accurately their, since there's more than one) 'interestingness' having been exposed only (relatively) recently is due to pitch class set theorists who invented the 'interval-class-vector' (scare-quotes because of course it's in only the loosest, most informal, mathematically-anathematical, way any kind of a vector). This intervalency is the thing which usually turns up in PC set theory as a comma separated list of numbers between a pair of angled brackets, such as <2,5,4,3,6,1>. It lists - in order - the frequency of occurence of differences between each pair of pitch classes in the set it's being used to characterise, where the differences range from 1 to 6 in the 12 tone system (because the shortest distance between any two 'hour points' on a clock is always going to be between 1 and 6).

History

Although it would have easily been possible for any (say) 15th century musician to notice that the tetrad comprising (say) the pitches B, C, E♭ and F carried (between B and C, B and E♭, B and F, C and E♭, C and F, E♭ and F) respective separations of 1, 4, 6, 3, 5, 2 semitones - which is to say exactly one of each possible separation between any pair of notes, it's not clear that this would have been considered in any way remarkable.

Indeed there seems little evidence that musicians - or even mathematicians - were considerate of the number of possible tetrachords (or chords of any size) possible within a universe of twelve pitches before the middle of the 19th century. Luigi Verdi's survey article of proto, pre-USAnian if you will, pitch class set theories "The History of Set Theory from a European point of view" is well worth a read in this regard. There are, by the way, 43 such tetrachords and only 4 of these have this property.

Note that when we say 43, it depends on what you are counting - in this case it's all of the distinct shapes, their reflections, and homometries. If you ignore reflections (i.e. if you accept the essential identity of congruent shapes regardless of mirroring) it drops to 29 since 15 of the quadrilaterals have (at least) bilateral symmetry, and the remaining 28 turn up as 14 asymmetric mirror-image pairs. If, further, you ignore homometries (differently-shaped tetrads with the same intervalency) it drops to 28 because 2 of those 14 pairs of mutually inversional tetrads - which happen to be the subject of this very post - carry the same interval-class distribution, which is to say one instance of each interval class 1 to 6.

The 'classic' case

The Four All-Interval Tetratonic sets out of 12
1 3 2 6 2 3 1 6 1 2 4 5 4 2 1 5
The above four PC sets applied as chords (with F ≡ Pitch Class 0)
PC Sets 4-Z15 and 4-Z29

1326 chord (rooted on 440Hz)

1245 chord (rooted on 440Hz)

Wherefore art thou audio?

So what exactly is so remarkable about these chords anyway? They don't sound all that great, especially when instantiated in their most compacted forms as a Fortean Prime Form Cluster (4-Z15 ≡ {0, 1, 4, 6} and 4-Z29 ≡ {0, 1, 3, 7}). One may, reasonably easily, see in the second of these forms certain jazziness since the root, minor 3rd, 5th and (the '1' being bumped up an octave) the minor 9th are applicable (sans the 7th). And an inversion of the first set (subtracting 1 from each element makes its second member a new root) {11, 0, 3, 5} carries a minor 3rd, a 4th and a major 7th in plain sight, as it were.

A pair of jazzy applications of 4-Z29A and 4Z-15A
A Displacement and an Inversion

But mainly the interest is in its very rarity as a musical (or geometric) object, in that out of all possible PC sets (351 distinct polygonal shapes, 223 distinct polygonal congruences, 200 distinct polygonal homometries) that one may pull out of a 12 pitch class universe, only these 4 (2 if you equivalence reflections, 1 if you equivalence intervalencies) have the property that the frequency distribution (or, more simply, 'counts') of differences between every single pair of pitch classes in the set occur exactly once.

Actually, the term 'all-interval set' is rather a weak description of the kind of object we're considering since, with enough pitch class pairs to play with (i.e. given sufficiently large k for a particular N), it's almost impossible to avoid all interval (classes) turning up between them. A better term would bring out not only the completeness of the coverage but also its parsimony, i.e. with exactly one instance of each. It's that property which makes these objects interesting, and this term doesn't really do it justice. The closest we seem to have is from Gamer and Wilson, in 2003, who recognise and define "a difference set (modulo n) to be a set of distinct integers c1, …, ck (modulo n) for which the differences ci − cj (for ij) include each non-zero integer (modulo n) exactly once" - but these are not the words you are looking for. Combinatorial Mathematics has the term "planar difference set", but this terminology would likely be completely opaque to a musician.

Such a property cannot simply happen for any set. Firstly, the things being counted are difference classes (i.e. the smaller of the two values |ci-cj| and N-1-|ci-cj| separating numbers ci and cj on an N-houred clock) between pairs of integers (ci, cj in the range 0 … N-1) drawn from the finite set N. In the above example, N is of course 12 and only (absolute) differences of 1 to 6 may turn up. In general, the number of possible values one may have for differences is N/2 for even N and (N-1)/2 for odd N. For instance, when N = 13 the differences from the 'top of the clock, at 0 [or any multiple of 13]' range from 1 - 0 to 6 - 0 clockwise, then 7 - 0 (having passed the halfway point of 6½) is the same distance or separation as 13 - 7 = 6; 8 - 0 is the same separation as 13 - 8 = 5, etc.

Secondly, the number of interval classes is not arbitrary. A set containing k pitch classes (i.e. a set Pk = { c1, c2, c3, … ck-1, ck }) can carry only k(k-1)/2 - a triangular number - pitch class differences |ci - cj| (i ≠ j). Thus if the distribution of these k(k-1)/2 differences is to be a k(k-1)/2 length list of 'all-exactly' 1s, it's clear that the only tonalities which can possibly carry these objects are modelled by 2, 3, 6, 7, 12, 13, 20, 21, 30, 31, etc where the corresponding all-interval k-sets are of dichords in 2 & 3, trichords in 6 & 7, tetrachords in 12 & 13, pentachords in 20 & 21, hexachords in 30 & 31 etc.

As it happens, 13 has a similar quartet:

The Four All-Interval Tetratonic sets out of 13
1 3 2 7 2 3 1 7 2 1 4 6 4 1 2 6

It's evident that the shapes are pretty similar. That they are in adjacent tonalities does not necessarily mean that 'all interval' shapes would be expected to be broadly similar.

Interval Strings

The white numerical annotations (along the polygonal edges) are simply the number of skips between each pitch class in the ordered set (the polygonal vertices). This interval skip, or interval string notation is a vastly superior method of denoting pitch class sets, far out-transparenting the structure-hiding and equivalence-hiding opacity of listing pitch class numbers between braces - which should only ever be needed when you're on the verge of committing music to paper (performers do - after all - usually need to know what actual notes to play).

Its superiority as an 'interval showcase' is achieved by first tabulating all of its k(k-1) substrings (k of length 1, k of length 2, k of length 3 … k of length k-2 and k of length k-1), e.g. for one of the modes of 2317, such as 1723 (a mode being just a rotation of the set's polygonal representation within its 'N hour clock' space, keeping its top, 'noon', slot occupied):

substrings of '1723' of length
123
117172
772723
223317
331231

Then we just 'sum' each substring (by adding up its digits):

substring sums
1810
7912
2511
346

It is then evident by inspection that each interval in the set 1, 2, … 10, 11, 12 turns up exactly once. You may satisfy yourself that this works also for the pattern 2146 and any of its rotations. Pick any other interval string consistent with 13 and you will easily see either missing or duplicated intervals in the sum table.

As a matter of (possibly minor - since the tonal universes are so small) interest, the tonal space 6 - in which the all-interval sets are perforce triads - is the only space where an all-interval set could be the same size as its complementary set. As it happens there are a pair of such sets, 132 and 123 (interval string-wise) which are indeed not only self-inverse but self-complementary. If you insist on explicit Pitch Class Set representations (where their inversional relationships are much less immediately evident), they are {0,1,4} and {0,1,3}. The heptaphonic space 7 also admits of a pair of all-interval sets 142 and 124 ({0,1,5} and {0,1,3}) - also clearly (from their interval string representations) mutual inverses.

More solutions

The next possible tonality where we could find an all-interval set would be where the triangular number is 10 (where k = 5), which would have to be half the number of possible interval classes carried by it (i.e. where N = 20 or 21). But it turns out that there are no such sets to be found in 20. There is, however a single pair of mutually inverse 5-sets in 21 and it is 2513A (and its inverse 3152A), in interval-string denotation (where 'A' stands for an interval of 10). These are PC sets {0, 2, 7, 8, 11} and {0, 3, 4, 9, 11} in what would be their prime form, had Forte considered tonalities other than dodecaphonic, with a corresponding intervalency (interval-vector »choke«) of <1,1,1,1,1,1,1,1,1,1>. We can draw their polygons, but cannot reasonably represent them on a musical staff without use of microtonal notations.

The only pair of all-interval pentachords in 21ville
2 5 1 3 A 3 1 5 2 A

By now, we can see a definite 'meat cleaver' shape common to many of these polygons (the triangles from 6 and 7, being so geometrically limited, could be said to resemble either 4-Z15 or 4-Z29).

Next up would be 30, but again there are no sets to be found. This is somewhat over-compensated for in 31 where we jump to 5 homometric pairs - 13278A and 87231A, 47215C and 51274C, 12546D and 64521D, 17324E and 42371E, and finally 13625E and 52631E. Here they are (though this time we'll place the inversions underneath rather than to the right), each carrying exactly one instance of interval classes 1 to 15:

Five pairs of all-interval hexachords in 31ville
1 3 2 7 8 A 8 7 2 3 1 A 4 7 2 1 5 C 5 1 2 7 4 C 1 2 5 4 6 D 6 4 5 2 1 D 1 3 6 2 5 E 5 2 6 3 1 E 1 7 3 2 4 E 4 2 3 7 1 E

13625E hexachord (rooted on 440Hz)

17324E hexachord (rooted on 440Hz)

12546D hexachord (rooted on 440Hz)

47215C hexachord (rooted on 440Hz)

13278A hexachord (rooted on 440Hz)

The polygons are ordered in, again, what would be their Fortean prime forms; i.e. interval strings are descending reverse alphabetically ordered horizontally (which is, essentially, how prime form is calculated) and vertically (the upper reverse-sorting before its corresponding lower inversion). Note that by 'reverse alphabetically' we mean the reversed strings are ordered descendingly. Blame Forte.

It is the author's fancy that the meat cleaver remains visible in one of these pairs.

Systematic Solutions

Jedrezejewski and Johnson's useful 2013 paper, The Structure of Z-Related Sets, presents polynomials capable of representing both pitch class sets and their consequent intervalic distributions. These derive from the realm of crystallography and Patterson Functions. Briefly, it means that a pitch class set {p1, p2, … pk-1, pk} - as usual of size k and drawn from a tonality of order n - and its inversion may be represented by a polynomial in x:

P(x; k, n) = xp1 + xp2 + … + xpk-1 + xpk

P-1(x; k, n) = P(x-1; k, n) = x-p1 + x-p2 + … + x-pk-1 + x-pk

where, without loss of generality, 0 ≤ p1 < p2 < … pk-1 < pk < n are the k pitch classes in the set and where the consequent interval distribution between those pairs of pitch classes is measured as the coefficients of the k(k-1) powers of x in the expression P(x; k, n)P-1(x; k, n). Note that all exponents of x are taken modulo n. Thus x-p2, for example, may be rewritten as xn-p2 to regain positive exponents.

In our particular case we seek pitch class sets which carry exactly one instance of each interval from 1 to n-1. For this purpose we don't particularly care whether or not the set is in prime form (since we can always turn it into its prime form after we have found it) and so we might as well fix the first two pitch classes as 0 and 1 - or in other words have our sets be at least in normal form (with pitch class 0 at the beginning) and with the shortest interval of 1 - between those two pitch classes - right at the beginning of the set. Thus we seek those particular P(x; k, n) looking like:

1 + x + xp3 + … + xpk-1 + xpk

with 3 ≤ p3 < … pk-1 < pk < n. We can assume p3 > 2 since we've already accounted for the interval of 1 which would otherwise appear twice due to x2 and x. And of course k fixes n (as either the even k(k-1) or the odd k(k-1) + 1) because we're looking specifically for the resultant interval polynomial P(x; k, n)P(x-1; k, n)

= (1 + x + xp3 + … + xpk-1 + xpk)(1 + x-1 + x-p3 + … + x-pk-1 + x-pk)
= (1 + x-1 + x-p3 + … + x-pk-1 + x-pk) + (x + 1 + x1-p3 + … + x1-pk-1 + x1-pk)
+ (xp3 + xp3-1 + 1 + … + xp3-pk-1 + xp3-pk) + … + (xpk + xpk-1 + xpk-p3 + … + xpk-pk-1 + 1)

which we would wish to have equal k + x + x2 + x3 + x4 + … + x-3 + x-2 + x-1 by finding the right k-2 values for the remaining pi

We note that when k = 7, the consequent embedding tonality will be either 42 or 43. All-interval sets from the latter would, one supposes, be especially interesting to fans of Harry Partch.

Here are the solutions we get for k = 3 to 9. The pi are the pitch classes in the set and the iString column presents the interval string representation of the set (letters A-Z representing intervals of 10 to 35) from which one may readily recover prime forms by rotating the largest letters to the end of the string, and thence unpacking into the explicit set, if desired.

k = 3, n = 6 k = 3, n = 7
p1 p2 p3 p4 p5 p6 p7 p8 p9 iString p1 p2 p3 p4 p5 p6 p7 p8 p9 iString
0 1 3   123 0 1 3   124
0 1 4   132 0 1 5   142
k = 4, n = 12 k = 4, n = 13
p1 p2 p3 p4 p5 p6 p7 p8 p9 iString p1 p2 p3 p4 p5 p6 p7 p8 p9 iString
0 1 3 7   1245 0 1 3 9   1264
0 1 4 6   1326 0 1 4 6   1327
0 1 6 10   1542 0 1 5 11   1462
0 1 7 9   1623 0 1 8 10   1723
k = 5, n = 20 k = 5, n = 21
p1 p2 p3 p4 p5 p6 p7 p8 p9 iString
no solutions 0 1 4 14 16   13A25
  0 1 6 8 18   152A3
k = 6, n = 30 k = 6, n = 31
p1 p2 p3 p4 p5 p6 p7 p8 p9 iString
no solutions 0 1 3 8 12 18   12546D
  0 1 3 10 14 26   1274C5
  0 1 4 6 13 21   13278A
  0 1 4 10 12 17   13625E
  0 1 6 18 22 29   15C472
  0 1 8 11 13 17   17324E
  0 1 11 19 26 28   1A8723
  0 1 14 20 24 29   1D6452
  0 1 15 19 21 24   1E4237
  0 1 15 20 22 28   1E5263
k = 7, n = 42 k = 7, n = 43
no solutions no solutions
k = 8, n = 56 k = 8, n = 57
p1 p2 p3 p4 p5 p6 p7 p8 p9 iString
no solutions 0 1 3 13 32 36 43 52   12AJ4795
  0 1 4 9 20 22 34 51   135B2CH6
  0 1 4 12 14 30 37 52   1382G7F5
  0 1 5 7 17 35 38 49   142AI3B8
  0 1 5 27 34 37 43 45   14M7362C
  0 1 6 15 22 26 45 55   15974JA2
  0 1 6 21 28 44 46 54   15F7G283
  0 1 7 19 23 44 47 49   16C4L328
  0 1 7 24 36 38 49 54   16HC2B53
  0 1 9 11 14 35 39 51   1823L4C6
  0 1 9 20 23 41 51 53   18B3IA24
  0 1 13 15 21 24 31 53   1C2637M4
k = 9, n = 72 k = 9, n = 73
p1 p2 p3 p4 p5 p6 p7 p8 p9 iString
no solutions 0 1 3 7 15 31 36 54 63 1248G5I9A
  0 1 5 12 18 21 49 51 59 14763S28E
  0 1 7 11 35 48 51 53 65 164OD32C8
  0 1 9 21 23 26 39 63 67 18C23DO46
  0 1 11 20 38 43 59 67 71 1A9I5G842
  0 1 12 20 26 30 33 35 57 1B86432MG
  0 1 15 23 25 53 56 62 69 1E82S3674
  0 1 17 39 41 44 48 54 62 1GM23468B

Odd tonalities appear to be favoured systems for all-interval chords. Except for the absence of solutions for 43 - almost as if it had been singled out. Also, it has not escaped our notice that the above solution for 21 appears to violate the Prime Power Conjecture. Answers on a postcard, please, as to why it does not.

We can just squeeze out one more table for the 6 (inversional) pairs of all-interval sets using 10 pitch classes, in a 91 tonality (there are none in 90). This now breaks at least one 'limit' (albeit an artificial one of our own making), which is to say the one involved in displaying the PC set's interval string. The alphabet is no longer big enough to carry one of the intervals and consequently an asterisk (*) is employed to represent an interval of 36 - just over the range provided by a letter Z.

k = 10, n = 91
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 iString
0139274956617781126IM75G4A
01413243038404573139B6825SI
0157273544677780142K89NA3B
0158182029435965143A29EG6Q
01610232634415355154D387C2*
01716275660687073169BT4823I
0111153136436583891A4G57MI62
0112152548576585871B3AN98K24
0119222432366576851I3284TB96
0119475254626879881IS5286B93
0127334963727484871Q6GE92A34
0137395158666982861*2C783D45

20180528

Unlucky 4 sum

From pitches …

C13 chord

A thirteenth chord is principally known as an all bells and whistles dominant which is expected to resolve to its tonic chord a fifth below. It usually turns up as a dominant seventh (say C7) topped off with its relative supertonic minor (which would be Dm) for something in F major.

If we ignore the key it's in, we see that this chord comprises seven pitches separated by - in sequence - 4, 3, 3, 4, 3, 4 semitones. A final 3 semitones would 'round it off' to the top keynote 24 semitones above the root note.

C13 chord

The chord's two-octave path is thus 4334343 - of length 7 and sum 24 (for a two octave span). We've got that eighth top note in parentheses to indicate that it's not intended to be included in the chord's 'definition' (the pitch class - of 0 - is already included).

C13 chord

Now let's look instead at a Cm13th chord, perhaps resolving to F minor. Its key-independent two-octave path is 3434343 (see below - again with a parenthesised closing note).

Cm13 chord
C13 chord

Finally let's consider a more adventurous Cmajor13#11 chord, with a Lydian flavour imparted by that sharpened 11th. Despite the F#, it can still resolve quite nicely to F. Its key-independent two-octave path is 4343433 (see below - again with a parenthesised closing note). This time, however, we're colour-coding the notes because we're going to switch things around a bit.

… to Pitch Class Sets

CM13#11 chord

To construct the pitch class set modelling this particular chord, we drop the second part of the chord (green note heads) by an octave - which essentially turns them into pitch classes alongside the first (red note headed) part.

pitch classing

We'll now slide the green pitch heads to the left, where we can see that they sit between the red pitch heads, and that none of the pitch classes are duplicated.

pitch class set

We now have a formal PC set, with the intervals - in semitones - between the PCs forming the key-independent interval path 2221221 (7 PCs in the set, summing correctly to 12).

Again, that final green-headed parenthesised note is not in the set - it's shown only to elucidate the final 'wraparound' step (of 1 semitone) back to pitch class 0.

The alert reader will note that the PC set resulting directly from CΔ13#11 is, in fact, the Lydian mode of the C Major (Ionian) scale. But that's not why we've seemingly drawn attention to its 'Lydianicity' by colouring the F# in blue. No - the real reason for drawing attention to the F# is because in order to get the prime form of this PC set, we have noticed that the set's largest interval skips (the three consecutive 2s) bring us to that F#. Since the prime form requires that the largest interval skips are to be placed at the end of the interval path, this means that the F# must become the prime form's pitch class zero (in other words, its first note). Accordingly (by transposing from C to F#) we obtain the prime form - with its interval path of 1221222 - shown below.

CΔ13#11 as an inversion of the F# Locrian mode (F#13♭5♭9♭13)
PC set in prime form

It's fairly easy to demonstrate that the PC sets which embody both the ordinary 13th and the minor 13th (with which we opened) are all exactly the same.

All of the C13ths discussed thus far 'prime form' (as Forte PC set 7-35 - the Locrian mode of the diatonic scale) - to some inversion of a Locrian, as the following two expositions show.

C13 as diatonic
C13 as an inversion of the E Locrian mode (Em13♭9♭13)
Cm13 as diatonic
Cm13 as an inversion of the A Locrian mode (Am13♭5♭9♭13)

Can this mean that all 13th chords are some inversion of the Locrian mode of the diatonic scale?

How many 13ths are there?

It seems reasonable to proceed with such an enumeration only if we have some formal definition of exactly what constitutes a 13th chord. One of the most obvious qualities of the chord would appear to be that it comprises 7 distinct pitch classes. Another would appear to be that they should be constructed by stacking up six successive major or minor thirds after the initial root note.

The interval paths of such chords will thus be formed from six 3s or 4s and terminated by whatever value would take the path sum up to 24. Six '3s or 4s', being the same as six '3 + (0s or 1s)', it's reasonably clear that 64 distinct interval paths - labelled from 333333, 333334, 333343, 333344, … to 444433, 444434, 444443, 444444 will capture all possibilities.

It's also easy to see that any path containing either three consecutive 4s or four consecutive 3s must be rejected since the pitch classes on either side of such jumps are bound to be exactly an octave apart, thus being the same pitch class and violating the principle of distinctness. There are many other routes, within this procedure, which result in the duplication of pitch classes.

It turns out, therefore, that - after filtering out all of the interval paths which would result in pitch class duplication - only 28 ways of stacking major and minor thirds to build up some kind of 7 note 13th chord remain. These are, in ascending order of minor-major-thirdiness:

33343441212213343334412131223443343212222143433342212131
33344341221213343343412221223443433213122143433432212221
33433441212222343344312222124333434131212243434332221221
33434341221222343433421221224333443131221243443333121221
33434431221312343434321222124334334221212244333432213121
33443342121222343443321312124334343221221244334332222121
33443432121312344333421221314334433222121244343333122121

Also shown, in red to the right of each 3|4 construction, is the interval path signature of the PC set which contains the 7 distinct PCs of the resulting chord. These are - of course - in normal form because interval path signatures give you that for nothing. A quick inspection should demonstrate that these are by no means all just variations of the seven modes of the diatonic scale. The presence of 3s in such signatures is enough to disabuse one of such notions. There are also several with runs of four 2s.

By rotating all 28 of the (red) PC set signatures into their prime forms (with their largest skips packing to the right, as per Forte), we quickly uncover the fact that there are four distinct PC sets in play here:

Four Prime Form PC Sets
1212213121222212212131221222
7-32A7-347-32B7-35
<3,3,5,4,4,2><2,5,4,4,4,2><3,3,5,4,4,2><2,5,4,3,6,1>
Harmonic MinorLocrian SuperIndian/EthiopianLocrian Diatonic
33343441212213334334412122223334434122121333434341221222
33434431221312334433421212223344343212131234334341222122
34344332131212343344312222123433344121312234343342122122
34433342122131344334321222213443433213122134343432122212
43334341312122433433422121224333443131221243343432212212
43443333121221433443322212124343334221213143433432212221
44333432213121443343322221214434333312212143434332221221

They occur in four groups of seven arrangements - the seven modes of each of their common prime PC sets. The first (which includes the Harmonic Minor scale, 2122131, in its fourth row) and third (including Indian and Ethiopian scales in its third and sixth rows) column pairs are asymmetric PC sets (inverses of each other). The second and fourth are both symmetric PC sets, the first being the prime form of the half-diminished scale (arguably the next most popular heptatonic division of the octave, encompassing as it does the hindi, melodic minor, overtone, javanese, and both locrian natural and super scales). The fourth is the various rotations (i.e. modes) of PC Set 7-35, i.e. Locrian-Diatonic. The three types of 13th chords we dealt with above are in boldface.

Relationships between 13ths and the Diatonic Modes

The 13th chord patterns in the fourth, diatonic, group are - starting from the top, and not based on any particular scale

  • 3343434 ≡ m13♭5♭9♭13 ['Locrian 13th']
  • 3433434 ≡ m13♭9♭13 ['Phrygian 13th']
  • 3434334 ≡ m13♭13 ['Aeolian 13th']
  • 3434343 ≡ m13 ['Dorian 13th']
  • 4334343 ≡ 13 ['Mixolydian 13th' = the standard 'dominant 7th' mode]
  • 4343343 ≡ Δ13 ['Ionian 13th' = the major 7th dominant series]
  • 4343433 ≡ Δ13#11 ['Lydian 13th']

But naturally, upon actual transcription, one must commit to a key - say C:

the Modal bases for 7 diatonic 13ths

Stepping away from the diatonic 13ths, if we based a 13th chord on the Ethiopian scale (row 6 column 3) - with its PC Set interval path signature 2212131 - we would generate a major/minor 3rd stacking of 4343334 ≡ Δ13♭13. The Indian scale (1213122 → 3433344) would yield a distinctly weird m13♭9♭11♭13. We feel reasonably certain that there will be a circumstance where every one of these 28 possible 13ths will sound fantastic.

Squeezed 'Thirteenths'

The 28 13ths above are constrained to be contained within 3 or 4 semitones of a double octave span. As such, their top notes will always be a 'true' 13th, possibly flattened. The notes within the chord are not subject to undue 'stress' and the 7ths, 9ths and 11ths turn up in their expected places - perhaps occasionally bumped sideways as flattened or sharpened creatures as it were. However if this 'thirteenth pegging' is relaxed, there are - technically speaking - eight further 13ths. The first two are rather 'squashed' as the top notes are 5 semitones down from a double octave, giving us a double-flattened 13th - which a musician will consider illegitimate (a double-flattened 13th being - enharmonically - just a plain old 5th, the chord's 'internal' 5th already being flattened).

333433
m13♭5♭♭7♭9♭11♭♭13
e.g. Cm13♭5♭♭7♭9♭11♭♭13 = C–E♭–G♭–A–D♭–E–G
334333
m13♭5♭9♭11♭♭13
e.g. Cm13♭5♭9♭11♭♭13 = C–E♭–G♭–B♭–D♭–E–G

We note that the initial four notes of the first form a full-diminished chord. It thereby already contains the 13th as a pitch class, albeit an octave lower (so not actually a 13th but a 6th). It's as if the chord has been put under so much compressive force that the 9th, 11th, 13th (and even the 7th) begin to crash into each other (as pitch classes). In both cases, the top flat 11th and double-flat 13th are effectively the major 3rd and major 5th of the tonic. The flat nine is really the only 'novelty' in these chords and to call these chords 13ths is rather stretching a point (actually the opposite - it's compressing a point). As pitch class sets, these are mutual inverses, the first being characterised as an interval path signature 1212123 (Forte's 7-31A) and the second (its inverse, 7-31B) as 1212132 (which would of course 'prime form path' as 2121213), both sharing interval vector <3,3,6,3,3,3>, 'maxing out' with their 6 minor thirds.

Stretched 'Thirteenths'

The remaining 6 chords are - in contrast - stretched, and their top notes are only two semitones below the double octave. Thus a dominant 7thness turns up at the top end, rather than in the middle, of the chord (where the internal 7thnesses are all Δ, i.e. major 7ths). The first couple of these are the antisymmetric pair (path signatures of 2131311 and 3131211, corresponding to prime form signatures of 1121313 (Forte 7-21A) and 1211313 (Forte 7-21B) respectively, sharing interval vector <4,2,4,6,4,1> with maximal major-thirdy content.

344344
mΔ13#11#13
e.g. CmΔ13#11#13 = C–E♭–G–B–D–F#–A#
443443
+Δ13#9##11#13
e.g. C+Δ13#9##11#13 = C–E–G#–B–D#–G–A#

In fact by now it's becoming rather difficult to 'spell' these as 13th chords, since the pitches tread on each others' toes so much. The above '+Δ13#9##11#13' is pure guesswork on this author's part.

The second pair of 'stretchy' 13ths have path signatures of 2221311 (prime path 1122213 ≡ Forte 7-30A) and 3122211 (prime path 1222113 ≡ Forte 7-30B) - again mutual inverses - with interval content <3,4,3,5,4,2>

434344
Δ13#11#13
e.g. CΔ13#11#13 = C–E–G–B–D–F#–A#
443434
+Δ13#9#11#13
e.g. C+Δ13#9#11#13 = C–E–G#–B–D#–F#–A#

There remain two further '13ths'. They each represent symmetric (self-inverting) PC sets, the first with a path signature of 3121311 (prime form path as 1131213 ≡ Forte 7-22, interval content <4,2,4,5,4,2>) and the second with path signature 2222211 (prime form path as 1122222 ≡ Forte 7-33, interval content <2,6,2,6,2,3> replete with major 2nds and major 3rds).

434434
Δ13#9#11#13
e.g. CΔ13#9#11#13 = C–E–G–B–D#–F#–A#
443344
+Δ13#11#13
e.g. C+Δ13#11#13 = C–E–G#–B–D–F#–A#

Terminological Conclusions

Are any of these eight stretched or squeezed chords really 13ths? Insofar as they comprise heptachords they are. Insofar as their construction involved upward skips of only major and minor thirds they are. But musically, they just aren't. They have too much internal compression or tension. This is due to not 'nailing down' - for the want of a better term - a (possibly flattened) 13th at the top of the runs of thirds and leaving the end of the chord flapping around in the breeze (as David Jones would say). Such freedom permits an absence of what one might expect of 'thirteenthness'.

So the 28 unstretched and unsqueezed 13ths above, distributed between only four PC Sets, are all (assuming fully populated ones) there really are. We note that this means there are only three distinct interval contents (two being identical since non-symmetric PC Sets' forms - A and B - always share a common interval vector) available to carry the various flavours of all 28. These are <3,3,5,4,4,2>, <2,5,4,4,4,2>, and <2,5,4,3,6,1>.

Interval classwise, then, we have that a 13th contains either 2 or 3 minor 2nds [or major 7ths], with - correspondingly - 5 or 3 major 2nds [minor 7ths] and (also correspondingly) 5 or 4 minor 3rds [major 6ths]. We also have that if the 13th contains only a single tritone then it must also contain the maximum number of perfect 4ths [or perfect 5ths] possible within a heptachord, i.e. 6, and must also contain 3 major 3rds [or minor 6ths]. Otherwise a 13th must contain 2 tritones and 4 each of major 3rds and perfect 4ths [or minor 6ths and perfect 5ths].

This means that if you elect a 13th chord constrained to containing only a single tritone then all else follows, i.e you cannot help but have 2 m2/M7, 5 M2/m7, 4m3/M6, 5 M3/m6 and 6 P4/P5, which further means that you have elected one of the 7 diatonic 13ths. If, however, you choose one with two tritones (the only remaining kind) then you have forced it to contain 4 P4/P5 and 4 M3/m6 and at least 4 m3/M6 and 2 m2/M7. The only choice you have left is a chord with 5 m3/M6, 3 M2/m7 and 3 m2/M7 or one with only 4 m3/M6 and 5 M2/m7 and 2 m2/M7.

It would appear, therefore, that there are - at most - two degrees of freedom available to anyone constructing a 13th chord from its interval content alone. Is that interesting or is that interesting?

20171122

Flipping Heaven

It has been a while since we looked at invertible scales and it might be worth a brief recap, starting with the Dorian mode of the diatonic pitch class set. This being the one with which we in the West are most familiar in its guises as the major and minor scales – or Ionian and Aeolian modes. From hereon in, we’ll abbreviate pitch class with PC.

By inversion, we mean reflection – or more accurately subtraction (see our earlier Pitch Axis Considered Harmful). By an invertible PC set, we specifically mean that the act of inversion – i.e. subtracting (modulo 12) each of its PCs from a certain fixed number - does not take you out of the set.

We start with the Dorian mode as it’s the most obviously invertible scale mode by anyone familiar with a piano, starting on a middle D and playing up and down the white keys either side of it. The piano keyboard is at its most obviously symmetric when you gaze at any D or A♭ key. If the D is PC 0 - usually represented on a twelve-hour clock-face as 12 o’clock (i.e. n00n!) - then the scale (which is to say the Dorian mode) seen as a seven note ‘motif’ is 0 2 3 5 7 9 10 (DEFGABC). Moving left (or anticlockwise) we also start at 0 but subtract and end up with the inverted sequence of negative numbers 0 –2 –3 –5 –7 –9 –10 (DCBAGFE) which, after adding 12 (an octave) to each (which is equivalent to adding 0, itself equivalent to doing nothing whatever to alter the essential nature of the inversion), gets us 0 10 9 7 5 3 2. And all of those numbers (and notes) remain a Dorian PC set’s PC numbers (and white keys).

dorian - the polar symmetric diatonic
interval skip pattern 2122212 D

That the fixed number one subtracts from may be different for different modes does not alter the essential invertibility of the set. For instance if one subtracts the major PC sequence (aka the Ionian mode 0 2 4 5 7 9 11) from 0, one would recover 0 -2 -4 -5 -7 -9 -11, which is 0 10 8 7 5 3 1 - decidedly not the same mode since, reordered, this is 0 1 3 5 7 8 10 – i.e. the Phrygian. The 2 and the 9 have been flattened to 1 and 8. To recover the same mode, one would have to subtract the Ionian PC numbers from 4, to yield 4 2 0 –1 –3 –5 -7 which – after adding a 12 (‘doing nothing’) where necessary to keep us in positive PC numbers – results in 4 2 0 11 9 7 5, the exact same Ionian PC numbers we started with – we just don’t start on the tonic note. There’s a table towards the end of Phrygian = subtonic – Dorian if you wish to see what happens to the modes if you subtract them from different values.

The invertible PC set however, regardless of mode, is the same particular ‘shape’ if you care to look at it as a polygon and – as such – does not alter at all. The shape just rotates around its centre. As such, its axis of symmetry is simply carried around with that rotation. There may of course be more than one axis of symmetry in other, non-diatonic, PC sets.

There are also scales which are not invertible such as the (pentatonic) Hirajoshi, the (hexatonic) Blues, the (heptatonic) Hungarian Major (which we’ve discussed), the (octatonic) Bebop Dominant Flat Nine and many, many more.

interval skip patterns
21414 321132 3121212 13122111
hirajoshi blues hungarian
major
bebop
dominant
flatnine

A Digressional Touch of Polemic

The 'interval skip pattern' notation, employed in the preceding diagrams, is simply a ‘key-signature’ independent representation of the semitone steps between each note of the scale, starting at its root or tonic note (twelve o'clock in the diagrams!), with the final step ‘returning’ you to the tonic (an octave above) regardless of whether or not the scale or mode is F Dorian, C# Hirajoshi or (if you’re one of those) a 432Hz-based-A Bebop Dominant Flatnine. This (completely scalable) interval skip pattern scheme works for systems other than twelve tone – indeed it works with any fixed octave-based microtonality and it doesn’t even need to be well-tempered (although the scales will sound different of course). Skip numbers simply have to total up to the number of microtones in your octave, and the number of skips (perforce not more than the number of microtones) must match the number of notes in your scale.

Other Heptatonic Flippers

The 2212221 skip pattern (of, specifically in that case, the Ionian mode of the ordinary ‘diatonic PC set’) simply rotates as 2122212 (Dorian), 1222122 (Phrygian), 2221221 (Lydian), 2212212 (Mixolydian), 2122122 (Aeolian) and 1221222 (Locrian) and carries its axis of symmetry around the clock as it does so. They are all regarded as the same PC Set (classified as Forte Number 7-35 in Allen Forte’s naming system). In that sense, modes are regarded as equivalences of the skip pattern, which is similarly regarded as a single pattern of interval skips, without being particularly bothered about which skip comes first.

For instance, with a seven note scale embedded within a twelve tone chromatic space, there are only two ways to have scales comprising only whole tone or semitone skips without two consecutive semitone skips. One of them is with three 2s and two 2s separated by 1s (they must, after all, add up to 12 in total). The only other possibility is four 2s and a single 2 separated by 1s. (Five 2s would leave only space for the forbidden two consecutive 1s). This alternative pattern (2222121, say) is also symmetric and also has seven modes. But it looks as if only six of them are in (reasonably, and variably so) common use.

various modes of interval skip pattern 2212122 (hindi)
javaneselocrian
natural
melodic
minor
hindiovertonelocrian
super

The axis of symmetry is marked with a yellow line. The Hindi scale, like the Dorian, is tonic-invertible in that subtraction from 0 will keep your PCs firmly inside the Hindi ‘mode’ of the PC set 7-34, the Forte Prime Form of which is known as the half-diminished scale (aka the Locrian Super in the above).

If you forego the restriction preventing your scales comprising only semitone or whole tone skips, you may permit yourself a sesquitonic (one and a half tone, or three semitone) step (or two). Such a pattern might be 1131213 – again, all of the numbers must add up to 12. This is PC set 7-22, turning up as the Double Harmonic scale. This PC set is invertible and here are three further modes which, due to historical and geographical accident, are also known as scales in their own right (despite their all being simply modes, or rotations, of each other).

three further modes of the symmetric sesquitonic scale PC set 7-22
hungarian minorgypsyoriental
interval skip
pattern
2131131
interval skip
pattern
1312131
interval skip
pattern
1312131

The distinctly non-PC (in the other sense) term ‘Gypsy scale’ is something that, alas, we can’t do very much about any time soon. Scale names are much too firmly embedded in musical language and history. Despite the obvious advantages of describing/classifying scales (and even modes) completely and unambiguously by numerical interval skip patterns, there’s pretty much no possibility of the more rational nomenclature prevailing any time soon. We just have to put up with the fact that the naming of musical scales is both haphazard and arbitrary. In short, the current space of scale names is clearly not scalable.

There are many other seven note PC sets. We (but we were by no means the first) counted 66 distinct (within modal, or rotational, equivalence) patterns back in that first post and we also noted that only 10 of those 66 were symmetric. In fact we observed that – regardless of the number of notes in a scale of any length (from 1 to 12 within the twelve-tone series) – the majority of PC sets are unsymmetric. Which is not to say that, worldwide, some musician has not come up with a scale which employs them (as, for example, the aforementioned Hungarian Major, demonstrates).

Octatonical Flips

How about eight note scales? We know there are 43 PC sets containing eight PCs, and that only 15 of them are symmetric. A few, like the heptatonic sets, can be constructed with the restriction that they contain only whole tone or semitone skips. But this time, because the scale is so crowded with notes, consecutive semitone skips are unavoidable. If three consecutive skips are whole tone then you must fit the remaining five notes – best case – within a span of six semitones (or fewer, if all three whole tone skips are not consecutive). If five skips are whole tone then that’s ten of the available PCs already accounted for, leaving you no room at all to fit in the other three PCs. Four 2s and four 1s seems to be a good compromise and the 21212121 pattern – which is doubly symmetric on two axes – is found in the usual pair of Jazz scales commonly called ‘octatonic’ – one of them starting with a whole tone, the other with a semitone. But there are other – single-axiswise symmetric – scales:

four symmetric octatonic scales - mostly bebop
modes with two consecutive semitones
bebop major flamenco
interval skip pattern
22121121
interval skip pattern
12112122
modes with three consecutive semitones
bebop minorbebop dominant
interval skip pattern
21221112
interval skip pattern
22122111

The two PC sets above are, respectively, 8-26 and 8-23, Forte-wise.

Counting - The maths bit

We already counted, right back at the beginning, how many three note, four note, five note etc scales were inversions of themselves - we reproduce the distribution here:

image

The bluish bars represent the minority of invertible sets and the reddish the uninvertible ones within each set size. The sequence for k-note scales is 1, 6, 5, 15, 10, 20, 10, 15, 5, 6, 1 for k=0 to 12. We didn't bother to chart the extremes (of 0 and 12) because, trivially there's only one way to have an empty musical scale (bound to be reflectionally symmetric), and also only one full 12 note chromatic scale - obviously very symmetric indeed.

If you add up all of these (bluish) numbers they total 96. Which is to say that, of the 352 (= 1 + 1 + 6 + 19 + 43 + 66 + 80 + 66 + 43 + 19 + 6 + 1 + 1) possible (differently 'shaped' rotationally equivalent) k-note scales (again, for k=0 through to k=12) representable inside a 12 hour clock, most of them (i.e. 256) are not reflectionally symmetric.

If you make the further equivalence between those shapes which are mirror images of each other (i.e. although you cannot rotate one to completely match the other, you can rotate its mirror image to match it) then those 352 shapes reduce to 224. As the 96 symmetric polygons already were 'equivalent in the mirror' they're not quite as rare amongst this reduced set.

If you remember your early geometry classes and remember your triangles, you may recall the difference between congruent triangles and similar triangles. In this regard, polygons are people too, as it were (notions of similarity also include not having to be the same size, but that's not relevant here). Additionally the general case of the haphazardly shaped polygon might remind you of those undistinguished scalene triangles. Isosceles triangles are analagous to our symmetric polygons.

More than 12 Notes

What about other microtonalities? Our '96', '224' and '352' counts are but one case (the '12' case) of well known integer sequences used to count bracelet or necklace arrangements. You may encounter the terms 'necklace' to model those patterns with rotational equivalences (our larger 352 case) and 'bracelet' to model a flippable necklace - i.e. patterns with the additional reflectional equivalences (the 224 case subset) and - still further - bracelets which are also symmetric.

The first two are, respectively and more formally, examples of Cyclic and Dihedral groups in Group Theory. The following tables show three entries in the Sloane Catalogue of Integer Sequences (the OEIS is the online version). The various headings for each sequence come from that catalogue.

12345678910111213141516
Number of n-bead necklaces with 2 colors when turning over is not allowed
also number of output sequences from a simple n-stage cycling shift register
also number of binary irreducible polynomials whose degree divides n
In music, a(n) is the number of distinct classes of scales and chords in an n-note equal-tempered tuning system
2346814203660108188352632118221924116
Number of necklaces with n beads of 2 colors, allowing turning over (these are also called bracelets)
23468131830467812622438068712242250
Number of necklaces with n beads and two colors that are the same when turned over and hence have reflection symmetry
2346812162432486496128192256384

Note that each number in the third sequence is the corresponding number in the first sequence subtracted from twice the number in the second sequence.

But we are modelling scales (or, more abstractly, PC sets) as k-sided, or k-pointed (it's the same thing - a PC set with k members) polygons picked out as sub-polygons of the 12-sided 'chromatic dodecagon'. It turns out that the last series can be rather nicely captured by the generating function:

Generating Function for Symmetric Polygon Totals in n-sized tuning systems

where the pn (for n = 1 and upwards) are the exact same values in the third sequence above. (For completeness, p0 = 1.) This generating function is given in the Sloane catalogue, but without that rather mysterious 1 we've slipped in, with a comma, before the z. What’s that all about?

In fact it is an application of a more general generating function of two variables, u and z, evaluated at the value of u = 1. This function is:

Generating Function for Symmetric Polygons of k-sized scales in n-sized tuning systems

The apparently unnecessary double summation tacked on to the end of that function is there for a reason - it allows you to pick out the individual counts for k-sized scales inside n-sized tuning systems for any n and for any k (perforce less than or equal to n). All you have to do is expand the rational polynomial as a power series in u and z. Which is pretty straightforward.

k-sized scale counts in 0 to 13 note tuning systems

In particular the penultimate line of the expression - representing the familiar 12-tone chromaticism - shows the coefficients of the individual powers of u, i.e. the a12 kuk, as the 1, 1, 6, 5, 15, 10, 20, 10, 15, 5, 6, 1, 1 we presented earlier (including the u0 and u12 cases). For example we see that there are 10 symmetric heptatonic (and of course pentatonic) scales within 12 tone chromaticism.

In general, for musical systems with an even number of divisions (like our familiar 12 tone), we have that:

P2m(u) = (1 + u + u2)(1 + u2)m-1

And that for musical systems with an odd number of divisions (e.g. 17 or 19 tone systems):

P2m+1(u) = (1 + u)(1 + u2)m

You may also wish to verify that - if you want to count all invertible scales, regardless of k, for each of those values of n, just set u = 1 (pn = Pn(1)) in the above.

The above 'triangular series' which appears as the ever lengthening lines of coefficients of u for increasing powers of z is also documented in the Sloane Catalogue as series A119963. Dr Petros Hadjicostas has attached a generating function Sum_{n,k >= 1} RE(n,k)*x^n*y^k = (1+x*y-x^2)*x*y/((1-x)*(1-x^2-x^2*y^2)) which, in our letterings (u for y and z for x), would translate to our G(u,z)-1/(1-z) which, taking into account our start at n = 0, is the same thing.

As a fun application, consider all of the triads available to a musician playing with the aforementioned 17 note system, which can be represented with a 17-gon. We pick out

P17(u) = (1 + u)(1 + u2)8

and expand powers of u to obtain

k-sized scale counts in a 17 note tuning systems

and we can immediately see (from the coefficient of u3) that 8 (out of the possible 40 from the well known dihedral symmetry enumerations) triads are self-inversional. Below is a diagram showing all 40, with the 8 self-inversional triads as isosceles triangles in blue and the remaining 32 non-inversional, as 16 mirror pairs, in salmon.