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:


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
Harmonic MinorLocrian SuperIndian/EthiopianLocrian Diatonic

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).

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

e.g. CmΔ13#11#13 = C–E♭–G–B–D–F#–A#
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>

e.g. CΔ13#11#13 = C–E–G–B–D–F#–A#
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).

e.g. CΔ13#9#11#13 = C–E–G–B–D#–F#–A#
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?


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

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)

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
interval skip
interval skip

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
interval skip pattern
modes with three consecutive semitones
bebop minorbebop dominant
interval skip pattern
interval skip pattern

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:


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.

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
Number of necklaces with n beads of 2 colors, allowing turning over (these are also called bracelets)
Number of necklaces with n beads and two colors that are the same when turned over and hence have reflection symmetry

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.


The Third Degree

As the previous article mentions, the following ‘motif’ (scare quotes because of what follows) begins with a long run of thirds:

This, the Berg Violin Concerto's 'defining tone row' is the one catalogued by the ‘34433443222’ interval class index in Wikipedia’s list of tone rows (hereinafter referred to as wLOTR), which cites this concerto as its principle reference. Fripertinger's more technically exciting Database on tone rows and tropes also indexes the works discussed here with the exact same tone row or interval sequence keys (it's not clear if the Wiki page is based on this). So it might be interesting to investigate how this particular sequence came to be the one elected to characterise the entire opus.

As a more general question, of all of the instantiations of a tone-row within a work, how does one decide which of them is ‘prime’? For it is a well-known problem, as Peter Castine reminds us, at the beginning of §2.9 in his Set Theory Objects (Europaïscher Verlag der Wissenschaften; Musicology, Peter Lang 1994):

“The main difficulty in set theoretic musical analysis is not so much that of recognizing relations between pc sets, it is deciding which notes in the score should be analyzed as pc sets. This is called segmentation. Once segmentation has been established, much of the remaining work is mechanical; [etc] …”

We should also consider Babbitt’s contributions to the concepts of serialism and what might be called ‘tone-rowism’. He writes (of Schoenberg - whom he knew - and of the nuances of translating German into English)

“The word Reihe bothered him because it became row. And to him row suggested left to right – something in a row – and that’s what it does connote. And that connotation, he thought, was part of all these misunderstandings about the twelve-tone notion having to do with some sort of thematic, motivic thing that went from left to right. It upset him, so he asked various friends about it.

[omitted paragraph on how series was suggested, and rejected, as translation for Reihe]

And I regret to tell you, I am guilty. I suggested the word set, which had absolutely no meaning in music as yet. It came out of mathematics (not that that pleased me particularly) and it seemed to be a neutral term. Of course, a set does not mean anything ordered, but if you append twelve-tone or twelve-pitch-class to the word set, then that implies an ordered set and that’s a very familiar structure, too, in abstract relation theory. So there we were.”

This is from chapter 1, “The Twelve Tone Tradition” of Milton Babbitt’s “Words About Music”, Dembski and Straus (eds), from the University of Wisconsin. You may read much of this book in google’s academicals, but not – unfortunately - this particular bleeding chunk since pages 11 and 12 are ‘not shown in this preview’. You’ll either have to take my word for it or find a real copy, but as Babbitt’s generally a fun read, it’s worthwhile.

So Babbitt claims that Schoenberg never intended that all twelve tones must be played out (exhausted if you will) in order, before you were allowed to proceed to any of its transformations (i.e. its repeats or retrogrades or inversions). Although this does not seem to square (advance pun warning) well with later attentions paid to the construction of P-R-I-RI grids, aka Babbitt Squares, it just means that these are possible playthings, not necessary ones - and a comparatively recent invention unused in the composition of the pair of works discussed here.

In any case. musicians may write what they please, even the (Babbittally argued) most intellectual of the Berg-Schoenberg-Webern group, Berg himself, to whose concerto’s tone-row ‘signature’ we now return.

Berg in Threes

In the score’s introduction (Universal Edition Philharmonia Partituren #426), “F.S.” (Friedrich Saathen) describes the above tone row (G B♭ D F# A C E G# B C# E♭ F) as being ‘the one from which the Concerto is made’ (presumably what we might call its prime ‘P0’ form). Attention is drawn to the G D A E, the four consecutive perfect fifths of the violin’s open strings, embedded within its head and also to the terminating tritone resulting from the last three (actually four, if you include a wraparound) whole tone steps. And of course the concerto begins with those four open strings. Clarinets and harp provide the interstitial B, F# and C in the first two bars. They also throw in some Fs, but as these could conceivably have come from some unheard previous row’s instantiation (n.b. F is at the end of the model above), we concede its legitimacy (no, we’re not serious).

One of the focal points of the concerto, in the second half, is at bar 195:

Berg Violin Concerto - Adagio Bars 195ff

where the soloist comes in (the p at the middle of bar 196) with the entire tone row (interval classes at the top, zeroed-out pitch classes at the bottom) :

- presumably intended to convey (the already) angelic little Manon Gropius heavenward. It’s damn’ poignant, but the ‘principal tone row’ seems not quite there. Its 34433443222 jump pattern appears to lack the fourth '2' present in the above 32222344334 ascent. However, this is due solely to omission of the final 'wraparound to initial' value in interval path representations of tone rows or of pitch class sets. Appending them results in 344334432222 (the 'ur-row' of F.S.) and 322223443344 (Manon's ascent), both final skips taking you to accumulated sums of 36 = 3×12.

These differential forms of both tone rows show that they are the same object (simple rotations of each other) in a way that explicit pitch class sequences make spectacularly opaque. Just try comparing {0,3,7,11,2,5,9,1,4,6,8,10} with {0,3,5,7,9,11,2,6,10,1,4,8} or, even worse, G-B♭-D-F#-A-C-E-G#-B-C#-E♭-F with A-C-D-E-F#-A♭-B-E♭-G-B♭-D♭-F in an attempt to work out if they're the same musical object.

Webern in Threes

Another piece of serialism, also from the B-S-W trinity, is indexed in both wLOTR and Fripertinger as 34343443431, intended to represent interval skips of the prime form P = +3-4+3-4+3-4-4+3-4+3-1 or its inversion I = -3+4-3+4-3+4+4-3+4-3+1.

This (unsigned) 'P' string is only 11 characters long. Fair enough; it represents, after all, skips between 12 pitches. But it already (accidentally) sums to 36 (a multiple of 12). If you take the signs (the directions of the interval skips up+ or down-) into account then they sum to (respectively) -6 and (for the 'I' string) +6 - thus telling you the size of the next skip to the beginning of a second instance of the tone row. In applications of tone rows - actual instantiations in real written down music - that final skip seldom matters because the next occurrence of the tone row is very likely not going to be a repeat of the exact same P (or I) form beginning with the same note as before, so the relevance of the 6 would be somewhat moot.

But for the purposes of tone row indexing, a better 'differential index' would be 3838388383B6 (include that 12th wraparound 6 with the index's digits summing to 72 = 6×12). This covers the explicit pitch class representation 0 3 e 2 t 1 9 5 8 4 7 6 (t=10, e=11). There's a second entry with an explicit pitch class representation of 0 9 1 t 2 e 3 7 4 8 5 6 (it's the inversion) covered by the exact same interval index of 34343443431 (of course it is the same, that second tone row's an inversion and the minus signs are absent). If you 'put the signs back' as it were, by using 8 for -4, 9 for -3 (and so on) then that second row would be indexed (adding the final 12th, wraparound, digit 6) as 949494494916 - again summing to 72. Because of the ups and downs of this tone row (a feature entirely lacking in Berg's constantly rising row) it's not quite so obvious that 3838388383B6 and 949494494916 are the same musical object as you cannot simply rotate (by shifting digits from its tail to its head) one index into the other. You actually have to 'do work' to notice that - digit by digit - all of the characters of each index sum to 12 (3+9, 8+4, … 3+9, B(=11)+1, 6+6), exactly what you'd expect of an inversion (regarded in PC set theory as equivalent).

It's extremely annoying that such indexing will work only by re-instituting all 11 intervals, foregoing the (incredibly useful) joy of having to contend with only 6 interval classes and thereby 'dis-integrating' the hitherto similar representations of (no longer visually identical) inversions. But when indexing tone rows by their internal intervals - where intervalic direction actually matters - it's unfortunately necessary. Pitch class sets, in contrast, may be safely indexed with interval class digits only since there are no sequences to contend with because pitch classes just 'stand there', motionless.

Regardless of indexing issues, we can see that Webern's tone row is much thirdier; almost, but not quite, as thirdy as you can get.

In wLOTR (loc cit) it’s the immediately preceding entry (if sorted by interval class, at the time of writing – there’s lots of possible room for expansion, so who knows how long this will stand). And the tone row entry that it falls under is indexed as 0 3 e 2 t 1 9 5 8 4 7 6 (or 03B2A1958476 in the alternative popular tone-row labelling scheme), which looks like this (if you regard the leading F# as the 0).

The interval class jumps (+ or – omitted as obvious, when you can actually see the directions) annotate the bottom of the illustration. Remember that, interval classwise, descending a major or minor sixth is the same as ascending (respectively) a minor or major third (and vice-versa). It’s why interval classes (which is one of the ways the wLOTR Wiki is indexed) are in the range 1 to 6 and not 1 to 11, as discussed above. I.e. the indexing is actually a string of ±3±4±4±3… where the ± is not shown but is to be taken as ‘understood’.

The work cited by that index (or indexed by that citation) is the second of Webern’s Drie Lieder (Op 18) of 1927(ish), "Erlösung" which begins thus:

Bars 1 to 6 of Webern's Erlösung

We’ve coloured up the first seven (red, green, blue, red, green, blue, red) of the tone-row (actually set) block instantiations. Apart from the slight ambiguity of which of two pitches in a guitar’s tremolo comes first, the sequence of pitch first-appearances across the instrumentation is – in all seven cases [except the last ‘interesting’ one] F# C F G# E G E♭ B D B♭ C# A.

One might notice the overlaps. For instance the first ‘green’ set begins before the earlier ‘red’ one has quite finished, and these overlaps continue as the piece progresses. But there’s no change in the particular sequence of pitch classes.

So, out of bars 1 to 3, one may pull out the first red (bars 1 and 2) and green (bars 2 and 3) blocks to confirm the (tremolos notwithstanding) sequence. You may painstakingly, if you wish, verify that the following blocks follow the same tone row (i.e. no inversions or retrogrades):

Erlösung's Tone Row

As the annotations show (interval jumps underneath the stemless abstractions), the intervals aren’t quite the same as the one ‘attached’ to this work in wLOTR. There's a 5 jump in the composition which does not appear in wLOTR's interval class index. And, of course, 0 3 e 2 t 1 9 5 8 4 7 6 (also from wLOTR) is not (from the above, assuming F# → 0) 0 6 e 2 t 1 9 5 8 4 7 3.

You can certainly see the similarity – it’s only at the edges (in bold underlining) that there is a difference

0 3 e 2 t 1 9 5 8 4 7 6
0 6 e 2 t 1 9 5 8 4 7 3

(although the 0 hardly needs to be underlined as its presence is demanded by the nature of the presentation). So what’s with this 3/6 head/tail swap? The worst possible place to happen for a lookup, wrecking both the tone-row and the interval-sequence index? The reference authority cited for this tone row is, in fact, a footnote in a journal article by the one and only David Lewin:

(from A Theory of Segmental Association in Twelve-Tone Music, D Lewin, Perspectives of New Music V#1 N#1, Autumn 1962) where one may clearly see the A after the initial F# and the terminal C in the upper (Φ0) line in contrast to the actualité of Webern’s C and F#, easily seen (multiple times, as if to nail it firmly into your head) in the above score-snippet.

The differential forms of the above Φ0 and I0 are 3838388383B6 and 949494494916 - hence their appearance (albeit with de-signed 3s and 4s) in wLOTR.

Lewin’s exposition is – by the way - quite legitimate (his article concerns hexachords and segmentation). As he says, earlier on (in regard to Schoenberg’s Op 36 Violin Concerto):

“These examples are of considerable value in cautioning us against the naïve but plausible assumption that all effective associative relations in such music as this must be presented explicitly. The reader is urged to keep this moral in mind throughout the sequel. Of course, the extent to which we will recognize any such relations, whether explicit or not, is heavily dependent on the extent to which the compositional presentation of the notes involved supports or obscures the abstract relation, and/or the extent to which the sonorities involved have been explicitly established as referential.” [our emph]

What appear to be the 'obvious' tone rows (or, equivalently, interval-class sequences) - by which we mean those sets or sequences readily available to a listener or a score-reader (bearing in mind Castine’s caution), in contrast to those other, rather abstruse, representations available to a reader of somewhat rarefied musical journals - may not be terribly useful when interrogating a database to discover them. Fortunately, other rather more instantly available - even to a non-musician - indexes such as the composer's name or the work's name are also present.