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 Drei 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.


The Third Way

This post was triggered by a brief exchange with Jan-Willem van Ree (on musescore) about the use of certain intervals in structured or otherwise constrained music. Having believed I had seen, long ago, examples of tone rows built exclusively from major and minor thirds (or their interval class equivalent, minor and major sixths), I’d formed the impression that either Webern or Berg had actively sought such sequences.

Dr van Ree kindly reminded me of the Berg Violin Concerto which includes the following run of eight successive thirds (abstracted and annotated below, with both note names and semitone interval classes between notes), and which finishes off with a triple whole tone tritone run. In the score’s introduction (Universal Edition Philharmonia Partituren #426), “F.S.” describes it as being the one from which the Concerto is made (presumably its prime ‘P0’ form - although it's not a strict 12 tone work).

Berg Violin Concerto 'Es ist Genug' Tone Row

Being the sort of thing I was looking for, the start of the sequence might be represented by the curved (red) path in the following transition-by-thirds (dark/light blue arrows for minor/major thirds respectively) diagram:

But with this we may seek circuits, i.e. closed paths. For instance one might start at E, move to G, then B, then to D, to F, to A♭, to C, to E♭, to F#, to B♭, to C# and back to the starting E. That’s a circuit of eleven pitches, visiting each note only once. The one missed is the A. Is there any way to visit all 12, and end up where we started?

Unhappily not. There's no path which can complete such a circuit. You cannot find a path from any note back to itself visiting every other exactly once on the way.

You may, however, visit all notes exactly once – you just don’t get to return home. Here is one such path, again the curved one, starting at C and finishing at B:

The length of the route is 3+3+3+4+3+3+3+4+3+3+3 = 35. If the diagram would allow you to step 1 semitone then you could make it home from B to C (a nicely cadential leading tone) at 36.

There are four such paths (or 48, 12 each, if you care about your start position, but in pitch class world we really don’t). The jump patterns (and path lengths thereof) are:

  1. 35 = 3+3+3+4+3+3+3+4+3+3+3(+1)
  2. 37 = 4+3+3+4+3+3+3+4+3+3+4(-1)
  3. 39 = 4+3+3+4+4+3+4+4+3+3+4(-3)
  4. 41 = 4+4+3+4+4+3+4+4+3+4+4(-5)

The numbers in parentheses at the end are the extra steps you'd need to return to your start position. If you insist on upward jumps only then you must change the last three 'homing' jumps to +11, +9 and +7.

Completenesswise, journey number 4 also has a resolving flavour with its dominant/tonic termination. Journey 3 doesn’t, but the final jump home is at least another third, it’s just in the ‘wrong’ (with this diagram) direction.

Another representation of such circuits is by drawing them on the 12 hour pitch class clock:

0369147A258B 047A258B3691 047A269158B3 048B37A26915

where the modulo 12 labelling describes the pitch class visit-order – all of them starting from 0 (at 12 o’clock) and tracing the black lines strictly clockwise in their 3 or 4 ‘hourly’ jumps, except for that final jump, back to 0 (A and B being 10 and 11 respectively). You may click on the coloured areas to watch the traversals.

It will not have escaped your attention that, plotted this way, bilateral polygonal symmetry is apparent. Musically, this simply admits that their retrogrades and inversions are identical (descending thirds instead of ascending ones). The inverse of a minor third is a major sixth (3 + 9 = 12), and vice versa (4 + 8 = 12). The differences are all interval class 3 or 4. The third clock diagram is distinguished green only because, unlike the others, the closing leap happens to be in that very interval class (+9 ≡ –3, modulo 12).

But once we’ve admitted the equivalent interval classes of sixths and thirds, we may dispense with the arrowheads, since we may now move from node to node in either direction (major 3rd being equivalent to minor sixth, minor third to major sixth, either rising or falling). We may also take the opportunity to dispense with note names and abstract to pitch classes. The following transition diagram results:

Where the A and the B represent pitch classes 10 and 11 (not notes A and B!).

Now, since one may move in either direction, the possibilities for complete circuits (visiting all twelve pitch classes exactly once in a closed loop) would appear to dramatically increase. As in fact they have, to 252 in sheer numbers. But as far as the actual patterns of such loops are concerned, where (musical) transposition of a circuit is the same as starting at a different point, retrograde is circuiting in the opposite direction, and inversion as reflection, etc, it turns out that there are only 11 distinct patterns. In ‘alphabetical’ order, these are

  1. 0362591A7B84 → +3+3-4+3+4+4-3-3+4-3-4(-4) ≡ +3+3+8+3+4+4+9+9+4+9+8(+8)
  2. 0362A147B859 → +3+3-4-4+3+3+3+4-3-3+4(+3) ≡ +3+3+8+8+3+3+3+4+9+9+4(+3)
  3. 0362A1958B74 → +3+3-4-4+3-4-4+3+3-4-3(-4) ≡ +3+3+8+8+3+8+8+3+3+8+9(+8)
  4. 0362A7B84159 → +3+3-4-4-3+4-3-4-3+4+4(+3) ≡ +3+3+8+8+9+4+9+8+9+4+4(+3)
  5. 0362B7A14859 → +3+3-4-3-4+3+3+3+4-3+4(+3) ≡ +3+3+8+9+8+3+3+3+4+9+4(+3)
  6. 0362B847A159 → +3+3-4-3-3-4+3+3+3+4+4(+3) ≡ +3+3+8+9+9+8+3+3+3+4+4(+3)
  7. 0369152A7B84 → +3+3+3+4+4-3-4-3+4-3-4(-4) ≡ +3+3+3+4+4+9+8+9+4+9+8(+8)
  8. 036A1952B748 → +3+3+4+3-4-4-3-3-4-3+4(+4) ≡ +3+3+4+3+8+8+9+9+8+9+4(+4)
  9. 036A259147B8 → +3+3+4+4+3+4+4+3+3+4-3(+4) ≡ +3+3+4+4+3+4+4+3+3+4+9(+4)
  10. 037B26A19584 → +3+4+4+3+4+4+3-4-4+3-4(-4) ≡ +3+4+4+3+4+4+3+8+8+3+8(+8)
  11. 037B2A691584 → +3+4+4+3-4-4+3+4+4+3-4(-4) ≡ +3+4+4+3+8+8+3+4+4+3+8(+8)

0362591A7B84 0362A147B859 0362A1958B74 0362A7B84159 0362B7A14859 0362B847A159 0369152A7B84 036A1952B748 036A259147B8 037B26A19584 037B2A691584

The above 'pitch class clock diagrams' show all 11 species of ‘dodecacircuits’ where every jump size is either ±3 or ±4 semitones. Musically, these steps are minor or major thirds (up or down) in pitch class (or, respectively, major or minor sixths, down or up). Of the 11 patterns, 10 turn up 24 times (rotations & reflections leaving their essential paths unaltered) and 1 turns up only 12 times (because it has two axes of symmetry) - the last shown, in dark blue. Thus accounting for the 252.

Two are coloured yellow; unlike the others, they are unsymmetric. The pair are, however, related – one being a reflection of the other (musical inversion) most easily perceived by flipping (say) the second one (0369152A7B84) in its horizontal (3 o’clock to 9 o’clock) axis (i.e. it cannot rotate into the other, only flip).

All others are symmetric – the brighter blue pair are so distinguished only because their single axis of symmetry happens to cut through an axis on whole hours (opposing dodecagonal vertices) rather than the majority whose axes of symmetry lie over half-hours (opposing dodecagonal edges). For instance the symmetry axis on 0362A147B859 is at 2 o’clock to 8 o’clock, but lies at 2:30 to 8:30 on 036A1952B748. That last one (number 8 in the above list, with the interval path +3+3+4+3-4-4-3-3-4-3+4 is depicted below:

Pitch class Sequence 036A1952B748 as a 12-Tone tone row

And at the end it will take the step of a major third to start the row afresh.

Notice that the ninth dodecagon, labelled 036A259147B8 (the first one in the third row here) is our old friend from the quartet of ‘third-up-only circuiters’ from before, labelled 047A269158B3. It turns up here because its final jump back (of –3) is in the interval class allowed by our new bidirectional interval jumping rule. To see that this is indeed the case, move the first character of the earlier label to the front, thus 047A269158B3 → 47A269158B30 (equivalent to rotating the polygonal pattern by one ‘hour’). Whereupon you now subtract 4 (or, equivalently modulo 12, add 8) to each character of this rotated label to restart it at zero. Thus 47A269158B30 → 036A259147B8, precisely the label of the ninth one above.

The labels above are chosen because they are – alphabetically – the ‘lowest valued’ ones of the (generally 24) possible ‘pitch class path’ labellings for each shape, i.e. with the longest runs of minor third runs at the start, then the major thirds, minor sixths and major sixths. Consequently the polygons drawn are the ones determined by that label.