The graphic below shows, among other things, the relative direction that every major (M), minor (m), and augmented (+) triad leans on the four-pan "scale." For example, in last month's post, I showed how the F-major triad tilts the "scale" 30° farther clockwise than the A-minor triad. On the graphic below, the F-major triad (FM) is located 30° farther clockwise than the A-minor triad (Am).
In this image, I have also used parts of Earth's geography as an analogy for this musical space. Each trio of major or minor triads that tilt in the same direction are associated with three port cities relatively near to one another on our globe. Two trios whose major or minor triads are no more than 30° apart match trios of cities on the same continent. Two trios whose major or minor triads are 60° apart are on opposite sides of a substantial body of water, which corresponds to an augmented triad. The triads in the first nine measures of my variation mostly hang out in South America, with the occasional jaunt over the Caribbean Sea to visit US cities on the Gulf of Mexico. But, after m. 9, a more potent wanderlust takes over.
When circumnavigating this space as smoothly as possible using exclusively major and minor triads, one alternates between a leg of the journey on land (a 30° difference) and a leg of the journey at sea (a 60° difference). Eight legs -- four each on land and sea -- can bring you back to where you started: 30° + 60° + 30° + 60° + 30° + 60° + 30° + 60° = 360°. (Like last month, there is that number 8 again, which is conveniently quite a frequent phrase-rhythm length.)
This geographic metaphor also provides a way to remember how voice leading interacts with triadic location, if you know your American mythology. The post last month introduced the general principle that, if you divide the number of degrees two major or minor triads differ by 30, you get the minimal number of semitones needed to change the notes of one triad to the other. So, in this case, one (semitone) if by land, and two (semitones) if by sea. Almost. (More on that later.)
Every leg of the circumnavigation can end in one of three triadic ports. That means that, starting from a specific part of the globe -- say, Salvador, Brazil -- and choosing either east or west (my variation's progression, which is highlighted in red and given a start/finish arrow in green in the graphic above, goes east, i.e. clockwise), there are 6,561 (3^8) different ways of smoothly circumnavigating the four-pan "scale" exclusively with major and minor triads such that you return to the eastern coast of South America.
There is a smaller set of globe-trotting itineraries that are more regular. The roots of two triads spanning a 60° seafaring leg can be apart by either a minor second, minor third, or perfect fifth. For example, as shown with last month's video, the first progression in my variation -- Am to EM -- moves 60° counterclockwise on the image above, west (and north) across the Caribbean Sea and Gulf of Mexico. These two roots -- A and E -- span (at its smallest) a perfect fourth. Two other possible moves 60° counterclockwise -- Am to AbM, and Am to CM -- would create root motions by minor second and a minor third, respectively. Therefore, every 60° leg can be further described by one of these three root motions; for example, Am to EM could be "60° P4." Direction of circumnavigation (clockwise (east) or counterclockwise (west)) and root interval (ascending or descending) need not be specified!
The roots of two triads spanning a 30° leg across land can either be the same or apart a major third. For example, as shown with last month's video, the fifth progression in my variation -- Fm to DbM -- moves 30° clockwise, east (and south) across the continent of South America. These two roots -- F and Db -- span a major third. Two other possible moves 30° clockwise -- Fm to AM, and Fm to FM -- would create a root motion by major third and no root motion at all, respectively. But now it appears that there are two different motions for "30° M3": Fm to DbM and Fm to AM. Again, 1 (semitone) if by land, and 2 (semitones) if by sea -- normally. For example, the cross-country Fm to DbM is fine: only 1 semitone of voice leading is needed (C-->Db). However, the cross-country Fm to AM is not: 3 semitones of voice leading are needed (F-->E, Ab-->A, C-->C#), even though Fm and AM differ by only 30°. This discrepancy is what the word "almost" in last month's and this month's posts both referred to.
However, this discrepancy is useful here, because it helps us distinguish the two types of "30° M3" from one another: one uses 1 semitone of voice leading ("30° M3") and the other uses 3 semitones of voice leading ("30° M3 3").
These six types of smallest legs have synonyms in neo-Riemannian theory:
Degrees apart
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Root distance
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Semitones of voice leading
(if not degrees/30°)
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Neo-Riemannian label
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30° (by land)
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P1
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P
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M3
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L
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||
3
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H
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||
60° (by sea)
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m2
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S
|
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m3
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R
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||
P4
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N
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The aforementioned "more regular globe-trotting itineraries" refer to those that alternate between the same kind of leg as they progress: each land leg is always the same one of the three possible, and each sea leg is always the same one of the three possible. This leads to nine (3 times 3) possible regular itineraries.
In his book, Cohn identified examples of seven of these nine. In my review, I identified an example of one of the missing two. YouTube audio of some of these examples are linked in the table below. (Some of these aren't complete circumnavigations, but all have enough to make the regularity of the trajectory clear.) The nine thick enclosures are the nine combinations of the three possible land legs with the three possible sea legs. Eight of the nine enclosures have at least one link. The one that does not -- the alternation between 60° m2 (S) and 30° M3 3 (H) -- is the one that my Paganini variation uses, because the roots of the triads that span 60° (e.g. AM and Bbm) differ by a minor second, and the roots of the triads that span 30° (e.g. Bbm and DM) differ by a major third, and require a minimum of three semitones of voice leading.
By land
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|||||||
30° P1 (P)
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30° M3 (L)
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30° M3 3 (H)
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|||||
By sea
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60° m2 (S)
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+m2
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-m3
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+P4
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-P4
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||
60° m3 (R)
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+m3*
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-m3
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-P4
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+m2
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60° P4 (N)
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+m2
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-m3
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|||||
* This one takes place over the entire slow introduction, in which each triad in the journey is potentially tonicized with other harmonies.
The subdivision within each thick enclosure indicate the directed transposition between every other triad. Clockwise or eastward journeys match the + symbol; counterclockwise or westward journeys match the - symbol. This is why I was drawn to Paganini's caprice in writing music that used the S-H combination, because it required a perfect-fourth root relationship between every other triad. The second half of Paganini's caprice features ascending-fourth root motion: AM, Dm, GM, CM. Creating a S-H progression simply required converting the Dm to DM -- which is not ideal, but other composers have distorted this music far more -- and interpolating the proper minor triads.
Cohn's book goes into much greater detail into these and other topics, and introduces a number of graphs and terms that I have mostly sidestepped. In fact, someone familiar with Cohn's ideas may be wondering why I appear to be reinventing wheels here. In part, it is because Quinn's four-pan "scale" allows us to make another connection between Paganini's original caprice and my variation that Cohn's concentration on consonant triads does not. I'll bring this up next month.
P.S. The S-L combination -- specifically, with the +m3 direction -- has become quite popular recently in mainstream Hollywood film music: I can think of three movies last year that made clear use of it, and here's an example from a movie this year that, like my progression, uses exactly 8 triads in its world tour, not quite ending up where it started but providing a fun ride nonetheless.
