Scalar transpositions

I'm currently writing a program for analysing tonal music, and ran into something I didn't know the answer to.

Basically, this has to do with scalar transpositions, i.e., transposing a given pitch P up or down by n scale degrees relative to some given scale S. For pitches that belong to that scale, this is trivial, of course.

But where things become complicated is when P has an accidental relative to the prevailing scale. For example, there may be an F# in the music in the prevailing scale of C major, as a modification of F (the 4th scale degree).   My current approach to handling this is to first measure the interval from the tonic to the given pitch, in this case, C to F#, which is an augmented 4th.  Then suppose we want to transpose up by 1 degree, which corresponds to an augmented 5th from C, so the result is G#. So far so good.

But where problems arise is when the number of steps in the transposition crosses perfect/imperfect intervals, for example, if, under the same transposition (up by 1 degree relative to C major) I have a note Eb, which is a minor 3rd from the tonic C, then what interval should be formed when I transpose it to a 4th? Should it be a perfect 4th, or a diminished 4th?  If a perfect 4th, does that mean both E and Eb would transpose to F under this transposition?

(P.S. Just to be clear, I'm talking here about scalar transposition, not chromatic transposition where the relative semitone distances between notes are maintained. By scalar transposition I mean that transposing, say, C-D-E up by 2 degrees (relative to C) would yield E-F-G, not E-F#-G#.)

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Replies

  • I'm not sure if there's a "correct" answer here, but I can tell you what Finale does.

    When you transpose a passage in Finale you have the choice of "chromatic" or "diatonic" transposition ("diatonic" being another less preferred term for "scalar", I just learned from Wikipedia...)

    For the diatonic transpositions, you are always assumed to be "in a key" and thus merely given a choice of scale degree movement within that key (up or down a unsion, second, third, etc.)

    If you are in C-Major and transpose Eb up a "fourth", you are given Ab.

    If you are in Bb-Major and transpose Eb up a "fourth", you are given A.

    It appears that the rule Finale is following is to move each note in a scale up a certain interval to the according note within the same scale.  If the initial note also has a flat or sharp (or doubles) then those extra steps are carried over into the transposed note, regardless of how awkward the spelling might be.

    So if you are in C and transpose E up a "second" it results in "F" (up the next degree of C-Major)

    If you are in C and transpose Eb up a "second" it results in "Fb" (up to F, the next degree of C-Major + half step down)

  • Thanks!

    Now I'm curious whether Finale distinguishes between major and minor keys. Extrapolating from your examples, if you are in C major, which is the relative major of A minor, what would happen to G#, which is an augmented 5th above C and the sharpened leading tone of A minor, up by, say, a 3rd? Would it give B or B#?  And would the result be different if you're in A minor?

  • In both Cmajor and A minor, a "third" transposition of G# gives you B#.

    I don't think Finale is doing anything fancier than moving the note up and down on the staff by a certain interval, followed by an instantaneous, yet additional, separate step to alter the tone up or down if there happened to be any accidentals attached to the initial note. 

  • HS,

    Forgive me, because I am in drink, but your post seems a little confusing as you are looking for a scalar solution and yet give an example of ordinary transposition with the raised 4th becoming a raised 5th on transposition of a maj.2nd. I think DMusick is right in saying there is no correct answer here. As you point out, in scalar transposition, a min AND maj. 3rd will become the sub-dominant when transposed up a tone. That is the only solution in a scalar transposition. But when transposing in scalar fashion, you will lose the intervallic integrity because in the example being cited, what starts as c,d,eflat,f in the root position, now becomes d,eflat,f,g. In other words, scalar transposing changes the functionality of the notes eg. the 7th will not be a leading tone anymore once shifted along because the interval structure of the scale is also shifted. This in effect modal transposition and may well hamper any analysis.

     If you are trying to analyse notes that are not in a scale, like e flat in c major then perhaps you need to take into account that the nature of the scale may be changing, modulating to a new centre perhaps or from major to minor. If so, then these notes should be considered as functioning/pivoting notes in/from the destination scale and not part of the original scale. One could then analyse the f sharp you mentioned as the leading tone to the dominant and not referenced to the original tonic.

    Not sure if I'm helping here because you no doubt know all of the above, but it's a few hours after wine o'clock, if no-one has chimed in by tomorrow, I'll see if I can  help anymore...hicc..

    Mike.

  • Thanks to all who responded.

    So I gather that diatonic transposition is only uniquely determined if all notes belong to the scale, and no accidentals are involved.  Which makes sense in retrospect, since some amount of human judgment would be required as well as the context of a particular melody line.

    For example, a motif A-G-F#-G in C major could potentially be treating the F# as a decorative accidental, like some kind of half turn or short mordent, so if one were to diatonically transpose it up by a 4th, one would write D-C-B-C rather than D-C-B#-C, so as to preserve the effect of the semitone descent and ascent.  But in other contexts, such as, say, C-D-E-F#-F, where the lowering of the raised 4th is an inherent feature of the motif, one would want to preserve the sharpness and render the transposed variant as, perhaps, F-G-A-B#-B.  (Though in this case one arguably would want to write the original as C-D-E-Gb-F instead, but then it's unclear whether one should or shouldn't transfer the accidental over to the result -- F-G-A-C-B or F-G-A-Cb-B. Probably the former, in order to preserve the semitone descent in the motif.)  In any case, it's clear that some amount of human judgment is required, and no single algorithm could produce the "best" transposition in all cases.

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