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How does one sign an ascending melodic minor scale using hexachordal solmization?

Here is an example, it's the famous Bourreé by Bach, from the Suite in E minor, BWV 996:

enter image description here

We know that Bach used this system of solmization, so the obvious question is, how would Bach sing this using solmization and teach it to his students?

The syllables that I've written, I'm fairly sure about. First we start in the hexachord on D, which is the natural hexachord here, and we have: re mi fa mi re. The note D sharp, is the cadential raised 7th degree, sung ut (or do... in Renaissance and earlier music, this would be "causa pulchritudinis"), so we stay in the same hexachord on D. Then follow re and mi. Now what?

One is tempted to mutate to the hexachord on B (♮), but this hexachord is not related to the hexachord on D, so this seems to me to be out of the question, since only hexachords on G (soft) and A (hard) are allowed in this transposition of the gamut. So this ascending tetrachord in melodic minor is causing me confusion. Descending is easy, we just proceed like in natural minor: (in soft hexachord) la sol fa (mutate to natural hexactord) la sol fa mi re.

To recapitulate, here we have...

Natural E minor (mutating on scale degree 4 from natural to soft hexachord, D🡒G): enter image description here

E minor (Dorian) (mutating on scale degree 5 from natural to hard hexachord, D🡒A): enter image description here

And, what I do not understand, ascending melodic E minor: enter image description here

What syllables should one sing in place of question marks?

Since this scale is not a subset of the transposed gamut, I suspect the syllables are going to be the same as in natural minor, just with a raised intonation (re mi fa re mi ♯fa ♯sol la), but I would like someone to confirm this to me (preferably with a quote from a source with authority) or tell me that I'm wrong and show me what is correct.

Another possibility that occurred to me is that I'm wrong about the allowed hexachords and that we temporarily treat this as modal mixture (pretend we are in E major and not E minor), and then the hexachord on B would be allowed as the hard hexachord. But what would be the point of mutation in that case? The raised sixth degree: re mi fa sol la re mi fa?

A third possibility that occurred to me is to sing it like a dorian scale, just with a raised seventh degree ("causa pulchritudinis"): re mi fa re mi ♯fa sol. I find it very practical but a bit odd.

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    How do we know that Bach used this system? It had basically broken down beyond recognition by his time -- as the analysis here implies.
    – phoog
    Sep 26, 2023 at 20:45
  • Watching with interest, though without knowledge. I wonder whether there was that much "sight-singing" going on at all, or whether we're reverse-projecting Kodaly practices? At any rate, this question might be helpful, though it's a broad overview. Sep 26, 2023 at 23:15
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    I know far too little about hexachordal solmisation (which in any case seems only applicable with great difficulty) to provide an answer, but jstor.org/stable/41639991?seq=8 (Schenkman, Walter, "The influence of hexachordal thinking in the organization of Bach's fugue subjects", Bach, vol 7, no. 3, 1976, p. 7-16.) provides some very interesting historical context on the topic of Bach and the "traditional" form of solmisation.
    – AlexJ
    Sep 27, 2023 at 2:30
  • Regarding phoog's comment, this is also a question I very much have in mind. Bach would have been aware of the more modern developments in music theory, and probably of the debate between Mattheson and Buttstedt regarding solmisation, though the dearth of theoretical works from his own hand (there's a few pages about figured bass realisation and that's about it) makes any more definitive statement about this or Bach's personal stance on the matter (and what he himself used in practice) a perilous proposition, at least without more in-depth research.
    – AlexJ
    Sep 27, 2023 at 2:55
  • @phoog That's an interesting question in itself, but goes far beyond the scope of this question. There is plenty of evidence that Bach used hexachord solmization. The most obvious evidence is the way he notated minor key signatures, often in "Dorian mode". There is also the introduction to the WTC, and the canon BWV 1078. Derek Remeš mentions more evidence to it in his disertation, as does Nicholas Baragwanath in his book about solfeggio. The hexachordal solmization is not restricted to Renaissance and early Baroque, it was used throughout the 18th century, well into the 19th.
    – Kresimir
    Sep 27, 2023 at 6:23

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