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I was researching techniques used in classical improvisation. And I keep coming across two terms. One is "Partimento". Which is an Italian method of improvising on a bass line. Another is "The Rule of Octaves" which is a way to harmonize bass lines.

If you look at the Partimento wiki you'll see it makes use of the Rule of the Octaves. Also I was reading on how the Bach family used bass lines for improvisation. And C.P.E Bach wrote a section about Rule of Octaves in his book "Essay on the True Art of Playing Keyboard Instruments".

But I know JS and C.P.E Bach were German and Partimento is an Italian method so I'm wondering if and how there is a connection with these two. Were the Bach family influenced by Partimento and was it the basis of their improvisation?

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My understanding is partimento is figured bass used for teaching. From that understanding the answer is yes. Here is a quote...

...[JS Bach's] students had to begin with the study of figured bass...

Source:

Title C.P.E. Bach Studies Cambridge Composer Studies Editor Annette Richards Edition illustrated, reprint Publisher Cambridge University Press, 2006 ISBN 0521836298, 9780521836296

https://books.google.com/books?id=smZc7Bs1ta0C

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Another book that may interest you is...

The Langloz Manuscript: Fugal Improvisation through Figured Bass (Oxford Early Music Series) by William Renwick Link: http://a.co/d/gtDUU5w

...Amazon offers a preview where the introduction has some discussion of Bach and figured bass teaching.

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    thanks that book's first chapter on "Partimento Fugue" does show a connection with Bach. I'm wondering how Bach found out about it. Was he influenced by the Italians somehow or was improvisation on figured bass just a thing "everyone" was doing at the time. – foreyez Oct 31 '18 at 17:04
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    Figured bass was used throughout Europe. Basically it was the harmony shorthand, sort of like the roman numerals used today. Keep in mind Bach was interested in international styles. Re. Italy, think of his "Italian" concerto which is an arrangement of Vivaldi. – Michael Curtis Oct 31 '18 at 17:23
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As Michael says, partimenti are really just figured and unfigured bass exercises for young musicians to realize at the keyboard.

As for a possible connection between Bach and the Italian style, consider the Italian Francesco Durante (1684–1755). In Chapter 16 of his Music in the Galant Style, Robert Gjerdingen discusses Durant's fifty-seven figured-bass partimenti, and then goes on to say that

J. S. Bach, Durante's contemporary, was aware of the Neapolitan's music and his high status. Bach had even copied one of Durante's masses for his own use (BWV Anh 26, 1727). In Bach's sonata for flute and harpsichord (BWV 1030) we can recognize a similar twofold presentation of the Fenaroli schema [that Durante had popularized].

Without getting into details, for the purposes of this answer "schema" is synonymous with "partimento." So this excerpt gives direct evidence of an Italian composer popularizing a partimento, Bach studying this composer's works, and then Bach using the partimento that this Italian composer popularized.

So as for wondering whether there is a connection between Bach and the Italian style: most definitely!

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