[seqfan] Fludd's sequences. James replies. [This one without bogus attachments.]

[Antti Karttunen]

[The following text is all from James Ingram, unless otherwise noted.
This is identical to previous mail, but without the GIF and other
attachments that were accidentally forwarded in the other mail.
I resend this as the previous mail might be caught by firewalls
or other filters.]


Subject: Re: [seqfan] Fludd's sequences. The debate continues.
   Date: Fri, 06 Jun 2003 15:24:01 +0200
   From: j.ingram at t-online.de (James Ingram)
     To: Antti Karttunen <Antti.Karttunen at iki.fi>


Antti wrote:

> Hopefully having remembered to ask everybody's permission,
> I will forward three latest e-mails from James Ingram
> and Pierre Lamothe concerning the Fludd's sequence
> http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082977
> and the table (now marked dead)
> http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082976
> 
> The original picture can be found at:
> http://highway49.library.yale.edu/photonegatives/oneITEM.asp?pid=39002036242312&iid=3624231

Dear Antti, Pierre, SeqFans,

Before I compare notes with Pierre, I'd like to say what I now really think
about the two number triangles. These things have been brewing slowly...

1. I no longer think there are any "deliberate mistakes".

2. The lower triangle:
This is, as I thought, about durations and time-modes. The strange looking
16 and 24 (in row 4) are - I now think - time modi in which the brevis (the
square duration symbol) contains 4 semibrevises (rhombus). So row 4 contains
2x4=8, 4x4=16, 3x6=18, 4x6=24. Probably, the missing 3x4=12 and 3x9=27 exist
(necessary for rows 5 and 6) but Fludd omitted them because he wanted a
triangle. 4x9=36 may also be missing.
I have to say that this solution is highly speculative, since only modus
perfektus (subdivisions of 3) and modus imperfectus (subdivisions of 2) are
well known. I have never seen a mode with groups of 4 before. Note that
Brahms also ran into trouble over having triplets of triplets (his
"eighth-notes" are shorter than his "sixteenth-notes"). Maybe this has
something to do with Fludd having a mode with subdivisions of 4s, and this
not having caught on. :-)
I think the 33 in row 6 is a copyists error (a pretty dumb copyist).

3. The upper triangle:
This contains the numbers necessary for completing a figured score like the
one being explained in the alcove above the table. The numbers are simply
the diatonic distances between two pitches. This is like a modern chart of
distances between cities. Fludd has omitted the dissonant intervals because
they are unnecessary for use in figuring such a score. The three octaves is
the range required for figuring singable scores (from the lowest bass to the
highest soprano).
I dont think the numbers are to be read simultaneously as chords. There are
no seventh chords or their inversions. The 6 would be used when the chord in
the score is a "first inversion": (reading upwards) E, G, C would be figured
from the lowest tone E: 3, 6. The bass tone never needs figuring because it
is always = 1.

As an exercise, I have had a go at transcribing Fludd's score.
There is a copy of my transcription at
http://www.iki.fi/~kartturi/fludd/fludd.gif [I made this copy on James's behalf, AK.]
Anyone can copy this as much as they like.

Unfortunately, the music is rather academic and no great revelation... :-)

Its interesting that labeling the intervals (i.e. figuring the score) makes
them easier to see, so things like weak parallel fifths become much easier
to spot. The top line of Fludd's score contains too many 12s for my taste.

---------------

On to Pierre: Apologies for any misunderstandings - my French is not very
good.

Pierre was obviously right about the numbers being diatonic intervals (see
below). I was wrong about bringing in the "standard chromatic keyboard", but
think the above explanation is more convincing (simpler, more elegant) than
involving chords in the table itself (especially chords involving sevenths).

---------
> James écrit : "Strange that the leading notes (si) are omitted in ut re mi
> fa sol la".
> 
> Il ne faut pas confondre l'échelle fixe ... A B C D E F G ... et l'hexacorde
> mobile ut re mi fa sol la qui sert à solfier et ne comporte ainsi à dessein
> qu'un seul demi-ton.

But why only a hexachord? I still dont understand why the seventh tone (si)
is being omitted. At school, we learned solfeggio with seven tones. Si
(sometimes called ti) is especially important because it has to be sung very
sharp (high).

--------

>> Probably Pierre is right that these are white-key numbers,
>> but I'm not convinced that seventh-chords are important enough in the
>> abstract.
> 
> 1) Je n'ai jamais considéré ces nombres comme des "white-key numbers"
> mais bien comme les degrés des divers modes qui occurent en se déplaçant sur
> l'échelle fixe, et ce en vertu d'un principe de congruité inscrit dans toute
> échelle musicale bien construite. Quand, plus tard, on a ajouté les
> altérations (on utilise ici seulement l'alternative b-mol ou b-carre), on a
> toujours strictement respecté cette congruence, distinguant par exemple
> quarte augmentée et quinte diminuée.

Okay. Maybe we can agree to call the numbers "diatonic" - with all the ups
and downs that that entails. Even in modern (non-keyboard) music, b-flat is
not the same pitch as a-sharp. Not even all b-flats have the same pitch.
Pitch symbols (like duration symbols) denote classes of pitches. The pitch
which sounds best always has to be found in rehearsals, and is highly
context dependent. (A si is always sharp, but how sharp?)

> 2) La superposition généralisée de quatre voix dans le contrepoint précède
> la reconnaissance comme accords des superpositions de notes. Ce qui est en
> réalité une "abstraction", ce sont ces entités spécifiques dénommées accords
> qui regroupent sous une même appellation diverses réalisations reconnues dès
> lors comme renversements d'un même accord.

I think this comes close to where Fludd was. But I cant pretend to have
understood you completely.
> 
> Ça suppose l'existence d'un principe de reconnaissance de l'état fondamental
> d'un accord. Si les considérations harmoniques ont eu une certaine
> importance pour fonder les triades, on a tôt fait de se rabattre sur la
> simple superposition de tierces pour construire le système des accords. Il
> faut éviter de regarder le passé avec les yeux d'aujourd'hui, et voir comme
> "abstract" le fondement concret du vocabulaire musical abstrait actuel.

Yes. And ditto. :-)).

all the very best

James