Fludd's sequences

Sat May 31 12:07:02 CEST 2003

(I beg a pardon for longish quotations...)

"N. J. A. Sloane" wrote:
> 
> Dear James,
> 
> I'm adding this sequence to the OEIS:
> 
> %I A082975
> %S A082975 1,2,3,4,6,9,8,16,18,24,16,24,36,54,81,33,48,72,108,162,243
> %N A082975 Fludd's triangle, read by rows.
> %C A082975 Related to A036561? It would be nice to have a precise definition. - njas
> %D A082795 Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.
> %H A082795 Robert Fludd, <a href="http://highway49.library.yale.edu/photonegatives/oneITEM.asp?pid=39002036242312&iid=3624231">Link to reference</a>
> %e A082975 1
> %e A082975 2 3
> %e A082975 4 6 9
> %e A082975 8 16 18 24
> %e A082975 16 24 36 54 81
> %e A082975 33 48 72 108 162 243
> %O A082975 0,2
> %K A082975 nonn,tabl
> %A A082975 Paul D Hanna (pauldhanna(AT)juno.com), May 30 2003
> %Y A082975 Cf. A047329.
> 
> Do you understand it?  I saw your message to Antti,
> but didn't really follow it
> 
> Can you tell me how the triangle is constructed?
> 
> (The 33 looks wrong to me)
> 
> Thanks!
> 
> Neil Sloane

Sorry Neil, SeqFanatics, Other interested people,
I forgot to forward the other mail from James (and no time
to ask his permission now... ;-), which follows below.

-- Antti

Subject: 
        Re: [seqfan] Music-related integer sequence. (In text format.)
   Date: 
        Thu, 29 May 2003 00:15:07 +0200
   From: 
        j.ingram at t-online.de (James Ingram)
     To: 
        Antti Karttunen <Antti.Karttunen at iki.fi>

Hi Antti,

Antti Karttunen wrote:

> with Pierre Lamothe's permission (see his site: http://www.aei.ca/~plamothe/
> ), 
> I forward also his explanation about this topic.

Pierre may well be nearer the truth than I was. My version gave more
emphasis to the verticals and diagonals than to the horizontals in the
table, but the existence of both E and F along the bottom diagonal gives me
a hard time....
If Pierre is right, I dont understand why the chords are not in root
position - the sequence 7n + (1 3 5 7). The rotation needs explaining.
Possibly Fludd thought the submediant important (i.e. these are "added
sixth" chords). And why are there just three octaves? There seems no good
reason for the table to be cut off horizontally at the top. Maybe this has
to do with the three towers and three sets of organ pipes on the right.
Maybe I havent understood Pierre's French well enough.

His theory that the numbers are white-key pitch degrees is coroborated in
the four arches at the top right, where 8 is the scale on g (one octave
higher than G); 6 is the scale on e; 3 is the scale on b, 5 is the scale on
d. Just below these arches are 3 x 7 organ pipes (not 3 x 12). Strange that
the scale starting on b is not an official mode...

But it may be wrong to look for an ultimate explanation. I suspect that this
is a "folly" showing the imperfection of all sublunary wisdom. It may even
have deliberate mistakes for that reason! A kind of philosophical Escher
drawing. :-) Music would be a great subject for such a satire....

Note that on the other drawing (the music of the spheres with monochord),
the sphere of the stars is located at 2/3 of the string (= 1 octave + a
fifth), and that everything above that is rather vague. The e, f and gg
positions dont seem to be related to anything. The gg position is actually
at a string length of zero (an infinitely high note)! Relates of course to
the gamma - gamut (eveything) :-))) - see the krystals remarks about zero
and infinity.

Other "mistakes" in the first drawing include
1. Having an F column to the right of the lowest G in the table - the gamma
denotes gamut - i.e. the whole (string). Having more is paradoxical.
2. The "torn" column on the left - seems not to make sense. Is it really
torn? Could it be put back correctly?

Physical human activity is represented at the bottom (seems to be a forge).
Music (the highest intellectual activity) at the top. Both seem doomed to
error. :-)

all for now

James 

> 
> 
>> Pierre Lamothe wrote:
>> 
>> Cher Antti,
>> 
>> 
>> Dans le contexte présenté, ce qui compte dans la suite 3 5 6 8 10 12 13 15 17
>> 19 20 22 24 ...
>> ou 7n + (3 5 6 8),  c'est la différence des termes, car c'est simplement une
>> mise en forme de tous
>> les accords septièmes possibles. À ce titre, elle ne devrait donc pas être
>> distinguée de sa translatée
>> 7n + ( 0 2 4 6 ) dont l'origine serait 0, alors que la première suite
>> enracine arbitrairement les accords
>> à -1 ou 6.
>> 
>> On se trouve ici dans le contexte d'une échelle fixe (sans altération) où les
>> divers modes sont obtenus
>> par translation de la tonique. La suite F G ... aaa est une suite de degrés
>> et non de largeurs d'intervalles.
>> Un accord septième, en terme de degrés, c'est la superposition de trois
>> tierces,. Sa formule relative en
>> degrés, dans l'état fondamental, est  0 2 4 6 ou, de façon équivalente ici, 6
>> 8 10 12. On a l'équivalence
>> suivante :
>> 
>> 3, 10, 17, 24 == Quinte
>> 5, 12, 19 == Septième
>> 6, 13, 20 == Fondamentale
>> 8, 15, 22 == Tierce
>> 
>> La suite 3 5 6 8, par exemple, représente un accord septième à l'état d'un
>> des renversements.
>> 
>> Les largeurs de tierce varient selon le point de départ de la série dans
>> l'échelle fixe. Les sept points de départ
>> (modulo octave) distincts engendrent les quatre types d'accord septième
>> suivants, selon la formule des tierces
>> exprimées en demi-tons tempérés.
>> 
>> 3 4 3 == min7
>> 3 3 4 == min7(5b)
>> 4 3 4 == Maj7
>> 4 3 3 == 7
>> 
>> 
>> Pierre

(Neil's mail continues:)
> 
> BTW, the other sequence is:
> 
> %I A047329
> %S A047329 1,3,5,6,8,10,12,13,15,17,19,20,22,24,26,27,29,31,33,34,36,38,40,41,43,
> %T A047329 45,47,48,50,52,54,55,57,59,61,62,64,66,68,69,71,73,75,76,78,80,82,83,
> %U A047329 85,87,89,90,92,94,96,97
> %N A047329 Congruent to {1, 3, 5, 6} mod 7.
> %K A047329 nonn
> %D A047329 Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.
> %H A047329 Robert Fludd, <a href="http://highway49.library.yale.edu/photonegatives/oneITEM.asp?pid=39002036242312&iid=3624231">Link to reference</a>
> %O A047329 0,2
> %A A047329 njas
> %E A047329 Fludd reference from Brendan McKay, May 27, 2003
> 
> but here there is no (mathematical) mystery
> 
>  Neil J. A. Sloane
>  AT&T Shannon Labs, Room C233,
>  180 Park Avenue, Florham Park, NJ 07932-0971
>  Email: njas at research.att.com
>  Office: 973 360 8415; fax: 973 360 8178
>  Home page: http://www.research.att.com/~njas/