[SeqFan] Re: Explain this old one...

[Antti Karttunen]

James Ingram (See his home page at: http://home.t-online.de/home/j.ingram/  )
has the following to say about this:

-------- Original Message --------
Subject: Re: Music-related integer sequence. [Fwd: Explain this old one...]
Date: Tue, 27 May 2003 15:19:47 +0200
From: j.ingram at t-online.de (James Ingram)
To: Antti Karttunen <Antti.Karttunen at iki.fi>

Hi Antti,

What an amazing picture!

> From the picture, the sequence
> 3 5 6 8 10 12 13 15 17 19 20 22 24...
> appears to be the union of four arithmetic sequences:
> 3+7n, 5+7n, 6+7n, 8+7n, for n>=0, and is definitely musical.
> But what is the significance of 3,5,6,8?

I think the sequence is actually
1 3 5 6 8 10 12 13 15 17 19 20 22 24

The 1 at the beginning being the name of the root tone of a major scale. The
root names are at the bottom of the upper number triangle (along the
diagonal). The scales are read upwards (as on the monochord hanging on the
column on the left). The numbers are then simply the (chromatic scale!)
numbers of the pitches of a 2-octave scale.
With G as the root, the sequence is
G, A, B, C, D, E, F#, G, A, B, C#, D, E, F#, G,
With C as the root, the sequence is
C, D, E, F, G, A, B, C, D, E, F#, G, A, B, C
Interesting that F# (=19) happens in the upper octave, rather than F (=18).
This may have something to do with modulating to the dominant where F# is
correct. ... I may be wrong, but I think its probably related to the cycle
of fifths, and that this diagram is supposed to show the relations between
the notes in different keys.

> Another puzzler here is the number triangle:
> 1
> 2  3
> 4  6  9
> 8 16 18  24
> 16 24 36  54  81
> 33 48 72 108 162 243
> ...
> which seems to possess mistakes at
> T(3,1)=16?, T(3,3)=24?, T(5,1)=33?,
> that is, if the pattern is supposed to be
> 1
> 2  3
> 4  6  9
> 8 12 18  27
> 16 24 36  54  81
> 32 48 72 108 162 243
> ...
> but perhaps there is simply a more complex formula?
> Paul

The duration symbols drawn between the rows of numbers seem to indicate that
this has to do with note durations rather than pitch. I think this triangle
probably represents symbol durations in different time-modi... but I'm not
an expert in old music. This drawing is so beautifully drawn and elaborate,
that I think mistakes are unlikely. :-)

Keep me posted if anyone has more to add!


pushed for time,

very best wishes

James