RE: [tuning] Re: CHALLENGE

I wrote,

>> { 5, 7, 12, 17, 29, 41, ... }

>These are the MOSs of the pure fifth generator. The continuation of
>this sequence can, I believe, be calculated by taking the
>denominators of the continued fraction convergents of log(3/2)/log
>(2), just like the process I demonstrated on Sunday with phi, etc.
>I'll try this on my computer tomorrow.

OK. To be consistent with previous messages I'll use the fourth instead of
the fifth. The continued fraction expansion of log(4/3)/log(2) begins

1/(2+1/(2+1/(2+1/(3+1/(1+1/(5+1/(2+1/(23+1/(2+1/(2+1/(1+1/(1+1/(55+1/(1)))))
)))))))))

The convergents are

1/2 = 1/2
1/(2+1/2) = 2/5
1/(2+1/(2+1/2)) = 1/(2+5/2) = 7/12
1/(2+1/(2+1/(2+1/3))) = 1/(2+1/(2+3/7)) = 1/(2+7/17) = 17/41
1/(2+1/(2+1/(2+1/(3+1/1)))) = 1/(2+1/(2+1/(2+1/4))) = 1/(2+1/(2+4/9)) =
1/(2+9/22) = 22/53

Pierre is correct that this misses many scales in the sequence.

To fix the shortfall, we allow each term in the expansion to step up from 1
to its final value before proceeding to the next term:

1/1 = 1
1/2 = 1/2
1/(2+1/1) = 1/3
1/(2+1/2) = 2/5
1/(2+1/(2+1/1)) = 1/(2+1/3) = 3/7
1/(2+1/(2+1/2)) = 1/(2+2/5) = 5/12
1/(2+1/(2+1/(2+1/1))) = 1/(2+1/(2+1/3)) = 1/(2+3/7) = 7/17
1/(2+1/(2+1/(2+1/2))) = 1/(2+1/(2+2/5)) = 1/(2+5/12) = 12/29
1/(2+1/(2+1/(2+1/3))) = 1/(2+1/(2+3/7)) = 1/(2+7/17) = 17/41
1/(2+1/(2+1/(2+1/(3+1/1)))) = 1/(2+1/(2+1/(2+1/4))) = 1/(2+1/(2+4/9)) =
1/(2+9/22) = 22/53

This should solve the challenge.

I will get back to Pierre's other points a bit later.

Pierre wrote,

>What is the meaning of

> << the 7-tone scale built from pure fifths is not proper >>.

>Is an historical point of view, a musical point of view, or only a
>mathematical point of view ?

It may be by coincidence that you came onto the list while we were
discussing very similar issues, in which case you should read perhaps the
last week of archives. By "propriety" we've been referring to Rotherberg
propriety, which means that specific interval size is a monotonic increasing
function of generic interval size. For example, the 7-tone Pythagorean scale
is not proper because the diminished fifth is smaller than the augmented
fourth. The reason I brought that up is that Jason Yust was looking for
generators that give Rothenberg-proper scales most often in the process of
stacking them to form larger and larger scales.

>For the moment, I suggest you look at these
>figures :

>http://www.aei.ca/~plamothe/pix/majmin1.gif
>http://www.aei.ca/~plamothe/pix/majmin2.gif

I'm sure there is much of great interest encapsulated in these diagrams, but
from a musical and historical standpoint, I must note that the "major mode"
of Ptolemy's Intense Diatonic, or your "gamme de Zarlino", is essentially a
fiction. At no point in the history of Western music, with the possible
exception of a Pythagorean schismatic approximation (i.e., using Pythagorean
diminished fourths to approximate 5:4s and augmented seconds to approximate
6:5s) used around 1420, did this scale have any importance whatsoever.
Zarlino may have referred to it as a conceptual ideal, but in practice he
proposed using 2/7-comma meantone temperament to overcome its shortcomings
-- most glaringly, its out-of-tune ii chord.

>With mystical preoccupations of Pythagore, the most
>consonant tones {5/4, 4/3, 3/2, 5/3} were identified to {81/64, 4/3, 3/2,
>27/16} to fit mathematical presupposition on the number 3.

Ouch -- we may have to rehash some lengthy arguments already in the archives
here, but suffice it to say that by all musicological indications, until
5-limit consonances began to be used in the 15th century and became the norm
in the 16th century, medieval Western music used thirds and sixths as
dissonances and valued their unstable quality in order to power the
resolutions to fifths, fourths, and octaves in which they were involved. For
those purposes, 81:64 and 27:16 are quite appropriate intervals, though one
would not notice if they were out-of-tune (in fact those Medieval theorists
who proposed latering these thirds and sixths suggested sizes that were
_further_ from the simple 5-limit ratios, rather than closer to them).
Morover, the "major" or "Ionian" mode was among the least used of the modes
in Medieval times, so your argument seems to fail on that basis as well.

>The definition of "proper mode of a set of tones" or "mode in a set of
>tones" distinguish, first, factorizable tones (by inferior tones in the
set)
>and prime tones. All the tones are prime in {9/8, 5/4, 3/2}. Only 5/4 is
>factorizable in {10/9, 9/8, 5/4, 3/2}. If there exist, in a set of tones, a
>chain of prime tones connecting 1 and 2, it's a proper mode of the set (by
>definition). The tones (1 4/3 3/2 2) are a mode in the set {1, 9/8, 4/3,
3/2}
>corresponding to the chain of prime tones 4/3 9/8 4/3.

I'm confused by this theory and what justification you could have for it,
but go on. Note, though, that the term "proper" is liable to be confused
with the Rothenberg propriety mentioned above.

>It seems to me that good scales result of balance between consonance and
>coherence.

On that we can agree. However, by restricting yourself to JI scales (with
pure, untempered ratios), you're missing many interesting possibilities
here, including the ones which have been important to Western music since
1500 (even the schismatic Pythagorean tuning of c.1420 mentioned earlier has
its consonant relationship best understood by considering it conceptually as
a schismatic temperament, say one where the fifth is tempered by 1/8 to 1/9
schisma).