TUNING digest 1494: Numerological Postings

Numerology is fun but finding counterexamples is just too easy, and
possibly distracts from the profound and elegant properties of the number=
s
themselves. =


For example, Paul Erlich's citation mentions the 12 apostles; most modern=

scholarship attributes the significance of the number 12 to the associati=
on
that early Jewish Christians would make with the number of Jewish tribes.=

So is the number 12 Christian or Jewish? =


The calendric significance of the numbers 7 and 12 is practically univers=
al
and are part of a shared Jewish and Christian cultural inheritance.

There is indeed a strong neo-platonic (and by extension, neo-pythagorean)=

element to post-Pauline Christianity, and the doctrine of the trinity and=

related doctrines (e.g. Pseudo-Dionysius's three heirarchies of three
orders of angels) are tempting to give importance to such numbers, but th=
e
number 10, which was identified as Jewish, is in fact the very Pythagorea=
n
sum (the Tetrachys) of the numbers one through four. Since medieval
Christianity and post-scriptural Judaism share the neo-platonic component=
,
this is not surprising. =


While all were aware of properties belonging to the pythagorean sequence =
of
12 fifths, the treatment of a scale type as X tones out of a set of Y
tones, interestingly, seems not to have been a major component of practic=
al
scale theory in any of the cultures discussed above (the classical Greek
systems were generally conceived as placeholders for varying interval
contents, pythagorean sequences being only one among many possibilities,
the later view of the harmoniae as keys rather than modes possibly an
exception to this point; Jewish and Christian cantillation seems to have
been diatonic, with at most one or two additions to a theoretical
pythagorean sequence, _if_ that was indeed the intonation in question, a
matter of high speculation), and in Europe well into the late middle ages=
,
instrumental representations of 12 tone sets in practice must have been
extremely rare.

''X out of Y theory'' was, however, a mainstay of Chinese theory and
practice (in the form 5+2 out of 12 or some larger pythagorean cycle) as
well as Indian sruti theory (5 or 6 or 7 out of 22, with the schisma
ignored possibly pythagorean), becomes important to fretting theory in
classical Islamic cultures (again, neo-platonic, and traditionally
hospitable to both Jewish and Christian musicians) and only later in
fretting and keyboard theory in Europe, or in Javanese/Balinese and
mainland southeast Asian seven tone sets. =

Johnny Reinhard wrote,

>I've been working on a new piece in a new tuning. It is for soprano,
>violin, viola, and cello and scheduled for performance at De Ysbreker
on
>October 13th in Amsterdam. It is largely tuned to Harmonic-17

Good luck with this piece and I hope to hear it someday. Are you
planning a second "Music of Johnny Reinhard" recording?

>However, when every
>single note is a 17-perspective, might the "-ness" resulting merely be
a
>manifestaion of the music's "17-ness"?

Not sure exactly what this means, but remember that individual notes can
have a numerical relationship only with other notes, and not standing on
their own. Certainly if a 1/1 is played as a drone, and other notes form
ratios of 17 with the drone, and the remaining notes are typically used
simultaneously with notes they form ratios of 17 with, then 17 will
define the harmonic norm of the piece and it can be said to project
17-ness, if such a thing is discernible at all. If there is no 1/1 in
the music, the 17s in the ratios for the other notes may fail to create
any actual ratios of 17 in the music, so projecting a sense of "17-ness"
in the sense it has been discussed would be difficult. On the other
hand, there are probably many simpler intervals in this tuning which
occur between notes which are all defined with 17 in the numerator, or
all with 17 in the denominator, and these intervals can easily come to
define the harmonic norm of the piece. Recall our earlier discussion
about Mayumi's Harmonic-13 piece -- Adam Silverman and I thought we
heard just 3/2 intervals in it, but you denied that the tuning had any.
When the details of the tuning finally became available, there were
several 3/2s to be found in it.

A lattice representation is more useful that a simple listing of ratios
relative to a single tonic for determining the harmonic resources of a
tuning. However, a 13-limit tuning may require at least 5 dimensions,
and a 17-limit tuning at least 6, so in many cases a single graphical
depiction of such tunings may not be able to clarify all its
relationships. However, I suspect that in the case of Harmonic-13 and
Harmonic-17, there may be simplifying features that could make a
graphical representation feasible. Care to provide us with the details?

The Harmonic-17 tuning I'm working with for my new work _Trespass_ uses
the following notes, based on a 1/1 fundamental:

1/1 17/18 17/16 34/31 19/17 17/15 20/17 17/14 22/17
cents: 0 99 105 160 193 217 281 336 447

17/13 24/17 26/17 17/11 28/17 34/20 62/34 34/17
464 597 735 754 864 919 1040 1095

While most people think about the 99 cent semitone that results from the
17/18 semitone found in the sequence of the overtone series (used most
cleverly by Vincenzo Gallilei, father of Galileo, to fret the archlute).

I was surprised to find a 400 cent major third - between 24/17 and 28/17.
I have not noticed any 3/2's. Can you map it out the way you think best?

re: a second album of my music - I'm still waiting for
the first album _Raven_ to be released in late September on a new label
called "The Stereo Society".

Johnny Reinhard
Director
American Festival of Microtonal Music
318 East 70th Street, Suite 5FW
New York, New York 10021 USA
(212)517-3550/fax (212) 517-5495
reinhard@idt.net
http://www.echonyc.com/~jhhl/AFMM

On Thu, 6 Aug 1998, Johnny Reinhard wrote:
>The Harmonic-17 tuning I'm working with for my new work _Trespass_ uses
>the following notes, based on a 1/1 fundamental:
>
> 1/1 17/18 17/16 34/31 19/17 17/15 20/17 17/14 22/17
^^^^^
>cents: 0 99 105 160 193 217 281 336 447

That's got to be 18/17, right Johnny?

>17/13 24/17 26/17 17/11 28/17 34/20 62/34 34/17
^^^^^ ^^^^^
>464 597 735 754 864 919 1040 1095

I.e. 17/10 and 31/17.

I think the best way to diagram this is as two intersecting Otonal and
Utonal identities, with a few members missing:

31/17
28/17
26/17
24/17
22/17
20/17
19/17
17/16 18/17
17/17
17/10
17/11
17/13
17/14
17/15
34/31


[snip]
>I have not noticed any 3/2's. Can you map it out the way you think best?

Hate to break it to you, Johnny, but there's a couple of lovely 3/2s
(4/3s): 17/10 to 17/15, and 18/17 to 24/17.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

Thanks Paul. When one is composing the theoretical wing of the mind takes
a back seat to the creation muse. :)

Johnny Reinhard
Director
American Festival of Microtonal Music
318 East 70th Street, Suite 5FW
New York, New York 10021 USA
(212)517-3550/fax (212) 517-5495
reinhard@idt.net
http://www.echonyc.com/~jhhl/AFMM

On Fri, 7 Aug 1998, Paul Hahn wrote:

> On Thu, 6 Aug 1998, Johnny Reinhard wrote:
> >The Harmonic-17 tuning I'm working with for my new work _Trespass_ uses
> >the following notes, based on a 1/1 fundamental:
> >
> > 1/1 17/18 17/16 34/31 19/17 17/15 20/17 17/14 22/17
> ^^^^^
> >cents: 0 99 105 160 193 217 281 336 447
>
> That's got to be 18/17, right Johnny?
>
> >17/13 24/17 26/17 17/11 28/17 34/20 62/34 34/17
> ^^^^^ ^^^^^
> >464 597 735 754 864 919 1040 1095
>
> I.e. 17/10 and 31/17.
>
> I think the best way to diagram this is as two intersecting Otonal and
> Utonal identities, with a few members missing:
>
> 31/17
> 28/17
> 26/17
> 24/17
> 22/17
> 20/17
> 19/17
> 17/16 18/17
> 17/17
> 17/10
> 17/11
> 17/13
> 17/14
> 17/15
> 34/31
>
>
> [snip]
> >I have not noticed any 3/2's. Can you map it out the way you think best?
>
> Hate to break it to you, Johnny, but there's a couple of lovely 3/2s
> (4/3s): 17/10 to 17/15, and 18/17 to 24/17.
>
> --pH http://library.wustl.edu/~manynote
> O
> /\ "Churchill? Can he run a hundred balls?"
> -\-\-- o
> NOTE: dehyphenate node to remove spamblock. <*>
>

Johnny Reinhard wrote:

>I was surprised to find a 400 cent major third - between 24/17 and
28/17.

Huh? Between 24/17 and 28/17 is a 28/24 = 7/6 = 267 cents.

Thanks to both Pauls for studying my projected tuning intended for an ever
impending deadline for a new composition - called "Trespass."

Thanks to Didier for pointing out that the last note of the Harmonic-17
scale is 32/17 (not 34/17). And I should have 18/17 (instead of my
dyslexic 17/18).

Paul Hahn is right to point out a Perfect Fifth from 17/10, but I hadn't
included this interval in my original list, though it is certainly implied
by my 20/17. Are there other intervals that _scream_ to be included?

Johnny Reinhard
Director
American Festival of Microtonal Music
318 East 70th Street, Suite 5FW
New York, New York 10021 USA
(212)517-3550/fax (212) 517-5495
reinhard@idt.net
http://www.echonyc.com/~jhhl/AFMM

On Mon, 10 Aug 1998, Paul Hahn wrote:

> On Mon, 10 Aug 1998, Paul H. Erlich wrote:
> > Johnny Reinhard wrote:
> > >I was surprised to find a 400 cent major third - between 24/17 and
> > 28/17.
> >
> > Huh? Between 24/17 and 28/17 is a 28/24 = 7/6 = 267 cents.
>
> I think Johnny meant between the 17/13 and the 28/17.
>
> --pH http://library.wustl.edu/~manynote
> O
> /\ "Churchill? Can he run a hundred balls?"
> -\-\-- o
> NOTE: dehyphenate node to remove spamblock. <*>
>

On Mon, 10 Aug 1998, Paul H. Erlich wrote:
> Johnny Reinhard wrote:
> >I was surprised to find a 400 cent major third - between 24/17 and
> 28/17.
>
> Huh? Between 24/17 and 28/17 is a 28/24 = 7/6 = 267 cents.

I think Johnny meant between the 17/13 and the 28/17.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

Johnny Reinhard wrote,

>Paul Hahn is right to point out a Perfect Fifth from 17/10, but I
hadn't
>included this interval in my original list

You did: 34/20 = 17/10.

On Mon, 10 Aug 1998, Johnny Reinhard wrote:
> Thanks to Didier for pointing out that the last note of the Harmonic-17
> scale is 32/17 (not 34/17).

I missed that--I'd wondered why there were only 16 pitches instead of
17. Adding that to my original diagram gives this:

31/17
28/17
26/17
24/17
22/17
20/17
19/17
17/16 18/17
17/17
32/17 17/10
17/11
17/13
17/14
17/15
34/31

> Paul Hahn is right to point out a Perfect Fifth from 17/10, but I hadn't
> included this interval in my original list, though it is certainly implied
> by my 20/17.

Umm, Johnny, here's your original list:

On Thu, 6 Aug 1998, Johnny Reinhard wrote:
>The Harmonic-17 tuning I'm working with for my new work _Trespass_ uses
>the following notes, based on a 1/1 fundamental:
>
> 1/1 17/18 17/16 34/31 19/17 17/15 20/17 17/14 22/17
>cents: 0 99 105 160 193 217 281 336 447
>
>17/13 24/17 26/17 17/11 28/17 34/20 62/34 34/17
>464 597 735 754 864 919 1040 1095

34/20 reduces to 17/10, and 62/34 reduces to 31/17. I mentioned this in
my first message on this thread. (Also, for those coming in late, the
17/18 should be 18/17, and the 34/17 should be 32/17.)

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

Johnny Reinhard wrote:
> Are there other intervals that _scream_ to be included?

There are 192 different intervals in the scale. The 3/2 occurs three
times, on degree 5, 10 and 16. Not to list all of them, these are the
ones with a denominator less than 10:

1 10/9 182.404 cents minor whole tone
1 9/8 203.910 cents major whole tone
2 8/7 231.174 cents septimal whole tone
1 7/6 266.871 cents septimal minor third
1 6/5 315.641 cents minor third
1 11/9 347.408 cents undecimal neutral third
2 5/4 386.314 cents major third
1 9/7 435.084 cents septimal major third
3 4/3 498.045 cents perfect fourth
2 11/8 551.318 cents harmonic augmented fourth
2 7/5 582.512 cents septimal tritone
2 10/7 617.488 cents Euler's tritone
1 13/9 636.618 cents
3 3/2 701.955 cents perfect fifth
1 14/9 764.916 cents septimal minor sixth
2 11/7 782.492 cents
2 8/5 813.686 cents minor sixth
2 13/8 840.528 cents tridecimal neutral sixth
1 5/3 884.359 cents major sixth
1 12/7 933.129 cents septimal major sixth
2 7/4 968.826 cents harmonic seventh
1 16/9 996.090 cents Pythagorean minor seventh
1 9/5 1017.596 cents just minor seventh
1 11/6 1049.363 cents 21/4-tone, undecimal neutral seventh
2 13/7 1071.702 cents 16/3-tone
1 15/8 1088.269 cents classic major seventh
1 17/9 1101.045 cents

This is the "diamond lattice" diagram of the scale. Numerators
horizontally,
denominators vertically.

1 3 5 7 9 13 17 21 25 29 33 37 41 45 49 53
1: 0 *
3: 0
5: 0 *
7: 0 *
9: 0
11: 0 *
13: 0 *
15: 0 *
17: * * * * * * * 0 * *
19: 0
21: 0
23: 0
25: 0
27: 0
29: 0
31: * 0

Manuel Op de Coul coul@ezh.nl