Yahoo Tuning Groups Ultimate Backup tuning Re: Pythagorean 3-speed bikes -- Monz's welcome emendation

Hello, there, Monz, and thank you very much both for a most due
Aristoxenian "friendly amendment" to my post on 3-speed Pythagorean
bikes, and a fascinating discussion on "least intervals."

First, I would very much like to emphasize the importance of your
point about choosing an interval varying somewhat from 7:6 _by ear_
for your composition _3+4_, and then, having settled on something
around 279 cents, selecting 75:64 as a convenient ratio.

Unfortunately, although I had read your account, my language neglects
this very characteristic Aristoxenian touch -- finding what sounds
right, and then integrating it into the intonational structure. My
emphasis was on "How neat that this fits a 5-limit RI model!" as if
the model were the basis for the choice, rather than the ratio a
convenient realization of an _aural_ judgment.

In contrast, my tricomma tuning was a theoretical construct designed
to solve a musical problem on a mathematical basis: "Come up with a
24-note Pythagorean tuning with both a 7-flavor and a 17-flavor."

Your experimental, directly interactive, "intracompositional" process
of trying 7:6, adjusting by ear to 279 cents, and then fine-tuning to
a 75:64 as a ratio at once satisfying to the ear and conceptually
elegant, is something left out of my narrative -- which could easily
be read to suggest that you set out in advance to emulate a 7-based
ratio with a complex 5-limit approximation.

In any future reference to the 75:64 saga, I'll be careful to tell the
full story, with much emphasis on the very audible process of
"fine-tuning" in the best Aristoxenian manner.

As for small intervals, or more specifically those with not-so-small
integer ratios, the monzisma at ~0.29 cents is larger than the kalisma,
the difference between 32:25 and the 12544:9801 interval equal to a
Pythagorean major third at 81:64 plus two 896:891 commas (the
difference between this third and 14:11). The kalisma has a ratio of
9801:9800, or ~0.1766 cents.

By the way, I wonder what might have happened had you chosen a
different rational expression for your "right-sounding" 279 cents, for
example 27:23 (~277.59 cents)? That ratio occurred to me by a certain
free association as the fifth complement of the 23:18 which I enjoy
quoting as an RI approximation for the regular major third of 17-tET
-- that is, the size that the minor third _would_ have in 17-tET if
the fifths were a pure 3:2.

The jots are interesting -- according to Scala, a kalisma at 9801:9800
is about 4.431350 jots; although the monzisma apparently involves a
rational ratio too large for my version of Scala for MS-DOS to report
in integer form, it returns a size of about 7.334990 jots.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

/tuning/topicId_22236.html#22236

> First, I would very much like to emphasize the importance
> of your point about choosing an interval varying somewhat
> from 7:6 _by ear_ for your composition _3+4_, and then,
> having settled on something around 279 cents, selecting
> 75:64 as a convenient ratio.
>
> Unfortunately, although I had read your account, my language
> neglects this very characteristic Aristoxenian touch --
> finding what sounds right, and then integrating it into
> the intonational structure. My emphasis was on "How neat
> that this fits a 5-limit RI model!" as if the model were
> the basis for the choice, rather than the ratio a convenient
> realization of an _aural_ judgment.

Hmmm... well, now that you mention it, I suppose I *have*
to admit that, with my experienced background in RI, I'm sure
that I *did* have in the back of my mind the idea that 75:64
was another possibility for the precise tuning of the pitch
I was seeking.

I wasn't *consciously* thinking that way, but I certainly
know that 75/64 is only a kleisma (~7 cents) higher than
7/6, and after I tried 19/16 as my second choice and decided
that it was too high, I probably drew my MIDI pitch-bend
into Cakewalk at the approximate level of 75/64 simply because
it would be the next logical choice in ratios that I would
think of. (I did, however, try about a dozen various
pitch-bend levels until settling on ~279 cents by ear.)

But in any case, 75/64 was much more an emulation of 19/16
(a whopping ~22.93 cents higher) than it was of 7/6 (only
~7.716 cents lower), which was what I found so surprising.

Your emphasis on my "Aristoxenian" aural experiments is
fundamental: 7/6 just didn't sound right, even tho it
was closer to the pitch I finally chose. 19/16, on the
other hand, could have been pressed into service and it
wouldn't have bothered my ears nearly as much.

I've still not been able to explain why this worked as it did...

> The jots are interesting -- according to Scala, a kalisma at
> 9801:9800 is about 4.431350 jots; although the monzisma
> apparently involves a rational ratio too large for my version
> of Scala for MS-DOS to report in integer form, it returns a
> size of about 7.334990 jots.

I used Microsoft Excel and got 7.334990057 jots for the monzisma.
Not quite sure what the limits of rounding error are in Excel,
but I'd say that that's a close match to Scala's value.

Note that Sauveur's "heptamerides" of 2^(1/301) are related
to the "jots", being simply a less accurate rounding. Saveur
chose this measurement partly because he also used "merides"
of 2^(1/43), and 7 * 43 = 301, so both of his units divided
evenly.

The "savart" was originally identical to this, but was later
"rationalized" to be 2^(1/300). Since the savart used to be
commonly used to indicate interval sizes in Indian music-theory,
this is yet one more complication to the endeavor to discover
exactly what srutis really mean in terms of intonation.

(Many thanks to John Chalmers for clarifying the history of
savarts.)

Another small interval that I find really interesting is
W.S.B. Woolhouse's 2^(1/730), or a division of 730-EDO,
which he used as his division of the virtual pitch continuum,
his analogy of "cents".

Woolhouse never gave this interval a name, so I can only
refer to it by its formula (unless I coin a term for it...
...a woolhousma?). 2^(1/730) is ~1.643835616 cents
or ~41.2369863 jots.

He chose 730-EDO because it gave amazingly close integer
approximations to all of the important JI intervals he was
analyzing and using in his calculations. More info is at:
http://www.ixpres.com/interval/monzo/woolhouse/essay.htm

Now, how 'bout it... since it's a measurement that's
actually in regular use by many of us here, let's rename
"MIDI pitch-bend unit", 2^(1/49152), to something more
user-friendly... a midipu?...

-monz
http://www.monz.org
"All roads lead to n^0"

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_22236.html#22239

> Now, how 'bout it... since it's a measurement that's
> actually in regular use by many of us here, let's rename
> "MIDI pitch-bend unit", 2^(1/49152), to something more
> user-friendly... a midipu?...

I took it upon myself to make "midipu" the official name
for this interval, and put it and all the other small intervals
Margo and I have been discussing into the Tuning Dictionary:

http://www.ixpres.com/interval/dict/midipu.htm
http://www.ixpres.com/interval/dict/jot.htm
http://www.ixpres.com/interval/dict/meride.htm
http://www.ixpres.com/interval/dict/heptameride.htm
http://www.ixpres.com/interval/dict/savart.htm

Many readers should find "Savart" especially useful, because
it has been used frequently by Danielou and others in describing
the Indian sruti system. Unfortunately, it has two slightly
different definitions...

-monz
http://www.monz.org
"All roads lead to n^0"

Re: Pythagorean 3-speed bikes -- Monz's welcome emendation

🔗mschulter <MSCHULTER@VALUE.NET>

🔗monz <joemonz@yahoo.com>

🔗monz <joemonz@yahoo.com>

🔗monz <joemonz@yahoo.com>