Ives stretched-8ve scales (was: Another BP linear temperament?)

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, August 01, 2001 7:55 PM
> Subject: [tuning-math] Re: Another BP linear temperament?
>
>
> Since the "orthodox" BP consonances are ratios of {3,5,7},
> and the triple-BP consonances are ratios of {3,5,7,11,13}, and
> their tritave-equivalents, and these have been referred to as
> "7-limit" and "13-limit", respectively, I think this could
> lead to confusion . . . let's abandon the term "limit" when
> leaving the octave-equivalent world and adopt more explicit
> mathematical terminology.

Hi Paul,

This caught my attention and reminded me of the webpage I made
examining Charles Ives's stretched-8ve scales.
http://www.ixpres.com/interval/monzo/ives/48tet.htm

On that webpage, I propose 5 different tunings for the three
types described by Ives, with equivalence-intervals of 15.00,
16.00, 15.50, 14.50, and ~14.94 Semitones.

Can you guys (Paul, Graham, Dave) explain some more detailed
mathematical analyses of these tunings?

love / peace / harmony ...

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

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> On that webpage, I propose 5 different tunings for the three
> types described by Ives, with equivalence-intervals of 15.00,
> 16.00, 15.50, 14.50, and ~14.94 Semitones.
>
> Can you guys (Paul, Graham, Dave) explain some more detailed
> mathematical analyses of these tunings?

Sorry Monz, I've got other things I should be doing at the moment.

BTW, do you suppose you are the only person on the planet that
routinely specifies intervals in semitones with two decimal places,
rather than cents? :-)

-- Dave Keenan

> From: Dave Keenan <D.KEENAN@UQ.NET.AU>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, August 02, 2001 8:10 PM
> Subject: [tuning-math] Re: Ives stretched-8ve scales (was: Another BP
linear temperament?)
>
>
> BTW, do you suppose you are the only person on the planet that
> routinely specifies intervals in semitones with two decimal places,
> rather than cents? :-)

Yep. It sure is taking a long time for everyone else to catch on! ;-P

I mean, really... if +/- 5 cents is pretty much an accepted margin
of error in practice, then do we really need anything more accurate
than 240-EDO?, let alone 1200-EDO (which is exactly what my "Semitones"
specify).

And if we *do* need more accuracy, why not use another widely accepted
standard with finer resolution, such as "cawapus", or even "midipus",
rather than having to deal with more decimal points after the cents?

I really like Semitones (with two decimal places) because isolating
the semitone component makes it very easy for me to see at a glance
how a pitch or interval relates to what I'm already familiar with
as a long-time practicing 12-EDO musician.

love / peace / harmony ...

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

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

> I mean, really... if +/- 5 cents is pretty much an accepted margin
> of error in practice, then do we really need anything more accurate
> than 240-EDO?, let alone 1200-EDO (which is exactly what
my "Semitones"
> specify).
>
> And if we *do* need more accuracy, why not use another widely
accepted
> standard with finer resolution, such as "cawapus", or
even "midipus",
> rather than having to deal with more decimal points after the cents?

Alas, I don't know what these are. For my own purposes, I often use a
612 system, which gives a sort of schisma as the basic unit. The
advantage over cents is that some pretty good approximations are
available where only integer values need to be remembered, for we have

2 = 612 s
3 = 969.997 s ~ 970 s
5 = 1421.02 s ~ 1421 s
7 = 1718.10 s ~ 1718 s
11 = 2264.67 s ~ 2265 s

So I end up remembering stuff like 9/8 = 104 s, 10/9 = 93 s (and of
course the equal tempered tone is exactly 102 s) and so forth. The
major fifth is 358 s, and the equally tempered fifth is exactly 357
s, flat by one schisma (= s.) The major third is 197 s, whereas the
equal tempered third is 204 s, sharp by 7 s.