Yahoo Tuning Groups Ultimate Backup makemicromusic 27.35 ET (nearly 44-CET) web page

I just added a page to my site devoted to this non-octave equal
temperament, which I found ca. 1999 by searching for the smallest ET
which would provide a good & distinct equivalent to 7/6, 6/5, 11/9,
5/4, and 9/7 (the 5 qualities of 3rds, some of us would agree)
without regard for the octave. I think Gary Morrison had explored
this one before me though.

I wrote "A Voice From Iraq" in this system, and a link to an mp3
excerpt from it is on the page.

The page includes a link for each scale step to a mp3 file
containing that step against middle C in sine waves (done in CSound
to .wav & then converted)

It's at
http://www.geocities.com/harold_fortuin/27Point35.html

Whomever else has written something in or about this temperament on
the web should let me know so I can link it in.

🔗Gene Ward Smith <gwsmith@...>

--- In MakeMicroMusic@yahoogroups.com, "harold_fortuin" <harold@m...>
wrote:

> I just added a page to my site devoted to this non-octave equal
> temperament, which I found ca. 1999 by searching for the smallest ET
> which would provide a good & distinct equivalent to 7/6, 6/5, 11/9,
> 5/4, and 9/7 (the 5 qualities of 3rds, some of us would agree)
> without regard for the octave. I think Gary Morrison had explored
> this one before me though.

I've never been convinced by 13/11, but I am fond of 14/11; it makes
especially good sense in the sixth version as 11/7. 19/16 is the
octave reduction of 19, and comes with 19/2, 19/4 and 19/8; 16/13 is
associated to 13/8, and hence 13/4, 13/2 and 13. Of course this
matters a lot less if you are ignoring octaves, but ignoring octaves
is ignoring the elephant in the livingroom. From that point of view,
the obvious way to accomplish the objective of distinguishing these
thirds with good values and not too much complexity is 31-equal, which
represents 7/6, 6/5, 11/9, 5/4 and 9/7 by 7, 8, 9, 10, and 11
respectively. Another meantone et is 55, which gives them to 12, 14,
16, 18, and 20; twice your values.

🔗Gene Ward Smith <gwsmith@...>

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:

From that point of view,
> the obvious way to accomplish the objective of distinguishing these
> thirds with good values and not too much complexity is 31-equal, which
> represents 7/6, 6/5, 11/9, 5/4 and 9/7 by 7, 8, 9, 10, and 11
> respectively. Another meantone et is 55, which gives them to 12, 14,
> 16, 18, and 20; twice your values.

This is getting a trifle theoretical; my apologizes, but the thread
originated here.

If we take the group generated by {7/6, 6/5, 11/9, 5/4, 9/7} we find
it can be generated by (the Tenney-Minkowski basis) {3/2, 5/4, 7/4,
11/4}, which is sort of neat, since it is a chord. Note this is four
generators, not five. A Fortuin division of the major fifth might be
defined as a division (possibly approximate) of the fifth into n
parts, with an eye to good values for 5/4, 7/4 and 11/4. If we check
for good Fortuin divisions, we find the division of the fifth into 16
parts (corresponding to an octave of 16/log2(3/2) = 27.35.) However,
right after it is the division into 18 parts, corresponding to 31-et;
note that we get this even if we ignore octaves and only want good
approximations to the Fortuin note group; moreover in terms of
relative error it is the champ. A similar situation arises with
dividing the fifth into 24 parts, which leads to 41-et whether we want
octaves or not. Also lurking out there in Fortuinland are 68, 72 and
even 152 equal.

🔗harold_fortuin <harold@...>

Gene,

Your obviously ahead of me in the theoretical aspects, but while I
know & love 31-ET, I was looking for fewer tones/octave than it, and
I wasn't approaching it from roots of 3, 5, etc. since I did not
need an exact match to one such interval.

This tuning also gives you errors on the 3rds that are quite similar
in size.

And the octave being off can be handled in some ways pretty well on
electronic devices, even with live performance--a change of patch or
pitch bend could give you the next higher octave of 27.35 ET in tune
with the lower via the appropriate shift--say down 29 cents from the
28th & subsequent tones.

In practice, the 5-3 (root position) triads in this temperament are
of course very clean, but the 6-3 (first inversion) ones are pretty
bad because the 6ths are not very good--a strange phenomena for us,
since we're used to more 'equivalent' 6ths that we have in clean
octave temperaments.

This issue with 6ths led me once to search for the best temperament
with fewer than 31 divisions that had the best possible 6ths, but
the results were less impressive--but we could formulate that
related problem as:

Find the temperament (ignoring the p5th & p octave) which has the
best match for
7/6, 6/5, 11/9, 5/4, 9/7,
14/9,8/5,18/11,5/3,12/7
and has less than 31 tones per octave.

Always in the service of making microtonal music from it all,
Harold

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...> wrote:
>
> From that point of view,
> > the obvious way to accomplish the objective of distinguishing
these
> > thirds with good values and not too much complexity is 31-equal,
which
> > represents 7/6, 6/5, 11/9, 5/4 and 9/7 by 7, 8, 9, 10, and 11
> > respectively. Another meantone et is 55, which gives them to 12,
14,
> > 16, 18, and 20; twice your values.
>
> This is getting a trifle theoretical; my apologizes, but the thread
> originated here.
>
> If we take the group generated by {7/6, 6/5, 11/9, 5/4, 9/7} we
find
> it can be generated by (the Tenney-Minkowski basis) {3/2, 5/4, 7/4,
> 11/4}, which is sort of neat, since it is a chord. Note this is
four
> generators, not five. A Fortuin division of the major fifth might
be
> defined as a division (possibly approximate) of the fifth into n
> parts, with an eye to good values for 5/4, 7/4 and 11/4. If we
check
> for good Fortuin divisions, we find the division of the fifth into
16
> parts (corresponding to an octave of 16/log2(3/2) = 27.35.)
However,
> right after it is the division into 18 parts, corresponding to 31-
et;
> note that we get this even if we ignore octaves and only want good
> approximations to the Fortuin note group; moreover in terms of
> relative error it is the champ. A similar situation arises with
> dividing the fifth into 24 parts, which leads to 41-et whether we
want
> octaves or not. Also lurking out there in Fortuinland are 68, 72
and
> even 152 equal.

🔗harold_fortuin <harold@...>

Whoops, I just realized that my "related problem" would require a
solution with a perfect octave, so,

accepting that some 3rds may be bad, and the P8 & P5 may be bad,
let's reformulate as

Find the temperament which has the
best match for
14/9,8/5,18/11,5/3,12/7

(which again are all 6ths)

--- In MakeMicroMusic@yahoogroups.com, "harold_fortuin"
<harold@m...> wrote:
>
> Gene,
>
> Your obviously ahead of me in the theoretical aspects, but while I
> know & love 31-ET, I was looking for fewer tones/octave than it,
and
> I wasn't approaching it from roots of 3, 5, etc. since I did not
> need an exact match to one such interval.
>
> This tuning also gives you errors on the 3rds that are quite
similar
> in size.
>
> And the octave being off can be handled in some ways pretty well
on
> electronic devices, even with live performance--a change of patch
or
> pitch bend could give you the next higher octave of 27.35 ET in
tune
> with the lower via the appropriate shift--say down 29 cents from
the
> 28th & subsequent tones.
>
> In practice, the 5-3 (root position) triads in this temperament
are
> of course very clean, but the 6-3 (first inversion) ones are
pretty
> bad because the 6ths are not very good--a strange phenomena for
us,
> since we're used to more 'equivalent' 6ths that we have in clean
> octave temperaments.
>
> This issue with 6ths led me once to search for the best
temperament
> with fewer than 31 divisions that had the best possible 6ths, but
> the results were less impressive--but we could formulate that
> related problem as:
>
> Find the temperament (ignoring the p5th & p octave) which has the
> best match for
> 7/6, 6/5, 11/9, 5/4, 9/7,
> 14/9,8/5,18/11,5/3,12/7
> and has less than 31 tones per octave.
>
> Always in the service of making microtonal music from it all,
> Harold
>
> --- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...> wrote:
> >
> > --- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
> > <gwsmith@s...> wrote:
> >
> > From that point of view,
> > > the obvious way to accomplish the objective of distinguishing
> these
> > > thirds with good values and not too much complexity is 31-
equal,
> which
> > > represents 7/6, 6/5, 11/9, 5/4 and 9/7 by 7, 8, 9, 10, and 11
> > > respectively. Another meantone et is 55, which gives them to
12,
> 14,
> > > 16, 18, and 20; twice your values.
> >
> > This is getting a trifle theoretical; my apologizes, but the
thread
> > originated here.
> >
> > If we take the group generated by {7/6, 6/5, 11/9, 5/4, 9/7} we
> find
> > it can be generated by (the Tenney-Minkowski basis) {3/2, 5/4,
7/4,
> > 11/4}, which is sort of neat, since it is a chord. Note this is
> four
> > generators, not five. A Fortuin division of the major fifth
might
> be
> > defined as a division (possibly approximate) of the fifth into n
> > parts, with an eye to good values for 5/4, 7/4 and 11/4. If we
> check
> > for good Fortuin divisions, we find the division of the fifth
into
> 16
> > parts (corresponding to an octave of 16/log2(3/2) = 27.35.)
> However,
> > right after it is the division into 18 parts, corresponding to
31-
> et;
> > note that we get this even if we ignore octaves and only want
good
> > approximations to the Fortuin note group; moreover in terms of
> > relative error it is the champ. A similar situation arises with
> > dividing the fifth into 24 parts, which leads to 41-et whether
we
> want
> > octaves or not. Also lurking out there in Fortuinland are 68, 72
> and
> > even 152 equal.

🔗Gene Ward Smith <gwsmith@...>

--- In MakeMicroMusic@yahoogroups.com, "harold_fortuin" <harold@m...>
wrote:

> This issue with 6ths led me once to search for the best temperament
> with fewer than 31 divisions that had the best possible 6ths, but
> the results were less impressive--but we could formulate that
> related problem as:
>
> Find the temperament (ignoring the p5th & p octave) which has the
> best match for
> 7/6, 6/5, 11/9, 5/4, 9/7,
> 14/9,8/5,18/11,5/3,12/7
> and has less than 31 tones per octave.

That's actually quite a different problem than before, since those
generate the entire 11-limit, and in paricular you have octaves. The
22 division would seem to be a good choice. For your original problem,
you've found the best solution--dividing the fifth into 16 parts.
However, dividing the fifth into 18 parts is not a lot more parts.

🔗Gene Ward Smith <gwsmith@...>

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "harold_fortuin" <harold@m...>
> wrote:
>
> > Find the temperament which has the
> > best match for
> > 14/9,8/5,18/11,5/3,12/7
> >
> > (which again are all 6ths)
>
> That gives a nonoctave note group generated by {18/11, 5/3, 12/7,
> 8/3}. Dividing the 5/3 into 22 parts (or a little less), giving an
> octave of 29.85, would seem to be right up your line.

If you add 11/7 to the mix, which I think makes a lot of sense, you
still get a sugroup--this time generated by {4,6,10,14,22}. The 10.5
division is an interesting possibility here, but of course any good
11-limit system will work.

🔗Aaron K. Johnson <akjmicro@...>

This is a good thread, however I think it best to take it to the 'tuning'
list, no? Unless you want to show us some music in this tuning... ;)

Gene, where ya been?

-Aaron

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

On Thursday 11 November 2004 01:17 pm, Gene Ward Smith wrote:
> --- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
>
> <gwsmith@s...> wrote:
> > --- In MakeMicroMusic@yahoogroups.com, "harold_fortuin" <harold@m...>
> >
> > wrote:
> > > Find the temperament which has the
> > > best match for
> > > 14/9,8/5,18/11,5/3,12/7
> > >
> > > (which again are all 6ths)
> >
> > That gives a nonoctave note group generated by {18/11, 5/3, 12/7,
> > 8/3}. Dividing the 5/3 into 22 parts (or a little less), giving an
> > octave of 29.85, would seem to be right up your line.
>
> If you add 11/7 to the mix, which I think makes a lot of sense, you
> still get a sugroup--this time generated by {4,6,10,14,22}. The 10.5
> division is an interesting possibility here, but of course any good
> 11-limit system will work.
>
>
>
>
>
>
>
> Yahoo! Groups Links
>
>
>

🔗Gene Ward Smith <gwsmith@...>

🔗harold_fortuin <harold@...>

The music in 27.35-ET at least is at
www.geocities.com/harold_fortuin/ExptVoiceIraq.mp3

which is linked from
http://www.geocities.com/harold_fortuin/27Point35.html

as well as from my Compositions page:
http://www.geocities.com/harold_fortuin/HFCompos.html
-------------

I will swap complete pieces on CDs with people on this list whose
music interests me.

And I will sell other audio/video recordings to others.

If you'd like to do either, e-mail me privately.

BTW, anyone else wanna help me put "Endangered Species" on your
local public access TV station?

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "Aaron K. Johnson"
> <akjmicro@c...> wrote:
> >
> > This is a good thread, however I think it best to take it to the
> 'tuning'
> > list, no? Unless you want to show us some music in this
tuning... ;)
>
> I need to post where Harold will read it, however.
>
> > Gene, where ya been?
>
> Should have something done soon. A lot of irons in the fire but I
> haven't been finishing up.

27.35 ET (nearly 44-CET) web page

🔗harold_fortuin <harold@...>

🔗Gene Ward Smith <gwsmith@...>

🔗Gene Ward Smith <gwsmith@...>

🔗harold_fortuin <harold@...>

🔗harold_fortuin <harold@...>

🔗Gene Ward Smith <gwsmith@...>

🔗Gene Ward Smith <gwsmith@...>

🔗Gene Ward Smith <gwsmith@...>

🔗Aaron K. Johnson <akjmicro@...>

🔗Gene Ward Smith <gwsmith@...>

🔗harold_fortuin <harold@...>

🔗Joseph Pehrson <jpehrson@...>