Perfect fifth E.T.

Does anyone know about P.F.E.T.(Cordier's system)at all in North
America? If anyone has experienced this tuning, I would like to hear
from them.I only red the book(in french),and it
sounded(!!!)interesting.

p-leger1@excite.com wrote...

>Does anyone know about P.F.E.T.(Cordier's system)at all in North
>America? If anyone has experienced this tuning, I would like to hear
>from them. I only red the book(in french),and it sounded(!!!)interesting.

I don't know of it, but maybe it's 7th-root-of-1.5 equal tuning? That's
where there are 7 equal steps to the perfect fifth, 12 equal steps to
a slightly stretched octave. The argument is, you _can_ have 12 perfect
fifths, and on a piano, at least, you stretch the octaves anyway. My
limited experience with this tuning tells me, however, that it isn't as
good as 12-tET for 5-limit music, since its thirds are worse.

-Carl

>>Does anyone know about P.F.E.T.(Cordier's system)at all in North
>>America? If anyone has experienced this tuning, I would like to hear
>>from them. I only red the book(in french),and it sounded(!!!)interesting.

>I don't know of it, but maybe it's 7th-root-of-1.5 equal tuning?

Yes, I think that's it -- I saw a book on that once.

>That's
>where there are 7 equal steps to the perfect fifth, 12 equal steps to
>a slightly stretched octave. The argument is, you _can_ have 12 perfect
>fifths, and on a piano, at least, you stretch the octaves anyway.

And psychoacoustically, melodic octaves are stretched a bit even on
perfectly harmonic instruments.

>My
>limited experience with this tuning tells me, however, that it isn't as
>good as 12-tET for 5-limit music, since its thirds are worse.

Major thirds are worse, major sixths are worse . . . but not by a noticeable
amount.

>>My limited experience with this tuning tells me, however, that it isn't as
>>good as 12-tET for 5-limit music, since its thirds are worse.
>
>Major thirds are worse, major sixths are worse . . . but not by a noticeable
>amount.

I think differences as small as a cent can be noticeable... coming across
as a change in the timbre of the instrument. Major sixths will different
by almost three cents. And major 10ths, traditionally some of the most
consonant intervals on the piano, will be worse by almost five cents.
Remember also that these differences should be applied quadratically to
the errors already committed by 12-tET.

Assuming that 5:3 is the target of the minor intervals (which, as recent
thread discusses, it may not be), what percent accuracy does this tuning
achieve at the 5-limit, by your reckoning, Paul?

-Carl

Carl wrote,

>Assuming that 5:3 is the target of the minor intervals (which, as recent
>thread discusses, it may not be),

Uhh . . . you mean 6:5? 5:3 is a _major_ third. And _both_ major and minor
thirds are affected equally by the possible target of 16:19:24 for the minor
triad -- since the major triad and the minor triad _each_ have _both_ a
minor third and a major third -- but we'll ignore the ratios of 19 for now.

>coming across
>as a change in the timbre of the instrument. Major sixths will different
>by almost three cents. And major 10ths, traditionally some of the most
>consonant intervals on the piano, will be worse by almost five cents.
>Remember also that these differences should be applied quadratically to
>the errors already committed by 12-tET.

Good points! The following calculation reflects all this.

>what percent accuracy does this tuning
>achieve at the 5-limit, by your reckoning, Paul?

Simce the octaves are tempered, we have to use an integer, not odd, limit of
accuracy. If we use 5-integer-limit, 12.000-tET has an accuracy of 87.243%,
while PFET has an accuracy of 78.767%, and if we use 6-integer-limit, 12-tET
has an accuracy of 88.739%, while PFET has an accuracy of 82.614% (since the
6:5 comes into the picture). As usual, we are assuming a hearing mechanism
with normally distributed errrors of standard deviation of 1% of the given
frequency. By comparison, the accuracies of 19-tET are 94.816% at the
5-integer-limit and 94.85% at the 6-integer-limit, while a 19-tone version
of pure fifths temperament would yield an accuracy of 73.064% at the
5-integer-limit and 73.033% at the 6-integer-limit.

>>Assuming that 5:3 is the target of the minor intervals (which, as recent
>>thread discusses, it may not be),
>
>Uhh . . . you mean 6:5?

Yes.

>what percent accuracy does this tuning achieve at the 5-limit, by your
>reckoning, Paul?

>Since the octaves are tempered, we have to use an integer, not odd, limit
>of accuracy. If we use 5-integer-limit, 12.000-tET has an accuracy of
>87.243%, while PFET has an accuracy of 78.767%, and if we use 6-integer-
>limit, 12-tET has an accuracy of 88.739%, while PFET has an accuracy of
>82.614% (since the 6:5 comes into the picture). As usual, we are assuming
>a hearing mechanism with normally distributed errrors of standard deviation
>of 1% of the given frequency. By comparison, the accuracies of 19-tET are
>94.816% at the 5-integer-limit and 94.85% at the 6-integer-limit, while a
>19-tone version of pure fifths temperament would yield an accuracy of
>73.064% at the 5-integer-limit and 73.033% at the 6-integer-limit.

Awesome! So my ears didn't deceive me, after all. Still, it doesn't
seem like the move from 5- to 6-limit should bring up PFET so much, since
6 adds all those octaves... are some intervals getting double-weighted,
like 6:4/3:2? Also, how are you dealing with the fact that some intervals
will be balanced by their inversions, like 3:2/4:3, effectively taking them
out of the measurement (?), while others, like 5:4 (lim8:5 >6), will not?

-Carl

Carl wrote,

>Awesome! So my ears didn't deceive me, after all. Still, it doesn't
>seem like the move from 5- to 6-limit should bring up PFET so much, since
>6 adds all those octaves...

6 adds all those octaves?

>are some intervals getting double-weighted,
>like 6:4/3:2?

Yes, and octaves are triple-weighted, 2:1/4:2/6:3. I think that's fair.

>Also, how are you dealing with the fact that some intervals
>will be balanced by their inversions, like 3:2/4:3, effectively taking them
>out of the measurement (?), while others, like 5:4 (lim8:5 >6), will not?

No need to seperately deal with that -- the results reflect the fact that
the integer limit is below 8 by not including 8:5 in the calculations; and
yes, there is some (not total) cancellation between 3:2 and 4:3, naturally .
. . hope I'm understanding you correctly . . .

>>Awesome! So my ears didn't deceive me, after all. Still, it doesn't
>>seem like the move from 5- to 6-limit should bring up PFET so much, since
>>6 adds all those octaves...
>
>6 adds all those octaves?

6:1, 6:2, 6:3.

>>are some intervals getting double-weighted, like 6:4/3:2?
>
>Yes, and octaves are triple-weighted, 2:1/4:2/6:3. I think that's fair.

I would rather not, but it isn't the end of the world.

-Carl

>>6 adds all those octaves?

>6:1, 6:2, 6:3.

Only 6:3 is an octave.

>>>are some intervals getting double-weighted, like 6:4/3:2?
>
>>Yes, and octaves are triple-weighted, 2:1/4:2/6:3. I think that's fair.

>I would rather not, but it isn't the end of the world.

Do you want to use _Tenney_ limit, rather than integer limit?

>>6:1, 6:2, 6:3.
>
>Only 6:3 is an octave.

Yeah, but the others are octaves + intervals we already had, which
is what I meant. I should have watched my literal meaning.

>>>Yes, and octaves are triple-weighted, 2:1/4:2/6:3. I think that's fair.
>>
>>I would rather not, but it isn't the end of the world.
>
>Do you want to use _Tenney_ limit, rather than integer limit?

That ought to be perfect.

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
> >>6:1, 6:2, 6:3.
> >
> >Only 6:3 is an octave.
>
> Yeah, but the others are octaves + intervals we already had, which
> is what I meant. I should have watched my literal meaning.

But that's good, right? I mean, you yourself brought up the major
tenth, which is an octave plus a major third -- we need to keep these
wider intervals under consideration.
>
> >>>Yes, and octaves are triple-weighted, 2:1/4:2/6:3. I think
that's fair.
> >>
> >>I would rather not, but it isn't the end of the world.
> >
> >Do you want to use _Tenney_ limit, rather than integer limit?
>
> That ought to be perfect.

Well, a Tenney limit high enough to include 6:5 will not only include
all the above intervals, but triple and quadruple octaves as well,
and a lot of inconsistent intervals too :(

>>>6:1, 6:2, 6:3.
>>>
>>>Only 6:3 is an octave.
>>
>>Yeah, but the others are octaves + intervals we already had, which
>>is what I meant. I should have watched my literal meaning.
>
>But that's good, right? I mean, you yourself brought up the major
>tenth, which is an octave plus a major third -- we need to keep these
>wider intervals under consideration.

Absolutely. But this microthread got started over my surprise at
how much better got, going from 5- to 6-limit. You'd think those
large intervals would contribute to a worsening of the score.

>>>Do you want to use _Tenney_ limit, rather than integer limit?
>>
>>That ought to be perfect.
>
>Well, a Tenney limit high enough to include 6:5 will not only include
>all the above intervals, but triple and quadruple octaves as well,
>and a lot of inconsistent intervals too :(

Oh, yeah, inconsistent intervals. Bummer. But we did get a good idea
as to what's going on. Thanks for doing the number-crunching!

-Carl