'Pure fifths' temperament, stretch etc.

I came across the 'pure fifths' on a piano-tech mailing list
attributed to Jim Coleman. Instead of dividing up the octave into 12
equal parts, you divide a pure fifth into 7 equal semitones then
extend this to obtain octaves stretched by about 3.4 cents (or 1/7
Pythagorean comma). Major thirds are thus stretched by about 1 cent
more than 12-ET.

Thus, not EDO but 'EDFi'.

Is there a general classification for this type of temperament where
the Pythagorean comma is borne not by the fifths alone but partly or
wholly by the octaves?

One might make the fifths and octaves equally out of tune: this would
imply 1/19 Pythagorean-comma tempered fifths (1.2c) since 12 of them
would absorb 12/19 comma leaving 1/19 comma each for 7 octaves. The
twelfth is pure: 'EDTw'.

I thought of that possibility pretty quickly but apparently someone
has beat me to it:

http://www.piano-stopper.de/html/homepe.html

The guy apparently says one can tune a piano using only pure
intervals. But the method is patented and he asks a three-figure sum
for it... Maybe you need to be able to play a twelfth and manipulate
the tuning hammer at the same time.

He has also some claim about the beating properties of the tuning,
presumably that would be fairly easy to work out as it is a regular
one modulo impure octaves.

I don't know if the 'pure fifths' proposal would be of musical
interest besides making pianos sound 'better' for some tastes (for
music with a lot of open fifths??), but it might be of mathematical
interest.

With the stretched partials on an actual piano the effect of 3.4 cent
wide octaves is not as bad as you might think. Then again the fifths
won't sound 100% pure due to the same effect. I read one comment to
the effect that in any case the beating of fifths is almost too slow
to hear for supposely 12-ET on a piano, since the tuning is stretched
in any case. Probably many tuners get fairly close to the 'Stopper
Tuning' without intending it.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
>
> I came across the 'pure fifths' on a piano-tech mailing list
> attributed to Jim Coleman. Instead of dividing up the octave into 12
> equal parts, you divide a pure fifth into 7 equal semitones then
> extend this to obtain octaves stretched by about 3.4 cents (or 1/7
> Pythagorean comma). Major thirds are thus stretched by about 1 cent
> more than 12-ET.
>
> Thus, not EDO but 'EDFi'.
>
> Is there a general classification for this type of temperament where
> the Pythagorean comma is borne not by the fifths alone but partly or
> wholly by the octaves?

We've been talking about these things quite a bit (especially on the
tuning-math list), for example under the rubric of 'TOP' tuning. The
TOP tuning of 12-equal has octaves stretched by 0.62 cents, as prime
2 and prime 3 are both helping to make the Pythagorean comma vanish.

> One might make the fifths and octaves equally out of tune: this
would
> imply 1/19 Pythagorean-comma tempered fifths (1.2c) since 12 of them
> would absorb 12/19 comma leaving 1/19 comma each for 7 octaves. The
> twelfth is pure: 'EDTw'.
>
> I thought of that possibility pretty quickly but apparently someone
> has beat me to it:
>
> http://www.piano-stopper.de/html/homepe.html
>
> The guy apparently says one can tune a piano using only pure
> intervals. But the method is patented and he asks a three-figure sum
> for it... Maybe you need to be able to play a twelfth and manipulate
> the tuning hammer at the same time.
>
> He has also some claim about the beating properties of the tuning,
> presumably that would be fairly easy to work out as it is a regular
> one modulo impure octaves.
>
> I don't know if the 'pure fifths' proposal would be of musical
> interest besides making pianos sound 'better' for some tastes (for
> music with a lot of open fifths??), but it might be of mathematical
> interest.
>
> With the stretched partials on an actual piano the effect of 3.4
cent
> wide octaves is not as bad as you might think. Then again the fifths
> won't sound 100% pure due to the same effect. I read one comment to
> the effect that in any case the beating of fifths is almost too slow
> to hear for supposely 12-ET on a piano, since the tuning is
stretched
> in any case. Probably many tuners get fairly close to the 'Stopper
> Tuning' without intending it.
>
> ~~~T~~~
>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> >
> >
> > I came across the 'pure fifths' on a piano-tech mailing list
> > attributed to Jim Coleman. Instead of dividing up the octave into
12
> > equal parts, you divide a pure fifth into 7 equal semitones then
> > extend this to obtain octaves stretched by about 3.4 cents (or 1/7
> > Pythagorean comma). Major thirds are thus stretched by about 1
cent
> > more than 12-ET.
> >
> > Thus, not EDO but 'EDFi'.
> >
> > Is there a general classification for this type of temperament
where
> > the Pythagorean comma is borne not by the fifths alone but partly
or
> > wholly by the octaves?
>
> We've been talking about these things quite a bit (especially on
the
> tuning-math list), for example under the rubric of 'TOP' tuning.
The
> TOP tuning of 12-equal has octaves stretched by 0.62 cents, as
prime
> 2 and prime 3 are both helping to make the Pythagorean comma vanish.

To clarify, this is the 3-limit TOP tuning of 12-equal. In higher
prime limits, *compressed* TOP tunings for 12-equal are found, and
actually sound better than you would think.