1/10 harrison comma meantone

Hey,

I've tuned 1/10 harrison comma meantone up (FYI-tempering 10 fifths
to make a 7/4). It is absolutely gorgeous, close to 1/4 comma, but
the tritones are subtly better, the fifths are also subtly better,
and the thirds are mostly so close to pure. Scala says it's close to
'meaneb742a' and 'meaneb1071a', as well as 'mean17'or 4/17 comma
meantone, and it competes against 12 out of 31 tet and wins, as far
as I am concerned, for having better fifths and tritones (not to
mention 7:4s). So dominant seventh chords are a bit more listenable
in 1/10 harrison comma.

Theoretically, it's generator 5th is virtually the same as 205 steps
of 353-tet (which is 696.884 while the true one is 696.883)

This tuning may have been mentioned here before, but alas, Yahoo
searching sucks, so I couldn't find a reference (does anyone see my
point here about *avoiding* redundency? For all I know, there might
be a huhe thread on this topic that I *could have* researched, but
we are stuck with knowledge that we can't access--like being at a
library without a catalogue!)

Anyway, enough rant-I tune it to Bb as a root so that the
traditional 'bad triads' from meantone--C#, F#, Ab, and B are the
ones that are, well, 'bad'.

Gorgeous for renaissance, folk, etc. try it out!

Best,
Aaron.

>I've tuned 1/10 harrison comma meantone up (FYI-tempering 10 fifths
>to make a 7/4). It is absolutely gorgeous, close to 1/4 comma, but
>the tritones are subtly better, the fifths are also subtly better,
>and the thirds are mostly so close to pure. Scala says it's close to
>'meaneb742a' and 'meaneb1071a', as well as 'mean17'or 4/17 comma
>meantone, and it competes against 12 out of 31 tet and wins, as far
>as I am concerned, for having better fifths and tritones (not to
>mention 7:4s). So dominant seventh chords are a bit more listenable
>in 1/10 harrison comma.
>
>Theoretically, it's generator 5th is virtually the same as 205 steps
>of 353-tet (which is 696.884 while the true one is 696.883)

Jorgensen in _Tuning_ gives it about half of page 94, along with discussion of Robert Smith (1749). He gives Harrison's fifth as 1.49441151.

Barbour's discussion is on pp40-44 of _Tuning and Temperament: A Historical Survey_. The Dover 2004 reprint (grab it immediately!) has finally repaginated the book and therefore also fixed the page-numbering problems of the index, which had plagued the two previous printings. ISBN = 0-486-43406-0.

Brad Lehman

--- In tuning@yahoogroups.com, "akjmicro" <akjmicro@c...> wrote:
> Hey,
>
> I've tuned 1/10 harrison comma meantone up (FYI-tempering 10 fifths
> to make a 7/4). It is absolutely gorgeous, close to 1/4 comma, but
> the tritones are subtly better, the fifths are also subtly better,
> and the thirds are mostly so close to pure. Scala says it's close to
> 'meaneb742a' and 'meaneb1071a', as well as 'mean17'or 4/17 comma
> meantone, and it competes against 12 out of 31 tet and wins, as far
> as I am concerned, for having better fifths and tritones (not to
> mention 7:4s). So dominant seventh chords are a bit more listenable
> in 1/10 harrison comma.

Another way to describe this tuning is that the fifth is 56^(1/8),
which is indeed close to 4/17-comma meantone.

> Theoretically, it's generator 5th is virtually the same as 205 steps
> of 353-tet (which is 696.884 while the true one is 696.883)

That's the next convergent after 31; the semiconvergents (giving the
MOS for this tuning) give a sequence Monz might recognize from the
recent discussion of 136-et and the huygens tuning for 11-limit
meantone: 5, 7, 12, 19, 31, 43, 74, 105, 136, 167 ... . The upshot is
that this is a fine tuning for 11-limit meantone in its huygens, or
31&43, incarnation; in fact it is a poptimal generator for huygens.

> This tuning may have been mentioned here before, but alas, Yahoo
> searching sucks, so I couldn't find a reference (does anyone see my
> point here about *avoiding* redundency? For all I know, there might
> be a huhe thread on this topic that I *could have* researched, but
> we are stuck with knowledge that we can't access--like being at a
> library without a catalogue!)

I hope the above has related it at least a little. If you aren't going
to simply use 31-et for huygens/meanpop and give up trying to
distinguish them, this fifth seems like a fine choice, since it gives
us those pure 7/4s.

> Gorgeous for renaissance, folk, etc. try it out!

The most interesting would be to find a piece with a lot of German
sixths. There doesn't seem to be a lot of point in making a big fuss
over exact 7/4s when they never actually occur.

You could go through the table of 9-limit consonances and see what you
turned up in this way:

3/2: Pythagorean, 3/2 fifth, 701.955 cents

5/4: 1/4-comma, 5^(1/4) fifth, 696.578 cents

6/5: 1/3-comma, (10/3)^(1/3) fifth, 694.768 cents

7/4: ~4/17 comma, 56^(1/10) fifth, 696.883 cents

7/5: ~2/9 comma, (56/5)^(1/6) fifth, 697.085 cents

7/6: ~5/19 comma, (112/3)^(1/9) fifth, 696.319 cents

9/7: ~5/17 comma, (224/9)^(1/8) fifth, 695.614 cents

I've mention the (224/9)^(1/8) fifth a few times since it is so close
to both the Wilson fifth and 69-equal. The (112/3)^(1/9) fifth is
quite interesting; it is very close to 81-et, and is 5-limit poptimal.
The tritonic fifth of (56/5)^(1/6) might be good for someone wanting
something a shade sharper than your septimal fifth of 56^(1/10), and
since the complexity of the tritone in meantone is 6, these are more
likely to occur than the pure 7/4s.

--- In tuning@yahoogroups.com, Brad Lehman <bpl@u...> wrote:

> >Theoretically, it's generator 5th is virtually the same as 205 steps
> >of 353-tet (which is 696.884 while the true one is 696.883)
>
> Jorgensen in _Tuning_ gives it about half of page 94, along with
discussion
> of Robert Smith (1749). He gives Harrison's fifth as 1.49441151.

That's the Lucy-tuning fifth of 600+300/pi cents; Aaron waa talking
about evenly distributing the error of the Harrison comma, which is
59049/57344, and which leads to a completely different kind of
meantone fifth.

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

> That's the Lucy-tuning fifth of 600+300/pi cents; Aaron waa talking
> about evenly distributing the error of the Harrison comma, which is
> 59049/57344, and which leads to a completely different kind of
> meantone fifth.

Not to mention it's a completely different kind of Harrison, come to
think of it.

--- In tuning@yahoogroups.com, Brad Lehman <bpl@u...> wrote:

/tuning/topicId_55450.html#55454

>
> Barbour's discussion is on pp40-44 of _Tuning and Temperament: A
Historical
> Survey_. The Dover 2004 reprint (grab it immediately!)

***Wow... that's news to *me*... great! (Of course, I paid quite a
bit of money for an old used copy... :)

J. Pehrson

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

/tuning/topicId_55450.html#55468

> --- In tuning@yahoogroups.com, Brad Lehman <bpl@u...> wrote:
>
> /tuning/topicId_55450.html#55454
>
> >
> > Barbour's discussion is on pp40-44 of _Tuning and Temperament: A
> Historical
> > Survey_. The Dover 2004 reprint (grab it immediately!)
>
> ***Wow... that's news to *me*... great! (Of course, I paid quite a
> bit of money for an old used copy... :)
>
> J. Pehrson

***Gheez... and this copy is only $13.95...

http://store.yahoo.com/doverpublications/0486434060.html

I think I paid well over $50 for my "out of print" find... :(

J. Pehrson