Extended pythagorean

Hi,

Slow as I am to realize things, I finally realized that pythagorean tuning is a fine tuning system. So I wrote a small literate programming file that demonstrateshow pythagorean tuning can be used to approximate 7- and
11-limit intervals, to perhaps enlighten others.

http://magnus.smartelectronix.com/temp/pythagorean-ji.txt

I'm actually hesitating whether I should post this on tuning-math (for fear of beside-the-point math attack replies) or tuning (for fear of being too technical), but I'll try tuning-math first. I am aware that this is a schismic tuning.

- Magnus

--- In tuning-math@yahoogroups.com, Magnus Jonsson <magnus@...> wrote:

> I'm actually hesitating whether I should post this on tuning-math (for
> fear of beside-the-point math attack replies) or tuning (for fear of
being
> too technical), but I'll try tuning-math first. I am aware that this
is a
> schismic tuning.

It's schismic or schismatic, and in the 7-limit "garibaldi". The
11-limit version I have listed as "garybald", an undignified name that
can hardly be considered official, but which at least suggests a
relationship to garibaldi. A more succinct name for it is 29&49, and
you could also call it 12&29 or 12&41. I'm tempted to call it
"magnus", and would like to know if you've actually written anything
in it.

It tempers out the commas {100/99, 225/224, 245/243}. As you note, you
can use a Pythagorean tuning, but if you are serious about the higher
limits, tuning fifths 1/2 to 2/3 cents sharp can help. 41 is an
excellent tuning for it, 193 a theoretical optimum of a kind.

On Wed, 14 Jun 2006, Gene Ward Smith wrote:

> --- In tuning-math@yahoogroups.com, Magnus Jonsson <magnus@...> wrote:
>
>> I'm actually hesitating whether I should post this on tuning-math (for
>> fear of beside-the-point math attack replies) or tuning (for fear of
> being
>> too technical), but I'll try tuning-math first. I am aware that this
> is a
>> schismic tuning.
>
> It's schismic or schismatic, and in the 7-limit "garibaldi". The
> 11-limit version I have listed as "garybald", an undignified name that
> can hardly be considered official, but which at least suggests a
> relationship to garibaldi. A more succinct name for it is 29&49, and
> you could also call it 12&29 or 12&41. I'm tempted to call it
> "magnus", and would like to know if you've actually written anything
> in it.

I haven't, so I don't think I qualify. I actually don't use traditional staff notation when I compose. I use a notation similar to chinese cipher and I'm getting fond of rational notation too, for when I need to se visually what's going on.

http://www.redshift.com/~dcanright/notatn/index.htm

> It tempers out the commas {100/99, 225/224, 245/243}. As you note, you

Okay, let me make sense out of these commas so that I understand them.
100/99, the difference between 10/9 and 11/10.
225/224, the difference between 45/32 and 7/5, or 15/16*15/16 and 7/8.
245/243, the difference between a pythagorean semitone and 5*7*7.

> can use a Pythagorean tuning, but if you are serious about the higher
> limits, tuning fifths 1/2 to 2/3 cents sharp can help. 41 is an

I care much more about lower limits than higher ones - I'm aiming for choir and I think it's hard enough to get a choir to differentiate between pythagorean and just thirds, and 16/9 vs 9/5 vs 7/4. It is my experience that sopranos tend to sing everything pythagorean whether it blends in or not.

> excellent tuning for it, 193 a theoretical optimum of a kind.

Aha, but I don't have an urge to close the chain. I like to know that it extends infinitely :).

- Magnus

--- In tuning-math@yahoogroups.com, Magnus Jonsson <magnus@...> wrote:

> Okay, let me make sense out of these commas so that I understand them.
> 100/99, the difference between 10/9 and 11/10.

> 225/224, the difference between 45/32 and 7/5, or 15/16*15/16 and 7/8.

More informatively, the difference between 15/14 and 16/15, and hence
the difference between 14/9 and 5/4 * 5/4. The augmented triad is
a 5/4 * 5/4 * 9/7 ~ 2 chord. Also, as you note, 15/16 * 15/16 ~ 7/8,
which means also 15/16 * 15/16 ~ 7/8, which tells you the 7th partial
is obtained by taking two of these compromise semitones down, and then
three octaves up.

> 245/243, the difference between a pythagorean semitone and 5*7*7.

Or (7/6)^2 ~ (6/5)(9/8).