Yahoo Tuning Groups Ultimate Backup tuning-math 270 equal as the universal temperament

Over on metatuning Graham had this to say:

"It doesn't matter if anybody can tell the difference, because there's
no evidence that four digit frequency ratios have any audible meaning.
The 13-limit is borderline..."

If "four digit frequency ratios" (which from context I take to mean
superparticular ones) have no audible meaning, it seems like a nifty
idea to temper them out. In the 7-limit this gives ennealimmal, and in
the 11-limit hemiennealimmal. If we take Graham's view, which I think
has something to be said for it, we go up to the 13-limit but no
farther. In the 13-limit, there are twelve four digit superparticular
commas; the kernel of all of these taken together is 270-equal. This
sort of fact I've discussed before; it does seem there is some
justification for considering 270-et to be a sort of universal
replacement for just intonation.

Incidentally, we have to stop here; if we try to add 729/728, the next
smallest comma, we find this is a step of 270-et.

🔗Paul Erlich <perlich@aya.yale.edu>

I don't infer that meaning at all. Instead, I think Graham was
talking about ratios that have meaning as harmonic simulteneities.
The most complex interval I could tune by ear was 17:13, so I infer
that in two-voice music, more complex frequency ratios have no
audible meaning -- and this is exactly the sense in which Graham
intended his statement above.

🔗Graham Breed <graham@microtonal.co.uk>

Gene Ward Smith wrote:

> If "four digit frequency ratios" (which from context I take to mean
> superparticular ones) have no audible meaning, it seems like a nifty
> idea to temper them out. In the 7-limit this gives ennealimmal, and in
> the 11-limit hemiennealimmal. If we take Graham's view, which I think
> has something to be said for it, we go up to the 13-limit but no
> farther. In the 13-limit, there are twelve four digit superparticular
> commas; the kernel of all of these taken together is 270-equal. This
> sort of fact I've discussed before; it does seem there is some
> justification for considering 270-et to be a sort of universal
> replacement for just intonation.

I did mean the complexity was too great (and obviously so). Probably harmonic commas can be heard for much smaller intervals -- particularly mistuned unisons. But commas aren't usually hidden within chords, are they? Such small commas should be melodically inaudible, especially if shared among a few chords. So an adaptive tuning scheme with these commas should be indistinguishable on a chord-by-chord basis from JI. In which case a 270 note system would be a replacement for JI if used with adaptive temperament.

13-prime limit is one place to stop. I think harmony works clearly up to the 9-limit. Then at the 11-limit you get exotic intervals like neutral thirds and seconds. The 13-limit is roughly more of the same, but you get 8:10:13 and 8:11:13 chords. These are theoretically important for a number of reasons:

- They start with a power of two, and so may have a strong virtual pitch.

- All intervals, including the implied octave, are larger than 7:8 (roughly a critical bandwidth).

- Along with 4:5:6, they're the only such chords with the first number less than 16.

Such chords may be useful in cadences, I still haven't decided. And slightly mistuned versions may work as well, if each interval is a better approximation to an 11-limit interval. But, anyway, they might be a reason for going to the 13-limit, at which point you may as well go to 15. It may even be possible to hear higher limits. With 16:19:24, you get that alternative tuning of a minor triad.

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> > If "four digit frequency ratios" (which from context I take to
mean
> > superparticular ones) have no audible meaning, it seems like a
nifty
> > idea to temper them out. In the 7-limit this gives ennealimmal,
and in
> > the 11-limit hemiennealimmal. If we take Graham's view, which I
think
> > has something to be said for it, we go up to the 13-limit but no
> > farther. In the 13-limit, there are twelve four digit
superparticular
> > commas; the kernel of all of these taken together is 270-equal.
This
> > sort of fact I've discussed before; it does seem there is some
> > justification for considering 270-et to be a sort of universal
> > replacement for just intonation.
>
> I did mean the complexity was too great (and obviously so).
Probably
> harmonic commas can be heard for much smaller intervals --
particularly
> mistuned unisons. But commas aren't usually hidden within chords,
are
> they?

64:63 is famous for being hidden within chords . . .

🔗Graham Breed <graham@microtonal.co.uk>

🔗Paul Erlich <perlich@aya.yale.edu>

Being just like JI (and ~2 cent errors don't bother me) doesn't
strike me as a sufficient qualification for being a/the "universal
temperament". In addition, multiple vals/breeds should be supported,
including some useful non-micro-temperaments.

My favorites among those include the ones that support
omnitetrachordal scales and have relatively low complexity, such as
meantone, double-meantone (injera), and pajara.

You'll find my suggested "universal tuning" here:

http://www.tonalsoft.com/enc/eqtemp.htm

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Over on metatuning Graham had this to say:
>
> "It doesn't matter if anybody can tell the difference, because
there's
> no evidence that four digit frequency ratios have any audible
meaning.
> The 13-limit is borderline..."
>
> If "four digit frequency ratios" (which from context I take to mean
> superparticular ones) have no audible meaning, it seems like a nifty
> idea to temper them out. In the 7-limit this gives ennealimmal, and
in
> the 11-limit hemiennealimmal. If we take Graham's view, which I
think
> has something to be said for it, we go up to the 13-limit but no
> farther. In the 13-limit, there are twelve four digit
superparticular
> commas; the kernel of all of these taken together is 270-equal. This
> sort of fact I've discussed before; it does seem there is some
> justification for considering 270-et to be a sort of universal
> replacement for just intonation.
>
> Incidentally, we have to stop here; if we try to add 729/728, the
next
> smallest comma, we find this is a step of 270-et.

If you try to add 729/728, then I believe you'll get 224-ET (a
schismic tuning, unlike 270), for a max. error <1.6c for 13 and 15-
limit consonances -- still a microtemperament by Dave Keenan's
standards. I think anyone would be hard pressed to distinguish 224
from 15-limit JI. (Dave and I have found that the notation for 224
is a bit simpler than 270.)

If we loosened our 13-limit requirements a bit to allow approximately
the same max error as Miracle in the 11 limit (~3.3c), then 130-ET
would be our universal near-JI replacement. (BTW, I've found the
semantics for a 130 notation to be quite elegant.)

Gene, I imagine that you might want to enlighten us as to which
superparticular commas taken together define 224 and 130,
respectively.

--George

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Being just like JI (and ~2 cent errors don't bother me) doesn't
> > strike me as a sufficient qualification for being
a/the "universal
> > temperament". In addition, multiple vals/breeds should be
> supported,
> > including some useful non-micro-temperaments.
>
> In that case I propose 196608-equal as the universal temperament.
> However, I was not calling 270 a universal temperament, but an all-
> purpose replacement for JI.

Then who came up with the subject line?

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> If you try to add 729/728, then I believe you'll get 224-ET (a
> schismic tuning, unlike 270), for a max. error <1.6c for 13 and 15-
> limit consonances -- still a microtemperament by Dave Keenan's
> standards. I think anyone would be hard pressed to distinguish 224
> from 15-limit JI. (Dave and I have found that the notation for 224
> is a bit simpler than 270.)

The list of 13-limit 4-digit superparticulars is consist with 270 and
only 270, so we know without checking at least one such comma won't
work with 224. In fact, there are two: 1001/1000, which we can readily
dispense with, and 2401/2400, which is the most important comma on the
list. Even so 224 is an interesting system.

> If we loosened our 13-limit requirements a bit to allow approximately
> the same max error as Miracle in the 11 limit (~3.3c), then 130-ET
> would be our universal near-JI replacement. (BTW, I've found the
> semantics for a 130 notation to be quite elegant.)

It would be one possibility. Others which have possibilities are 111,
494 and the amazing 311.

> Gene, I imagine that you might want to enlighten us as to which
> superparticular commas taken together define 224 and 130,
> respectively.

13-limit superparticulars consistent with 224 are as listed here:

540/539, 625/624, 729/728, 1716/1715, 2080/2079, 3025/3024,
4096/4095, 4225/4224, 4375/4374, 6656/6655, 9801/9800, 10648/10647,
123201/123200

I've already pointed out 1001/1000 and 2401/2400 are missing.

13-limit superparticulars belonging to 130 are

243/242, 351/350, 364/363, 441/440, 540/539, 676/675, 729/728,
1001/1000, 1716/1715, 2080/2079, 2401/2400, 4096/4095, 4225/4224,
9801/9800

Four digit superparticulars missing from the list are 3025/3034,
4375/4374, 6656/6655, 10648/10647 and 123201/123200.

In both cases, the comma list is consistent only with the
corresponding equal temperament. However, the reduced list for the
11-limit in both cases corrrespond to a linear temperament, not an
equal temperament. We have

130-et consistent
<<12 34 20 30 26 -2 6 -49 -48 15||
[<2 4 7 7 9|, <0 -6 -17 -10 -15|]
TOP generators 600.225 83.277
in terms of 130, 1/2 and 9/130; 9/130 splits the difference between
22/21 and 21/20

224-et consistent octoid
<<24 32 40 24 -5 -4 -45 3 -55 -71||
[<8 13 19 23 28|, <0 -3 -4 -5 -3|]
TOP generators of 150.034 16.238
In terms of 224, 1/8 and 3/224

I named the 224 temperament octoid some time back; the first turns up
on a lot of my 7 and 11 limit lists, but I don't think it's been named.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > If we loosened our 13-limit requirements a bit to allow
approximately
> > the same max error as Miracle in the 11 limit (~3.3c), then 130-
ET
> > would be our universal near-JI replacement. (BTW, I've found the
> > semantics for a 130 notation to be quite elegant.)
>
> It would be one possibility.

If I may elaborate, we all know how wonderful 72-ET is at the 11-
limit. I would also like to point out that there is an elegance with
72-ET semantics in that there are separate accidentals for primes 5,
7, and 11:

1deg 5-comma (80:81)
2deg 7-comma (63:64)
3deg 11-diesis (32:33)

With 130-ET there are also separate accidentals for primes 5, 7, 11,
and 13:

1deg 7-comma less 5-comma (aka 5:7-kleisma, 5103:5120)
2deg 5-comma (80:81)
3deg 7-comma (63:64)
4deg 11-diesis less 5-comma (aka 55-comma, 54:55)
5deg 13-diesis (1024:1053); also 5-comma plus 7-comma (35:36)
6deg 11-diesis (32:33)

It might be argued that the 17-comma (4096:4131) or 17-kleisma
(2176:2187) might be used for 1deg130. In selecting sagittal symbols
for ETs such as this, Dave and I have concluded that, since ratios of
17 are much less popular than 7/5 and 10/7, a 5:7 kleisma symbol
would be the best choice for 1deg. (Besides, 130-ET is not 17-limit
consistent.)

> Others which have possibilities are 111,
> 494 and the amazing 311.

A problem that I have with 111 is that it's neither 1,5,25 nor 1,7,49-
consistent, which can result in notational discrepancies between 111-
ET and JI. Also, ratios of 5 and 7 are not easily distinguished by a
111 mapping, since the 5-comma and 7-comma are each 2 degrees of 111.

On the other hand, 494 is excellent, and I believe that it would also
satisfy Paul's personal requirements for a "universal tuning."

Yes, 311 is amazing -- I only wish that 255879:256000 vanished in
this one so that 25/16 and 13/10 could be notated using the same
accidental.

Dave and I failed to find a "perfect" ET into which JI could be
mapped and thereby notated using that division's notation. We
therefore decided to notate JI (at several levels of precision)
without reference to any ETs.

The foregoing is just to point out a few things we discovered when we
actually went about deciding how to notate these tunings.

> > Gene, I imagine that you might want to enlighten us as to which
> > superparticular commas taken together define 224 and 130,
> > respectively.
>
> [which you did]

Thanks!

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> In that case I propose 196608-equal as the universal temperament.

Not a good choice: it's not 5-limit consistent, nor 1,3,9-consistent,
plus 7th harmonic has error of >43% of a degree, etc., etc.

But you weren't really serious about this, were you?

And I can hear Dave telling me that I have better things to do, which
reminds me that I'm not really here. :-|

Please ignore this message. ;-)

--George