22TET

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Carl,

Looking at your 22TET JI approximation, it occurs to me that it will all
fit pretty neatly into the 11 limit. You have 31/16, and, by implication
its inversion 32/31, but 33/32 is only 1 cent off, and is already present
in the distance between the 4/3 and the 11/8 (or 3/2 and 16/11). Likewise
40/33, while a couple of cents further off than 23/19 that you use, also
uses this interval down from the 5/4. A 25/22 (11/10 down from 5/4) is just
2 cents worse than your 17/15. While the 600 cent interval is always a
problem in low limit JIs, 99/70 (7/6 down from 33/20), though not exactly a
pretty ratio, nails it. Here are my suggestions in your format, with
inversions filled in:

> 1- 55 33/32 (1)
> 2- 109 16/15 (3)
> 3- 164 11/10 (1)
> 4- 218 25/22 (3)
> 5- 273 7/6 (6)
> 6- 327 40/33 (6)
> 7- 382 5/4 (4)
> 8- 436 9/7 (1)
> 9- 491 4/3 (7)
>10- 545 11/8 (6)
>11- 600 99/70 (0)
>12- 655 16/11 (6)
>13- 709 3/2 (7)
>14- 764 14/9 (1)
>15- 818 8/5 (4)
>16- 873 33/20 (6)
>17- 927 12/7 (6)
>18- 982 44/25 (3)
>19- 1036 20/11 (1)
>20- 1091 15/8 (3)
>21- 1145 64/33 (0)
>22- 1200 2/1 (0)

I understand that you may be using different criteria for your
approximations, so I just offer these as another possible interpretation
and look forward to your paper.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
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SMTPOriginator: tuning@eartha.mills.edu
From: Steven Rezsutek
Subject: Re:
PostedDate: 09-01-98 23:56:17
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I would be very surprised if any temperament significantly
different to 22 equal supported decatonic scales. Vanishing
septimal commas is a difficult trick to pull off.

The octave is the best interval to define a harmonic scale around.
Once you've done that, the octave should be very accurate.
Although you should think about the octaves, it would be annoying
if "the same" chords in different registers were out of tune with
each other.

I tend to think that low primes should be tuned best. The more
complex harmony you use, though, the more accuracy is required
for _all_ intervals.

Carl Lumma wrote:

> It just stands to reason that there are all kinds
> of effects, probably most of them un-discovered, based on the "anomalies"
> of JI.

Remember that there are a lot of temperaments out there! Most of
what you can do in JI should work schismically, as all the commas
are there.

> 1) Conventional theory is full of holes. The tritone is spelled as an
> augmented forth in certain contexts, and as a diminished fifth in others,
> but it's the same pitch. All the enharmonics should be thrown out in 12.

No, the tritone is not always the same pitch. It can be 7/5, 10/7,
25/18 or 36/25. The first two do not arise in a diatonic context.
So, 25/18 is an augmented fourth and 36/25 a diminished fifth. The
difference between them, 63 cents, is not negligible. 12-equal
enharmonies are irrelevant as diatonic scales work in all
meantones, and the notation reflects this.

When Paul Erlich says "scale steps" he obviously means the 7 note
diatonic scale. I didn't have any trouble with this when I read
the paper first.


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From: mr88cet@texas.net (Gary Morrison)
Subject: Re: TUNING digest 1293 jloffink
PostedDate: 10-01-98 22:22:22
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