Re: 13 note just intonation

i believe that the following tuning system yields just intonation in all
twelve keys, each key being unique, and a plausible circularity of fifths.

C = 1/1

C# = 16/15

D = 9/8

D# = 32/27

E = 5/4

F = 4/3

F# = (45/32 64/45)

G = 3/2

G# = 8/5

A = 27/16

A# = 16/9

B = 15/8

muenda qwa sahure wrote,

>i believe that the following tuning system yields just intonation in all
>twelve keys, each key being unique, and a plausible circularity of fifths.

>C = 1/1

>C# = 16/15

>D = 9/8

>D# = 32/27

>E = 5/4

>F = 4/3

>F# = (45/32 64/45)

>G = 3/2

>G# = 8/5

>A = 27/16

>A# = 16/9

>B = 15/8

This tuning system does not provide traditional just tuning (i.e., six
consonant 5-limit triads) for even one key, let alone twelve. The only
purpose for just intonation is to provide consonant harmony; most harmonies
in most keys in this tuning will be dissonant.

Paul!
I temporarily return for a little while!
As the traditions that have investigated and use just intonation are:
Ancient Greece
Medieval Persia
India (ragas)
All these cultures were not concerned with harmony. It is melody. It is one
place where both Lou Harrison and Myself agree. It was harmony (so-called)
that caused the soulless industrial efficiency of temperment to come about. A
final solution
On one hand you accept intervals, some being 17 cents off (7/5), in reference
to 22 ET as consonant but dismiss JI triads where one of the factors is off
buy a comma. Just where is the dividing line. Your statement implies that the
Pythagorean triad is too dissonant to use.
As there is no consensus as to what is dissonant on even the most basic
level ,we
can only assume that that it is not a black and white question. Anyway what is
accomplished by figuring out such categorizing. I can think of only the
following
1.The ability to gradually add or subtract tension by pitch means only
2. For certain humorous effects such as alternating the extremes for humorous
effects

Hindemith, a composer who I have respect for, worried about such things, and
this would lead to him ending many of his pieces with Major Chords. This
aspect of his work I find unfortunate in that It doesn't really fit quite
often sound completely out of place!

On the other hand, there are other JI tunings I would use instead!

"Paul H. Erlich" wrote:

>
> The only
> purpose for just intonation is to provide consonant harmony; most harmonies
> in most keys in this tuning will be dissonant.

North American Embassy of Anaphoria Island
http://www.anaphoria.com

Kraig Grady wrote,

>As the traditions that have investigated and use just intonation are:
>Ancient Greece
>Medieval Persia
>India (ragas)
>All these cultures were not concerned with harmony.

I think it is most likely that the mathematical theorists of the time only
knew how to deal with rational divisions (and correctly recognized the
importance of the 2:1, 3:2, and 4:3 for melody) while the musicians could
easily have been using all kinds of quasi-tempered schemes. No one knows
what ancient Greek music sounded like, certainly not subcommatic variations
within the scale. There is evidence that in ancient India sruti #2 (out of
22) was tempered, implying that they recognized the syntonic comma, but not
the diaschisma, as a pitch difference.

>On one hand you accept intervals, some being 17 cents off (7/5), in
reference
>to 22 ET as consonant

Actually, in my paper I state that the interval 600 cents is _dissonant_ but
as part of a 4:5:6:7 triad the ear knows exactly what is meant and most of
the dissonance goes away.

>but dismiss JI triads where one of the factors is off
>buy a comma. Just where is the dividing line.

A comma is 22 cents. So the dividing line is somewhere between 17 and 22
cents, say 20 cents. (just kidding)

>Your statement implies that the
>Pythagorean triad is too dissonant to use.

No, I was just implying that if you're going to use triads with fifths or
thirds that are off by a comma, then there's really no point in using JI --
JI maximizes consonance, while you can use, say, 15-tET to get fifths and
major thirds that are off by almost a comma, and 12-tET to get major and
minor thirds that are off by the better part of a comma. Certainly one need
not use an ET to get these effects, but there's an infinite varienty of
tuning that you can use to get these effects and JI offers nothing special
in this regard.

>From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
>Kraig Grady wrote,
>
>>As the traditions that have investigated and use just intonation are:
>>Ancient Greece
>>Medieval Persia
>>India (ragas)
>>All these cultures were not concerned with harmony.
>
>I think it is most likely that the mathematical theorists of the time only
>knew how to deal with rational divisions (and correctly recognized the
>importance of the 2:1, 3:2, and 4:3 for melody) while the musicians could
>easily have been using all kinds of quasi-tempered schemes.

By "of the time," I assume you are refering to the first two of Kraig's
examples, since Indian raga music is with us today. In fact, Indian
musicians today continue to use and argue over just versus tempered
intervals. Certainly the ancient Greeks and Persians knew of the existence
of irrationals (perhaps John Chalmers could provide some details), and
Aristoxenus spoke to their use in scales. Though some kind of "harmony" in
the broadest since may exist in these musics (as with the Indian drones), I
think Kraig's point is a good one. While you may think that JI is only good
for maximally consonant harmonies, other musicians may have found just
intervals good for other reasons.

Bill

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^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
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"Paul H. Erlich" wrote:

> .
>
> I think it is most likely that the mathematical theorists of the time only
> knew how to deal with rational divisions (and correctly recognized the
> importance of the 2:1, 3:2, and 4:3 for melody) while the musicians could
> easily have been using all kinds of quasi-tempered schemes.

Actually you had Aristoxenus who proposed the octave divided into what i
believe was 60 equal parts which he believed is the limit of our perception of
ratios. this is what you propose.

> No one knows
> what ancient Greek music sounded like, certainly not subcommatic variations
> within the scale. There is evidence that in ancient India sruti #2 (out of
> 22) was tempered, implying that they recognized the syntonic comma, but not
> the diaschisma, as a pitch difference.

In working against drones, intonation becomes even more important. I can't
imagine how any theorist could make a mistake like that!

>
>
> >On one hand you accept intervals, some being 17 cents off (7/5), in
> reference
> >to 22 ET as consonant
>
> Actually, in my paper I state that the interval 600 cents is _dissonant_ but
> as part of a 4:5:6:7 triad the ear knows exactly what is meant and most of
> the dissonance goes away.
>
> >but dismiss JI triads where one of the factors is off
> >buy a comma. Just where is the dividing line.
>
> A comma is 22 cents. So the dividing line is somewhere between 17 and 22
> cents, say 20 cents. (just kidding)
>
>

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com