Clearer

>>There is no such thing as the odd limit of a pitch -- all intervals can
be >>constructed above and below any pitch. What do you really mean?

Odd limit of the interval, sorry.

>>}Idea "A" seems to suggest that we temper the octave the most, the fifth the
>>}next most, and so on up. But we can't do this without wrecking tonality,
>>}since the low identities provide that, as explained in idea "B".
>>
>>}Both A and B are really one idea, in that they're both caused by the way
>>}superparticular fractions get closer to eachother, so there's no
>>}"contradiction", true. But when tempering, it presents a trade-off
>>}situation, one which can't be well-addressed by just making the de-tuning
>>}inversely proportional to limit.
>>
>>None of the above makes any sense to me. Perhaps you have something in mind
>>that you'd like to clarify, but insofar as you're relying on (B), I honestly
>>can't figure it out. Moreover, I don't see anything special about
>>superparticulars.

The superparticulars are just all the intervals between adjacent members of
the series. You're supposed to be able to detune a 3/2 more than an 13/12,
since there's less low-numbered ratios near the 3/2 to get confussed with,
right? That's idea A. It suggests that if you've got a certain amount of
tempering to do, you should do it more on the low identities.

Yet the high identities rely on the low ones for context in music. High
identities work in a tonality, and its the lower identities that play the
bigger role in establishing tonality. That's idea B. It's the cons to
tempering the lower identities more. While they alone can stand more
tempering than the higher identities, how well the higher identities to
work depends on the strength of the lower ones.

>>How did you think I arrived at this list, if not by insisting that the
>>candidates must be better in the 7-limit than 12 at the 5-limit?

I see I made a mistake as to what your criteria were for the two candidate
lists. I don't have the charts, but I didn't think that 22, 26, and 27
were as good at the 5-limit as 12...

>>Can you suggest a way to make the paper less confusing on this point?

Yes.

Instead of...

"if we wish to move to the 7-limit without decreasing the overall level of
accuracy from that of 12-equal at the 5-limit"

Say...

"if we want our new tuning to approximate the 7-limit at least as well as
12-equal approximates the 5-limit"

Instead of...

"but now only two are still more accurate than 5-limit 12-equal"

Say...

"but now only two are still better at the 7-limit than 12 equal is at the
5-limit."

>>}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.
>>
>>First of all, it's an interval, not a pitch.

This time my use of pitch is just fine. The intervals are the aug 4th and
the dim 5th. I meant that, starting the two intervals on the same bottom
pitch, the "two" upper pitches are the same.

>>Secondly, the augmented fourth and diminished fifth _sound_ different in
>>context, and one must resolve outwards while the other must resolve inwards.

I agree. Playing Jazz got me to understand enharmonics in 12, especially
the difference between the aug 4 and the dim 5. I can see the reason why
someone might prefer to use enharmonics and proper spellings of intervals
for certain work in 12. I just don't prefer using them for the type of
work I do in 12.

>>}Why you consider it a type of fifth in your
>>}example was, in any case, not clear to me.
>>
>>Because you have to count 5 scale steps to get from the lower note to the
>>upper note, inclusive.

You mean scale degrees, not steps?

If my scale is, in 12...

C C# E F# G# B C

..then the tritone is the 4th degree.

>>I did provide such a definition -- "step" means a step in the scale, not a
>>step in the tuning. Without a basic, "generalized diatonic" scale, the
>>concept is meaningless.

I'm with you on the need for a generalized diatonic scale, but I missed the
definition. I don't see why a fifth is ever anything but the fifth degree
of a scale. The tritone isn't part of the "unaltered diatonic" scale, so I
can't see how it can be a type of 5th or 4th degree in that scale.

>>Wow, I thought my definition was fine. It certainly seems robust to me. How
>>would you suggest I improve it? Perhaps by spelling out every single
interval
>>occuring in each of the "generalized diatonic" scales?

Maybe.

How would you define diatonic?

How do you define dissonace, as in "characteristic dissonance", at a given
odd limit? An interval whose ratio involves odd numbers (after factors of
two are taken out) bigger than the limit?

>>}I noticed that. But why is it Microsoft's fault? Did you used some
>>}automatic numbering scheme? I hate that trend in software nowadays! It's
>>}not already so easy that you can't do it yourself?
>>
>>I couldn't figure out how to turn off the auto-numbering in Word 97.

You can't just type the numbers in by hand? I'm sure you can turn off the
auto-numbering with a short spelunk thru the preferences thingy. I hate
Word to death, and I just removed it from my computer, or I'd tell you how
exactly.

>>Henry Cow was a British group whose music was very composed, a bit
Zappa-like
>>but much deeper emotionally (I feel).

I'll keep my eyes peeled.

>>Have you seen or heard Phish live?
>>There's no comparison.

Unfortunately, no.

>>I'm sure you can transcribe all Partch music in 72TET and not lose anything
>>in the modulatory effects.

Oh, maybe so. At the cost of a pitch set twice the size of Partch's
(roughly, but the pitch set used varies from work to work).

>>}Please Note: How the intervals are heard depends on context!!! I'm just
>>}throwing in my analysis here because sometimes a particular context seems
>>}to be assumed when giving rational approximations for the 22TET
intervals...
>>
>>what context are you assuming???

None in particular. That was the point.

Bill Alves writes...

>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.

Thanks for work. This seems quite a bit more elegant to me. Truth be
told, elegant was what I was trying to avoid. I said that I left out the
inversions to reveal structure in the scale. Maybe this was misleading. I
didn't pick the approximations as a scale. I was just trying to show what
you might wind up hearing if you're not careful.

Carl


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From: jpff@maths.bath.ac.uk
Subject: Re: Our instruments
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Bob Lee wrote:

> I had always assumed that 7/5 was the natural ratio for an augmented fourth,
> and that 10/7 was the natural ratio for the diminished fifth. What do you
> mean when you say they "do not arise in a diatonic context"?

Traditional diatonic music uses 7-note scales and 5-limit harmony.
In this context tritones are dissonant. The simplest 5-prime
limit ratios are 25/18 and 36/25, or (-1 -2 2)H and (2 2 -2)H.
The latter comprises two minor thirds 6/5 or (1 1 -1)H. That
means it must be a kind of fifth.

The difference between these quintal tritones is just over a 19th
of an octave. That means they're well represented in 19-equal
and the difference is melodically important. Staff notation is
a consistent way of notating quintal intervals, so the distinction
is preserved.

When a tritone is used in isolation, or is treated as consonant,
the best tuning is septimal. The difference between 10/7 and
7/5 is only 35 cents. 7/5 is chosen as the augmented fourth
because it's the smaller. However, as this isn't a diatonic
interval, there's no standard way of notating it. The obvious
example is a 4:5:6:7 chord. For 7/5 to be an augmented 4th,
7/4 must be an augmented sixth. If you're writing septimal music
in meantone temperament or notation, you have to be aware of this.
Unfortunately, most composers do treat 10/7 and 7/5 as enharmonies,
so the usual notation of 7/4 is as a minor seventh. You have to
work out from the context whether a 7th chord should be quintal or
septimal.


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From: "Paul H. Erlich"
Subject: b5th ratios
PostedDate: 13-01-98 21:27:51
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