Pythagorean scales

>What I had in mind was the potentiality of using the circle of 7:4s instead
of 3:2s.

I invented a formula that will help you a lot (I someone else know this
trick, email me so I stop claiming I invented it, but I am pretty sure that
I am the first one to use it). Here goes:

{ (x,y) E N* | x>y} ---> x= 1, 2, 3... ; y= 2, 3, ...
n E Z ---> n= ..., -2, -1, 0, 1, 2, ...

1 <= (x^n)/(y^p) < y
where p= | n*log(x) / log(y) |

| | means interger part of a number.

So, for your question, x=7, y=2

Say you want to find the 5th interval of the "7/4 stack" serie, to build a
Pythagorean scale.

n=5, p= | 5*log(7) / log(2) |
p= | 14.036...|
p= 14

7^5 / 2^14 = 1.0258...



n=45, p= | 45*log(7) / log(2) |
p= | 126.33 |
p= 126

7^45 / 2^126 = 1.2579...

It works, the final number will always be smaller than y (2 in our exemple,
so we needn't worry about normalizing intervals to a 2/1)

If anyone would like to have the demonstration for this theorem, email me.

Here are some arbitrary values:
The human can ear from 20 30 000 Hz,
The smallest interval one can perceive is 2 cents,
The maximum number of beatings per second is 20, because 20 Hz is the lower
limit of audition.
We cannot hear beats that are further than 16 seconds apart (because the
slowest tempo is 30 and the longest note is the square (2 whole notes))

I anyone disagrees with these values, email me.
But if everyone agrees, then these values become the list's standard.

Bye.


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>A posting in the list is
>
>>I recognize the sound of only two tunings -
>>one is 12-tone equal-temperament and the
>>other is not.
>
>This is a reasonable sentiment for a milatary general, but for a keyboard
>muscian, it only makes evident the lack of tonal perspective.

William Sethares just sent me a private mail to explain that
he meant this statement just for laughs. Well not quite, he
told me something which I also find true, he said:

> this is the kind of attitude that I get from musicians
> whenever talk about alternate tunings comes up

What I also find is that musicians and audience always cannot tell a
difference between music played in tune and 12-TET (thinking
that they are the same). But they can tell "modified" tunings.

An example is my wife commented Gary's 19-TET piece sounding like
a band of small children playing their cheap recorders (Sorry, Gary).

> If you do not recognize a mean-tone tuning, or a Well temperament,
>you are really missing an important perspective on what 12 tone ET actually
>sounds like.

I do not know what to say someone has missed if someone cannot notice that
Heifetz, and a few others, play 100% in tune - on certain pieces.

> As far as what is "in tune"; if "in tune" describes that which is
>arranged harmoniously, it is a mathematical fact that equally tempered
>keyboards are out of tune.
> Violinists and other non fixed pitch instruments don't play in equal
>temperament very long if left to themselves .The critical musical ear
>naturally seeks the harmonious placement of pitches to form purer intervals
>than what is found in 12 tone equal temperament.
>regards,
>Ed Foote
>Precision Piano Works
>Nashville, Tn.
>

Not just one string quartet member confessed on the "bowed-strings"
newsgroup that they spend 99% of their time on intonation. Harpists
and harpsichordists complain of their having to tune their own
instruments. They forget that violinists and singers have to tune
each and every note in real time. it is natural that we spend 99%
of our time on intonation.

Intonation is of course not just meant for "harmonious placement".
It is meant for creating the sentiments of joy, sorrow, love, and
every possible sentiment of an artist's endeavour.

If you are still not bored, please congratulate me over my 6-year-
old daughter's winning yesterday second prize in an open schools
piano competition, grade 3, among 54 competitors. Before, she won
third prize at the same competition grade one at four, and another
third prize in grade two at five. She has absolute pitch, but does not
realise that my violin intonation is different from the piano. She
tries to correct me even, sometimes. But when my Faure Sicilienne
sounded so good, she bounced up and down like a ball.

Regards,
Linus Liu.



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