equal-beating vs. 12-tET

All this controversy over whether Bach could have tuned
12-tET, and whether the theorists of his day knew the
difference between equal-beating and equal steps. I know
somebody (Aaron?) posted an equal-beating pythagorean
from Scala, dunno if it was this one...

0: 1/1 0.000 unison, perfect prime
1: 100.034 cents 100.034
2: 199.519 cents 199.519
3: 299.800 cents 299.800
4: 399.516 cents 399.516
5: 500.017 cents 500.017
6: 599.941 cents 599.941
7: 699.322 cents 699.322
8: 799.504 cents 799.504
9: 899.127 cents 899.127
10: 999.540 cents 999.540
11: 1099.381 cents 1099.381
12: 2/1 1200.000 octave

Every fifth beats the same here, and in no interval class
does this tuning differ from 12-tET by even a whole cent.
Whoop-dee-doo.

Incidentally, Norman Henry apparently has tuned this way
all his life (as a professional tuner).

-Carl

on 12/20/03 3:53 PM, Carl Lumma <ekin@lumma.org> wrote:

> All this controversy over whether Bach could have tuned
> 12-tET, and whether the theorists of his day knew the
> difference between equal-beating and equal steps. I know
> somebody (Aaron?) posted an equal-beating pythagorean
> from Scala, dunno if it was this one...

I've asked 3 times now, but now one has explained to me how there can be
such a thing as equal-beating pythagorean if pythagorean means strictly 3:2
fifths rather than a broader class.

-Kurt

>
> 0: 1/1 0.000 unison, perfect prime
> 1: 100.034 cents 100.034
> 2: 199.519 cents 199.519
> 3: 299.800 cents 299.800
> 4: 399.516 cents 399.516
> 5: 500.017 cents 500.017
> 6: 599.941 cents 599.941
> 7: 699.322 cents 699.322
> 8: 799.504 cents 799.504
> 9: 899.127 cents 899.127
> 10: 999.540 cents 999.540
> 11: 1099.381 cents 1099.381
> 12: 2/1 1200.000 octave
>
> Every fifth beats the same here, and in no interval class
> does this tuning differ from 12-tET by even a whole cent.
> Whoop-dee-doo.
>
> Incidentally, Norman Henry apparently has tuned this way
> all his life (as a professional tuner).
>
> -Carl
>
>
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>> All this controversy over whether Bach could have tuned
>> 12-tET, and whether the theorists of his day knew the
>> difference between equal-beating and equal steps. I know
>> somebody (Aaron?) posted an equal-beating pythagorean
>> from Scala, dunno if it was this one...
>>
>> 0: 1/1 0.000 unison, perfect prime
>> 1: 100.034 cents 100.034
>> 2: 199.519 cents 199.519
>> 3: 299.800 cents 299.800
>> 4: 399.516 cents 399.516
>> 5: 500.017 cents 500.017
>> 6: 599.941 cents 599.941
>> 7: 699.322 cents 699.322
>> 8: 799.504 cents 799.504
>> 9: 899.127 cents 899.127
>> 10: 999.540 cents 999.540
>> 11: 1099.381 cents 1099.381
>> 12: 2/1 1200.000 octave
>>
>> Every fifth beats the same here, and in no interval class
>> does this tuning differ from 12-tET by even a whole cent.
>> Whoop-dee-doo.
>>
>> Incidentally, Norman Henry apparently has tuned this way
>> all his life (as a professional tuner).
>
>I've asked 3 times now, but now one has explained to me how there
>can be such a thing as equal-beating pythagorean if pythagorean
>means strictly 3:2 fifths rather than a broader class.

Usually pythagorean means strictly 3:2, but sometimes it means
any chain of near (usually positive) fifths. "Positive" is
Bosanquet's terminology for fifths which are sharper than 700
cents. In this case the fifths wind up being negative (meantone)
fifths, but I was probably influenced in my use of terminology
by the fact that the Scala command to generate these is
"pythag".

Now, doesn't that just clear it all up? :)

-Carl

> 0: 1/1 0.000 unison, perfect prime
> 1: 100.034 cents 100.034
> 2: 199.519 cents 199.519
> 3: 299.800 cents 299.800
> 4: 399.516 cents 399.516
> 5: 500.017 cents 500.017
> 6: 599.941 cents 599.941
> 7: 699.322 cents 699.322
> 8: 799.504 cents 799.504
> 9: 899.127 cents 899.127
> 10: 999.540 cents 999.540
> 11: 1099.381 cents 1099.381
> 12: 2/1 1200.000 octave
>
>Every fifth beats the same here, and in no interval class
>does this tuning differ from 12-tET by even a whole cent.
>Whoop-dee-doo.

Incidentally, I thought I remembered this. Manual posted it
here an age ago...

>Date: Fri, 10 Oct 1997 19:48:06 +0200
>From: Manuel.Op.de.Coul@ezh.nl
>To: tuning@eartha.mills.edu
>Subject: Another version of 12-tET
>Message-ID: <C125652C.0061B07D.00@notesrv2.ezh.nl>
>
>To celebrate the 1200th digest, I will present another version of
>12-tET:
> 0: 1/1 0.000000
> 1: 1662005/1568693 100.0340
> 2: 1760309/1568693 199.5188
> 3: 1865285/1568693 299.7998
> 4: 1975877/1568693 399.5162
> 5: 2093975/1568693 500.0176
> 6: 10223/7229 599.9410
> 7: 2349463/1568693 699.3218
> 8: 355633/224099 799.5039
> 9: 2636887/1568693 899.1276
> 10: 399193/224099 999.5407
> 11: 2960239/1568693 1099.381
> 12: 2/1 1200.000
>
>Before you think that I went crazy, I didn't invent this. I just
>calculated the ratios for the sake of it. This tuning was used in pianos
>by the English "Best Factory Tuners" in 1840.
>The idea becomes clear by showing the beat frequencies of 3/2 for the
>octave above middle C:
>
> 0: 0.000: -1.1930
> 1: 100.034: -1.1930
> 2: 199.519: -1.1930
> 3: 299.800: -1.1930
> 4: 399.516: -1.1930
> 5: 500.017: -1.1930
> 6: 599.941: -1.1930
> 7: 699.322: -1.1930
> 8: 799.504: -1.1930
> 9: 899.127: -1.1930
> 10: 999.540: -1.1930
> 11: 1099.381: -1.1930
> 12: 1200.000: -2.3859
>
>Now someone can persuade his piano tuner to set this tuning and let us
>know if it's an improvement.

-Carl

on 12/20/03 5:34 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> All this controversy over whether Bach could have tuned
>>> 12-tET, and whether the theorists of his day knew the
>>> difference between equal-beating and equal steps. I know
>>> somebody (Aaron?) posted an equal-beating pythagorean
>>> from Scala, dunno if it was this one...
>>>
>>> 0: 1/1 0.000 unison, perfect prime
>>> 1: 100.034 cents 100.034
>>> 2: 199.519 cents 199.519
>>> 3: 299.800 cents 299.800
>>> 4: 399.516 cents 399.516
>>> 5: 500.017 cents 500.017
>>> 6: 599.941 cents 599.941
>>> 7: 699.322 cents 699.322
>>> 8: 799.504 cents 799.504
>>> 9: 899.127 cents 899.127
>>> 10: 999.540 cents 999.540
>>> 11: 1099.381 cents 1099.381
>>> 12: 2/1 1200.000 octave
>>>
>>> Every fifth beats the same here, and in no interval class
>>> does this tuning differ from 12-tET by even a whole cent.
>>> Whoop-dee-doo.
>>>
>>> Incidentally, Norman Henry apparently has tuned this way
>>> all his life (as a professional tuner).
>>
>> I've asked 3 times now, but now one has explained to me how there
>> can be such a thing as equal-beating pythagorean if pythagorean
>> means strictly 3:2 fifths rather than a broader class.
>
> Usually pythagorean means strictly 3:2, but sometimes it means
> any chain of near (usually positive) fifths. "Positive" is
> Bosanquet's terminology for fifths which are sharper than 700
> cents. In this case the fifths wind up being negative (meantone)
> fifths, but I was probably influenced in my use of terminology
> by the fact that the Scala command to generate these is
> "pythag".
>
> Now, doesn't that just clear it all up? :)
>
> -Carl

But you weren't the only one using the words that way, though I still wonder
whether scala usage ends up being the "culprit". It was Aaron (AKJ) who
brought up this term in the thread "Bach, Werckmeister & Co.", as follows:

on 12/12/03 6:35 AM, Aaron K. Johnson <akjmicro@comcast.net> wrote:
> True, however, if one does an equal-beating pythagorean, the result is so
> close to equal temperament as to be indistinguishable to the ear...still,
> this is very painstaking to do by ear without an accurate way of measuring
> the sought after beat rate...
>
> -Aaron.

I can't find the original post containing the scala scale through.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_50205.html#50206

> on 12/20/03 3:53 PM, Carl Lumma <ekin@l...> wrote:
>
> > All this controversy over whether Bach could have tuned
> > 12-tET, and whether the theorists of his day knew the
> > difference between equal-beating and equal steps. I know
> > somebody (Aaron?) posted an equal-beating pythagorean
> > from Scala, dunno if it was this one...
>
> I've asked 3 times now, but now one has explained to me how there
can be
> such a thing as equal-beating pythagorean if pythagorean means
strictly 3:2
> fifths rather than a broader class.
>
> -Kurt
>

***Hello Kurt!

Well, this is what confused me too at first... but I believe I recall
that the term pythagorean on this list became used for a range of
tunings surrounding and including 3:2. I can't recall what the
actual boundaries are, however...

Joseph P.