Turkish Tunbur - Spirals of Fifths and Rational Asymptotes

Consider a descending Spiral of Fifths
that begins at the "octave," ratio 2/1.

1st descending 3/2 is
4/3 = 498.0 ¢.

2nd descending 3/2 is
16/9 = 996.1 ¢.

Then --

8th descending 3/2 is
8192/6561 = 384.4 ¢
or approx. 5/4 = 386.3 ¢.

******************************

22nd descending 3/2 is
34359738368/31381059609 = 156.99 ¢
or approx. 12/11 = 150.64 ¢.

One may also derive this ratio by reducing 9/8 by 2 commas:
(9/8)/(531441/524288)^2 = 34359738368/31381059609

******************************

34th descending 3/2 is
18014398509481984/16677181699666569 = 133.53 ¢
or approx. 13/12 = 138.57 ¢.

One may also derive this ratio by reducing 9/8 by 3 commas:
(9/8)/(531441/524288)^3 =
18014398509481984/16677181699666569

******************************

Al-Farabi clearly distinguishes between locating the frets on an ud,
and locating the frets on a tunbur. The former moveable frets
were always located by dividing strings into various fractional
lengths with a ruler. In contrast, the latter moveable frets were
and are determined by ear! This crucial distinction tells us that
even the seemingly most complicated Spirals-of-Fifths ratios have
been tuned by ear for thousands of years.

******************************

For an example of tunbur (tanbur) Spirals-of-Fifths ratios, please
visit --

http://www.turkmusikisi.com/calgilar/tanbur/tanbur.htm

On the bottom of this page, there is a table with tunbur fret
locations, which results in the following scale:

1. 1/1
2. 256/243
3. 2187/2048
4. 65536/59049 (10th descending 3/2)
5. 9/8
6. 32/27
7. 19683/16384
8. 8192/6561 (8th descending 3/2)
9. 81/64
10. 2097152/1594323 (13th descending 3/2)
11. 4/3
12. 1024/729
13. 729/512
14. 262144/177147 (11th descending 3/2)
15. 3/2
16. 128/81
17. 6561/4096
18. 32768/19683 (9th descending 3/2)
19. 27/16
20. 8388608/4782969 (14th descending 3/2)
21. 16/9
22. 4096/2187
23. 243/128
24. 1048576/531441 (12th descending 3/2)
25. 2/1

******************************

Finally, for tunburs with 32 frets per "octave," please visit --

http://www.uam.es/personal_pdi/filoyletras/jsango/MapaTone/Tanbur.htm

An analysis of the first six-fret cluster gives the following ratios:

1. 5th descending 3/2 is
256/243 = 90.2 ¢.

2. 7th *ascending* 3/2 is
2187/2048 = 113.7 ¢
or approx. 16/15 = 111.7 ¢.

3. 34th descending 3/2 is
18014398509481984/16677181699666569 = 133.53 ¢
or approx. 13/12 = 138.57 ¢.

4. 22nd descending 3/2 is
34359738368/31381059609 = 156.99 ¢
or approx. 12/11 = 150.64 ¢.

5. 10th descending 3/2 is
65536/59049 = 180.5 ¢
or approx. 10/9 = 182.4 ¢.

6. 2nd *ascending* 3/2 is
9/8 = 203.4 ¢.

******************************

Cris Forster, Music Director
www.Chrysalis-Foundation.org

--- In tuning@yahoogroups.com, Cris Forster <76153.763@c...> wrote:
> Consider a descending Spiral of Fifths
> that begins at the "octave," ratio 2/1.

Why does an octave get scare quotes and a fifth doesn't?

> 1st descending 3/2 is
> 4/3 = 498.0 ¢.
>
> 2nd descending 3/2 is
> 16/9 = 996.1 ¢.
>
> Then --
>
> 8th descending 3/2 is
> 8192/6561 = 384.4 ¢
> or approx. 5/4 = 386.3 ¢.

Schismatic temperament.

> 22nd descending 3/2 is
> 34359738368/31381059609 = 156.99 ¢
> or approx. 12/11 = 150.64 ¢.

If we map 3, 5, and 11 in this way, we are mapping the 11 using
+22 fifths. This suggests we may as well map 7 using -14 fifths, since
the optimized tunings are very close (rms tunings give 702.23 cents vs
702.18 cents with the 7 included.) This gives us what Paul Erlich
suggested could be called the "garibaldi" temperament. 94-et, with a
fifth of 702.13 cents, works well for this this, but 135-et, which
gives 702.22 cents, is even better for the no-7s aspect.

> 34th descending 3/2 is
> 18014398509481984/16677181699666569 = 133.53 ¢
> or approx. 13/12 = 138.57 ¢.

This maps 13 to -33 fifths. Why not +20 fifths instead?

If we call the temperament with your mapping for 13, and including the
mapping for 7 "tunbar", then tunbar has a comma basis
{225/224,351/350,385/384,572/567}. It's both more complex and less
accurate than the alternative I suggested. On the plus side, it has
the comma 351/350, which my alternative does not have. This comma has
a 7 in it, but taking the ratio with 225/224 gives 625/624, which
factors as 2^(-4) 3^(-1) 5^4 13^(-1). It is the ratio between 25/24
and 26/25, and tells us that in tunbar these are identified.

Dear Mr. Smith,

Let's play Jeopardy!

Answer: When practicing musicians tune their own instruments.

What is musical mathematics?

Answer: When theoretical musicians tune other people's instruments.

What is mathematical music?

Cris Forster, Music Director
www.Chrysalis-Foundation.org

> If we call the temperament with your mapping for 13, and including the
> mapping for 7 "tunbar", then tunbar has a comma basis
> {225/224,351/350,385/384,572/567}. It's both more complex and less
> accurate than the alternative I suggested.

--- In tuning@yahoogroups.com, "c_ml_forster" <76153.763@c...> wrote:

> What is mathematical music?

You weren't tuning an instrument, you were defining a temperament.
That does, of course, involve math--and math was what you were doing.
What else do you imagine was involved in your calculations?

circle of fifths--color wheel of art theory.

anyone know the correspondence, or can figger it?

assumption: A=440 equals primary red

chiptruth@excite.com

--- On Thu 06/23, Gene Ward Smith < gwsmith@svpal.org > wrote:
From: Gene Ward Smith [mailto: gwsmith@svpal.org]
To: tuning@yahoogroups.com
Date: Thu, 23 Jun 2005 20:08:12 -0000
Subject: [tuning] Re: Turkish Tunbur - Spirals of Fifths and Rational Asymptotes

<html><body>

<tt>
--- In tuning@yahoogroups.com, Cris Forster <76153.763@c...> wrote:<BR>
> Consider a descending Spiral of Fifths<BR>
> that begins at the "octave," ratio 2/1. <BR>
<BR>
Why does an octave get scare quotes and a fifth doesn't?<BR>
<BR>
> 1st descending 3/2 is<BR>
> 4/3 = 498.0 �.<BR>
> <BR>
> 2nd descending 3/2 is<BR>
> 16/9 = 996.1 �.<BR>
> <BR>
> Then --<BR>
> <BR>
> 8th descending 3/2 is<BR>
> 8192/6561 = 384.4 �<BR>
> or approx. 5/4 = 386.3 �.<BR>
<BR>
Schismatic temperament.<BR>
<BR>
> 22nd descending 3/2 is<BR>
> 34359738368/31381059609 = 156.99 �<BR>
> or approx. 12/11 = 150.64 �.<BR>
<BR>
If we map 3, 5, and 11 in this way, we are mapping the 11 using<BR>
+22 fifths. This suggests we may as well map 7 using -14 fifths, since<BR>
the optimized tunings are very close (rms tunings give 702.23 cents vs<BR>
702.18 cents with the 7 included.) This gives us what Paul Erlich<BR>
suggested could be called the "garibaldi" temperament. 94-et, with a<BR>
fifth of 702.13 cents, works well for this this, but 135-et, which<BR>
gives 702.22 cents, is even better for the no-7s aspect.<BR>
<BR>
> 34th descending 3/2 is<BR>
> 18014398509481984/16677181699666569 = 133.53 �<BR>
> or approx. 13/12 = 138.57 �.<BR>
<BR>
This maps 13 to -33 fifths. Why not +20 fifths instead?<BR>
<BR>
If we call the temperament with your mapping for 13, and including the<BR>
mapping for 7 "tunbar", then tunbar has a comma basis<BR>
{225/224,351/350,385/384,572/567}. It's both more complex and less<BR>
accurate than the alternative I suggested. On the plus side, it has<BR>
the comma 351/350, which my alternative does not have. This comma has<BR>
a 7 in it, but taking the ratio with 225/224 gives 625/624, which<BR>
factors as 2^(-4) 3^(-1) 5^4 13^(-1). It is the ratio between 25/24<BR>
and 26/25, and tells us that in tunbar these are identified.<BR>
<BR>
<BR>
</tt>

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--- In tuning@yahoogroups.com, "Mikal De Valia" <chiptruth@e...> wrote:
>
>
> circle of fifths--color wheel of art theory.
>
> anyone know the correspondence, or can figger it?

The idea goes back to Newton, who decided there should be seven colors
to the spectrum, to correspond to the notes of a diatonic scale.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Mikal De Valia" <chiptruth@e...>
wrote:
> >
> > circle of fifths--color wheel of art theory.
> >
> > anyone know the correspondence, or can figger it?
> >
> > assumption: A=440 equals primary red

Ah, one of my favorite "trivia" questions!

You're close -- it's A-flat:
/tuning/topicId_56687.html#56714
"A" actually sounds "orange." :-)

> The idea goes back to Newton, who decided there should be seven
colors
> to the spectrum, to correspond to the notes of a diatonic scale.

Since the visible spectrum corresponds to somewhat less than
an "octave" of frequencies, I'm sure that the idea has occurred to
many that doubling or halving the frequency of light would, if we
could see it, be perceived as maintaining the same color-class,
analogous to octaves of a tone being in the same pitch-class.

But that still leaves open the question of what color should
correspond to what pitch. There is a way of determining this, using
the principle of octave equivalence, effectively eliminates the
element of subjectivity:

/tuning/topicId_56687.html#56717

--George

http://www.Chrysalis-Foundation.org/Turkish_Comma_Notation.pdf

--- In tuning@yahoogroups.com, Cris Forster <76153.763@c...> wrote:
> Consider a descending Spiral of Fifths
> that begins at the "octave," ratio 2/1.
>
> 1st descending 3/2 is
> 4/3 = 498.0 ¢.
>
> 2nd descending 3/2 is
> 16/9 = 996.1 ¢.
>
> Then --
>
> 8th descending 3/2 is
> 8192/6561 = 384.4 ¢
> or approx. 5/4 = 386.3 ¢.
>
> ******************************
>
> 22nd descending 3/2 is
> 34359738368/31381059609 = 156.99 ¢
> or approx. 12/11 = 150.64 ¢.
>
> One may also derive this ratio by reducing 9/8 by 2 commas:
> (9/8)/(531441/524288)^2 = 34359738368/31381059609
>
> ******************************
>
> 34th descending 3/2 is
> 18014398509481984/16677181699666569 = 133.53 ¢
> or approx. 13/12 = 138.57 ¢.
>
> One may also derive this ratio by reducing 9/8 by 3 commas:
> (9/8)/(531441/524288)^3 =
> 18014398509481984/16677181699666569
>