A (mere?) speculation

A Speculation
=============

Consider the octave 22 Hz-44 Hz. Twenty two "shruti-s" will comprise this octave.

The frequency of each "shruti" will be cycles in whole numbers -- no fractions. If you take twenty two "shruti-s" to the octave, this is the only octave where we get "shruti-s" as cycles in whole numbers.

Their ratios will be:

22 Hz ..... 1/1 ....... 1.0000000...... 0 cent
23 ..... 23/22 ..... 1.0454545...... 76.95640490365868 cents
24 ..... 24/22 ..... 1.0909090...... 150.63705850063062
25 ..... 25/22 ..... 1.1363636...... 221.3094853649131
26 ..... 26/22 ..... 1.1818181...... 289.209719404554
27 ..... 27/22 ..... 1.2272727...... 354.54706023140557
28 ..... 28/22 ..... 1.2727272...... 417.5079641043682
29 ..... 29/22 ..... 1.3181818...... 478.2592517883298
30 ..... 30/22 ..... 1.3636363...... 536.9507723654654
31 ..... 31/22 ..... 1.4090909...... 593.7176300994936
32 ..... 32/22 ..... 1.4545454...... 648.6820576352434
33 ..... 33/22 ..... 1.5000000...... 701.9550008653874
34 ..... 34/22 ..... 1.5454545...... 753.6374671356506
35 ..... 35/22 ..... 1.5909090...... 803.8216779692029
36 ..... 36/22 ..... 1.6363636...... 852.5920593660184
37 ..... 37/22 ..... 1.6818181...... 900.0260963899831
38 ..... 38/22 ..... 1.7272727...... 946.195073767546
39 ..... 39/22 ..... 1.7727272...... 991.1647202699413
40 ..... 40/22 ..... 1.8181818...... 1034.995771500078
41 ..... 41/22 ..... 1.8636363...... 1077.7444631769436
42 ..... 42/22 ..... 1.9090909...... 1119.4629649697558
43 ..... 43/22 ..... 1.9545454...... 1160.199763277761
44 ..... 44/22 ..... 2.0000000...... 1200 cents

Is this cycles-in-whole-numbers the reason why the lower end of the human hearing range is [close to] 22 Hz?

No doubt, the ratios obtained above will give rise to several interesting scales and observations.

Help from mathamaticians on the Tuning Group, please?

Regards,
Haresh.

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:

> A Speculation
> =============
>
> Consider the octave 22 Hz-44 Hz. Twenty two "shruti-s"
> will comprise this octave.
>
> The frequency of each "shruti" will be cycles in
> whole numbers -- no fractions. If you take twenty two
> "shruti-s" to the octave, this is the only octave where
> we get "shruti-s" as cycles in whole numbers.
>
> Their ratios will be:
> <table snipped>
>
> Is this cycles-in-whole-numbers the reason why the
> lower end of the human hearing range is [close to] 22 Hz?

no. the low end of the hearing range varies with
each individual, and also for one individual the bottom
end gets higher and higher with age and exposure to
loud sounds.

i've tested my own hearing on my computer and have
been able to distinguish pitch as low as 16 Hz.
below that, it just sounds like rhythmic clicks.

> No doubt, the ratios obtained above will give rise to
> several interesting scales and observations.
>
> Help from mathamaticians on the Tuning Group, please?

http://launch.groups.yahoo.
com/group/tuning_files/files/monz/bakshi_srutis-as-hz.gif

OR

http://tinyurl.com/6og8e

(you have to be signed in to Yahoo)

you can see by looking at the pitch-height graph that
this scale has very little in common with most scales
that have been described for Indian music.

while it does have a perfect 3:2 "5th", and does give
notes which make a good "minor-3rd" and "tritone", it
lacks anything resembling a "major-2nd" (tone),
"major-3rd", or "perfect-4th".

-monz

Haresh BAKSHI schrieb:

> A Speculation
> =============
> > Consider the octave 22 Hz-44 Hz. Twenty two "shruti-s" will comprise this octave.
> > The frequency of each "shruti" will be cycles in whole numbers -- no fractions. If you take twenty two "shruti-s" to the octave, this is the only octave where we get "shruti-s" as cycles in whole numbers.
> > Their ratios will be:
> > 22 Hz ..... 1/1 ....... 1.0000000...... 0 cent
> 23 ..... 23/22 ..... 1.0454545...... 76.95640490365868 cents
> 24 ..... 24/22 ..... 1.0909090...... 150.63705850063062
> 25 ..... 25/22 ..... 1.1363636...... 221.3094853649131
> 26 ..... 26/22 ..... 1.1818181...... 289.209719404554
> 27 ..... 27/22 ..... 1.2272727...... 354.54706023140557
> 28 ..... 28/22 ..... 1.2727272...... 417.5079641043682
> 29 ..... 29/22 ..... 1.3181818...... 478.2592517883298
> 30 ..... 30/22 ..... 1.3636363...... 536.9507723654654
> 31 ..... 31/22 ..... 1.4090909...... 593.7176300994936
> 32 ..... 32/22 ..... 1.4545454...... 648.6820576352434
> 33 ..... 33/22 ..... 1.5000000...... 701.9550008653874
> 34 ..... 34/22 ..... 1.5454545...... 753.6374671356506
> 35 ..... 35/22 ..... 1.5909090...... 803.8216779692029
> 36 ..... 36/22 ..... 1.6363636...... 852.5920593660184
> 37 ..... 37/22 ..... 1.6818181...... 900.0260963899831
> 38 ..... 38/22 ..... 1.7272727...... 946.195073767546
> 39 ..... 39/22 ..... 1.7727272...... 991.1647202699413
> 40 ..... 40/22 ..... 1.8181818...... 1034.995771500078
> 41 ..... 41/22 ..... 1.8636363...... 1077.7444631769436
> 42 ..... 42/22 ..... 1.9090909...... 1119.4629649697558
> 43 ..... 43/22 ..... 1.9545454...... 1160.199763277761
> 44 ..... 44/22 ..... 2.0000000...... 1200 cents
> > Is this cycles-in-whole-numbers the reason why the lower end of the human hearing range is [close to] 22 Hz?

Definitely no. They happen to be whole numbers because
Hertz=cycles per seconds, and nothing in the human brain
relates to the 86,400th part of this planet's revolution.
Human perception (sometimes i'm not sure about mine)
proceeds in considerably shorter units than seconds.

> > No doubt, the ratios obtained above will give rise to several interesting scales and observations. > > Help from mathamaticians on the Tuning Group, please?

Sorry to disappoint you. I'm no mathematician, just another
speculator...

If you rotate this harmonic series to start at 36, you get
13 srutis to a fifth (inverting the fourth 36/27). Here are
the classical sruti scales:

The lower part is always

1/1 10/9 43/36 23/18 3/2
0 cent 182 308 424 702

Ouch, that fourth! Now the upper parts:

Shadja grama:

31/8 17/9
941 1101

Madhyama grama:

5/3 17/9
884 1101

and Gandhara grama:

5/3 11/6
884 1049

Now the same for a pure 4/3 in 9 srutis, the scale rotated
to 27:

Lower part:

1/1 31/27 34/27 4/3 40/27
0 293 399 498 680

Shadja grama:
44/27 50/27
845 1067

Mahyama grama:
43/27 50/27
806 1067

Gandhara grama:

43/27 16/9
806 996

> > Regards,
> Haresh.

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:
> A Speculation
> =============
>
> Consider the octave 22 Hz-44 Hz. Twenty two "shruti-s" will
comprise this octave.
>
> The frequency of each "shruti" will be cycles in whole numbers --
no fractions. If you take twenty two "shruti-s" to the octave, this
is the only octave where we get "shruti-s" as cycles in whole
numbers.
>
> Their ratios will be:
>
> 22 Hz ..... 1/1 ....... 1.0000000...... 0 cent
> 23 ..... 23/22 ..... 1.0454545...... 76.95640490365868
cents
> 24 ..... 24/22 ..... 1.0909090...... 150.63705850063062
> 25 ..... 25/22 ..... 1.1363636...... 221.3094853649131
> 26 ..... 26/22 ..... 1.1818181...... 289.209719404554
...
> 43 ..... 43/22 ..... 1.9545454...... 1160.199763277761
> 44 ..... 44/22 ..... 2.0000000...... 1200 cents
>
...
> No doubt, the ratios obtained above will give rise to several
interesting scales and observations.
>

Am I wrong if I say it'd be an octave divided in 22 equal-beating
shrutis?

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:
> A Speculation
> =============
>
> Consider the octave 22 Hz-44 Hz. Twenty two "shruti-s" will
comprise this octave.
>
> The frequency of each "shruti" will be cycles in whole numbers --
no fractions. If you take twenty two "shruti-s" to the octave, this
is the only octave where we get "shruti-s" as cycles in whole numbers.
>
> Their ratios will be: ...
>
> No doubt, the ratios obtained above will give rise to several
interesting scales and observations.

Hi Haresh,

I have found the 22-division the most useful one below the 29-
division for mapping JI tones. However, one problem I have found
with taking consecutive harmonics (such as you have done) is that the
resulting scale is not a constant structure, e.g., the interval of
3:4 from 24/22 to 32/22 is 8 steps, while the same interval from
33/22 to 44/22 is 11 steps.

I submit, for your consideration, the following scale (in .scl file
format), which not only has the constant-structure property, but
which also may be more harmonically useful in that it contains ten
perfect fifths:

! secor22ji29.scl
!
George Secor's 22-tone just intonation, 29-limit otonality on 4/3
! 15-limit otonality on 1/1
! 7-limit scale 16:18:20:21:24:25:28:32 available starting on 4/3 and
5/3
22
!
25/24
13/12
105/96
9/8
7/6
29/24
5/4
125/96
4/3
11/8
17/12
35/24
3/2
19/12
13/8
5/3
27/16
7/4
11/6
15/8
23/12
2/1

--George