Yahoo Tuning Groups Ultimate Backup tuning Arithmetic divisions of a string

Dear brother, I have divided not the whole of the string into 72 to parts, but only the distance between the fundamental tone and its octave. This arithmetic division yields the previously mentioned 72-tone rational tuning when you calculate the relative frequencies:

365 hz/360 hz = 73/72,
375 hz/360 hz = 25/24,
480 hz/360 hz = 4/3,
etc...

all independent of a pitch standard, and all derived with 5 hz increments from 360 hz onward.

Can you then calculate what happens when you divide the string's whole lenght into 72? You will arrive at the 36 of the tones within the octave comprised by the 72-tone system I have already given.

Cordially,
Ozan
----- Original Message -----
From: Mohajeri Shahin
To: tuning@yahoogroups.com
Sent: 24 Aralık 2005 Cumartesi 6:45
Subject: RE: [tuning] Bir teli 12 eşit parçaya böldüğümüzde çıkan sonuç

Dear ozan

But the system you mentioned is not system of equally dividing the length of string to 72 part. Also , what is the relationship between frequency-difference of 5 hz and equal divisions of string as you know that relationship between them is stright line :

F1 , f2,f3,………2f1

F1=f1

F2 = f1 +5

F3=f2+5=f1+10

L (f2f1)= f1/f2 = f1/f1+5

L(f3f2) = f2/f3 = f1+5/f1+10

You know that logic of systems such as 72/72 ….. 72/36 is equally dividing the length of string , which farabi used for his intervallic experiments.

Shaahin Mohaajeri

Tombak Player & Researcher , Composer

www.geocities.com/acousticsoftombak

My tombak musics : www.rhythmweb.com/gdg

My articles in ''Harmonytalk'':

www.harmonytalk.com/archives/000296.html

www.harmonytalk.com/archives/000288.html

My article in DrumDojo:

www.drumdojo.com/world/persia/tonbak_acoustics.htm

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:
>
> Dear brother, I have divided not the whole of the string into 72 to
>parts, but only the distance between the fundamental tone and its
>octave. This arithmetic division yields the previously mentioned 72-
>tone rational tuning when you calculate the relative frequencies:
>
> 365 hz/360 hz = 73/72,
> 375 hz/360 hz = 25/24,
> 480 hz/360 hz = 4/3,
> etc...
>
> all independent of a pitch standard, and all derived with 5 hz
>increments from 360 hz onward.

If I may rudely intereject -- This is not what you get when you
divide the string into equal parts; rather, you're using equal
frequency increments, a very different thing.

> Can you then calculate what happens when you divide the string's
>whole lenght into 72? You will arrive at the 36 of the tones within
>the octave comprised by the 72-tone system I have already given.

I beg to differ. Dividing the string's length into 72 equal parts
will yield this set of frequency ratios relative to the open string:

72/71
36/35
24/23
18/17
72/67
12/11
72/65
9/8
8/7
36/31
72/61
6/5
72/59
36/29
24/19
9/7
72/55
4/3
72/53
18/13
24/17
36/25
72/49
3/2
72/47
36/23
8/5
18/11
72/43
12/7
72/41
9/5
24/13
36/19
72/37
2
72/35
36/17
24/11
9/4
72/31
12/5
72/29
18/7
8/3
36/13
72/25
3
72/23
36/11
24/7
18/5
72/19
4
72/17
9/2
24/5
36/7
72/13
6
72/11
36/5
8
9
72/7
12
72/5
18
24
36
72

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 25 Aralï¿½k 2005 Pazar 4:07
Subject: [tuning] Re: Arithmetic divisions of a string

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:
> >
> > Dear brother, I have divided not the whole of the string into 72 to
> >parts, but only the distance between the fundamental tone and its
> >octave. This arithmetic division yields the previously mentioned 72-
> >tone rational tuning when you calculate the relative frequencies:
> >
> > 365 hz/360 hz = 73/72,
> > 375 hz/360 hz = 25/24,
> > 480 hz/360 hz = 4/3,
> > etc...
> >
> > all independent of a pitch standard, and all derived with 5 hz
> >increments from 360 hz onward.
>
> If I may rudely intereject -- This is not what you get when you
> divide the string into equal parts; rather, you're using equal
> frequency increments, a very different thing.
>

Enlighten me Paul, I'm an ignorant in this area. What happens when you
physically divide the lenght of a string tuned to 360 hz up to its octave
into 12 equal parts? You are, in effect, dividing the distance between 360
hz to 720 hz into equal increments, no?

> > Can you then calculate what happens when you divide the string's
> >whole lenght into 72? You will arrive at the 36 of the tones within
> >the octave comprised by the 72-tone system I have already given.
>
> I beg to differ. Dividing the string's length into 72 equal parts
> will yield this set of frequency ratios relative to the open string:
>
> 72/71
> 36/35
> 24/23
> 18/17
> 72/67
> 12/11
> 72/65
> 9/8
> 8/7
> 36/31
> 72/61
> 6/5
> 72/59
> 36/29
> 24/19
> 9/7
> 72/55
> 4/3
> 72/53
> 18/13
> 24/17
> 36/25
> 72/49
> 3/2
> 72/47
> 36/23
> 8/5
> 18/11
> 72/43
> 12/7
> 72/41
> 9/5
> 24/13
> 36/19
> 72/37
> 2
> 72/35
> 36/17
> 24/11
> 9/4
> 72/31
> 12/5
> 72/29
> 18/7
> 8/3
> 36/13
> 72/25
> 3
> 72/23
> 36/11
> 24/7
> 18/5
> 72/19
> 4
> 72/17
> 9/2
> 24/5
> 36/7
> 72/13
> 6
> 72/11
> 36/5
> 8
> 9
> 72/7
> 12
> 72/5
> 18
> 24
> 36
> 72
>
>

And what do you get when you divide the distance between a tone and its
octave into 72 aritmetically equal parts on a string?

Cordially,
Ozan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:
>
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning@yahoogroups.com>
> Sent: 25 Aralýk 2005 Pazar 4:07
> Subject: [tuning] Re: Arithmetic divisions of a string
>
>
> > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...>
wrote:
> > >
> > > Dear brother, I have divided not the whole of the string into
72 to
> > >parts, but only the distance between the fundamental tone and its
> > >octave. This arithmetic division yields the previously mentioned
72-
> > >tone rational tuning when you calculate the relative frequencies:
> > >
> > > 365 hz/360 hz = 73/72,
> > > 375 hz/360 hz = 25/24,
> > > 480 hz/360 hz = 4/3,
> > > etc...
> > >
> > > all independent of a pitch standard, and all derived with 5 hz
> > >increments from 360 hz onward.
> >
> > If I may rudely intereject -- This is not what you get when you
> > divide the string into equal parts; rather, you're using equal
> > frequency increments, a very different thing.
> >
>
>
> Enlighten me Paul, I'm an ignorant in this area. What happens when
you
> physically divide the lenght of a string tuned to 360 hz up to its
octave
> into 12 equal parts? You are, in effect, dividing the distance
between 360
> hz to 720 hz into equal increments, no?

No, you are not. (I felt justified in being rude here since Mohajeri
has been pointing out this very difference since almost his first
post here, while you have posted some technical scientific
discussions on the relationship between string length, tension, and
frequency.) Since frequency is inversely proportional to string
length, you get the following frequencies in Hz:

360*24/23 = 375.65
360*24/22 = 392.73
360*24/21 = 411.43
360*24/20 = 432
360*24/19 = 454.74
360*24/18 = 480
360*24/17 = 508.24
360*24/16 = 540
360*24/15 = 576
360*24/14 = 617.14
360*24/13 = 664.62
360*24/12 = 720

> And what do you get when you divide the distance between a tone and
its
> octave into 72 aritmetically equal parts on a string?

1
144/143
72/71
48/47
36/35
144/139
24/23
144/137
18/17
16/15
72/67
144/133
12/11
144/131
72/65
48/43
9/8
144/127
8/7
144/125
36/31
48/41
72/61
144/121
6/5
144/119
72/59
16/13
36/29
144/115
24/19
144/113
9/7
48/37
72/55
144/109
4/3
144/107
72/53
48/35
18/13
144/103
24/17
144/101
36/25
16/11
72/49
144/97
3/2
144/95
72/47
48/31
36/23
144/91
8/5
144/89
18/11
48/29
72/43
144/85
12/7
144/83
72/41
16/9
9/5
144/79
24/13
144/77
36/19
48/25
72/37
144/73
2

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

> Enlighten me Paul, I'm an ignorant in this area. What happens when
you
> physically divide the lenght of a string tuned to 360 hz up to its
octave
> into 12 equal parts? You are, in effect, dividing the distance
between 360
> hz to 720 hz into equal increments, no?

[PA]
No, you are not. (I felt justified in being rude here since Mohajeri
has been pointing out this very difference since almost his first
post here, while you have posted some technical scientific
discussions on the relationship between string length, tension, and
frequency.) Since frequency is inversely proportional to string
length, you get the following frequencies in Hz:

54/54
54/53
54/52
54/51
54/50
54/49
54/48

[OZ]
Ah, I'm beginning to see my mistake. Thanks for elucidating. So, I can
safely conclude that mandals in a Turkish Qanun are affixed according to
this fashion:

(nut)===(mandals)===============================(string)
...|..\..\..\..\..\..\../../../../../../
...0..1..2.3..4..5.6..7..8..9.10.11.12

0-6=100 cents
0-12=200 cents

If the open string is tuned to 260 Hz, the 6th mandal is affixed at:

260 x 1.0594631 = 275.460 hz

The 12th is affixed at:

260 x 1.122462 = 291.840 hz

The distances in between are divided equally and arithmetically into 6 parts
each:

0: 260 hz (1/1)
1: *789,300,000/781,916,667 = 262.455 hz (52491/52000)
2: *789,300,000/774,533,334 = 264.957 hz (264957/260000)
3: *789,300,000/767,150,001 = 267.507 hz (267507/260000)
4: *789,300,000/759,766,668 = 270.107 hz (270107/260000)
5: *789,300,000/752,383,335 = 272.757 hz (272757/260000)
6: *789,300,000/745,000,002 = 275.460 hz (13773/13000), 100 cents

Taking the 6th mandal as the natural tone at 0 cents, we have:

6: 275.460 hz (1/1)
7: 278.061 hz (52491/52000)
8: 280.712 hz (264957/260000)
9: 283.413 hz (267507/260000)
10: 286.168 hz (270107/260000)
11: 288.976 hz (272757/260000)
12: 291.840 hz (13773/13000) 100 cents

The results are indistinguishable from 72-edo:

0: 1/1
1: 52491/52000
2: 264957/260000
3: 267507/260000
4: 270107/260000
5: 272757/260000
6: 13773/13000
7: 278061/260000
8: 35089/32500
9: 21801/20000
10: 35771/32500
11: 18061/16250
12: 1824/1625

Step size is 16.6667 cents
1: 16.270: 1: 16.6667 cents, diff. 0.023790 steps, 0.3965 cents
2: 32.696: 2: 33.3333 cents, diff. 0.038243 steps, 0.6374 cents
3: 49.278: 3: 50.0000 cents, diff. 0.043318 steps, 0.7220 cents
4: 66.023: 4: 66.6667 cents, diff. 0.038602 steps, 0.6434 cents
5: 82.926: 5: 83.3333 cents, diff. 0.024468 steps, 0.4078 cents
6: 100.000: 6: 100.0000 cents, diff. 0.000000 steps, 0.0000 cents
7: 116.268: 7: 116.6667 cents, diff. 0.023935 steps, 0.3989 cents
8: 132.695: 8: 133.3333 cents, diff. 0.038303 steps, 0.6384 cents
9: 149.273: 9: 150.0000 cents, diff. 0.043609 steps, 0.7268 cents
10: 166.021: 10: 166.6667 cents, diff. 0.038747 steps, 0.6458 cents
11: 182.926: 11: 183.3333 cents, diff. 0.024460 steps, 0.4077 cents
12: 200.000: 12: 200.0000 cents, diff. 0.000000 steps, 0.0000 cents
Total absolute difference : 0.337680 steps, 5.6280 cents
Average absolute difference: 0.028140 steps, 0.4690 cents
Root mean square difference: 0.031669 steps, 0.5278 cents
Highest absolute difference: 0.043609 steps, 0.7268 cents

> And what do you get when you divide the distance between a tone and
its
> octave into 72 aritmetically equal parts on a string?

[OZ]

Very interesting.