Yahoo Tuning Groups Ultimate Backup tuning to monzo and petr : RE: [tuning] Re: what EDO to simulate pythagorean scale? and about 530-edo

Hi monzo and petr

As I told before:

For any EDO , showing an interval (X) near to Pythagorean comma :

If we have a fifth as 700+((X)/12) , so it is a good approximation for pythagoean scale or chains of 3/2.

For example :

Is 154-EDO a good approximation for pythagoean scale?

In 154-EDO we have third degree equal to 23.37662 cent as comma , so our fifth must be 701.94805 but it isn't , fifth is 701.2987 cent.

So it isn't a good EDO to approximate pythagoean scale.

But what about 412-EDO , showing a degree of 23.30097 cent?

The fifth must be 701.94174 and we have it as 241st degree , so a good simulation for pythagoean scale. This is also true for 53 , 253 , 306 , 171, …. EDO.

Last day I tried with edos , finding that :

1- some edos with pc=0 like 359-edo after 359 chains of fifth will reach to 1200 cent ( using scala commands : extend and reduce) also correct for 171 , 53 , 453 , 1024 , 306 , ….

2- For Some edo with PC not zero ( negative) we see that:

- 513-EDO after 171(513 / 3) chains of fifth will reach to 1200.

- 990-edo after 330 chains of fifth……..

- 1190-edo after 595 chains of fifth……

3- For Some edo with PC not zero ( positive) we see that:

- 1188-EDO after 1188 chains of fifth

- 552-edo after 552 chains of fifth we see 1200 cent

4- For pythagorean scale or chiains of 3/2 which JFC=1 and PC=0 ,

- we see that after 665 cycle we will get a cent of 0.076 ( or reach to 1200.076) which present the best edo for simulation.

- we see that after 1024 cycle we will get a cent of 1.921 ( or reach to 1201.921) which present another good edo for simulation.

- we see that after 971 cycle we will get a cent of 1198.306 which present another good edo for simulation.

- we see that after 53 cycle we will get a cent of 3.615 ( or reach to 1203.615) which present another good edo for simulation.

These edos have PC=0 and FJC----> 1

So I found that the chain of 3/2 will show the best edos for simulation. Compare my FJC and PC with these result:

cardinalities of edos with PC=0 and FJC----> 1are degrees of chains of 3/2 which approximate 1200 cent .

and about 530-edo , according to above statements :

petr is right , considering n*53 , the greater ( n) , the worst our simulation.

Shaahin Mohajeri

Tombak Player & Researcher , Microtonal Composer

My web siteوب سايت شاهين مهاجري <http://240edo.tripod.com/>

My farsi page in Harmonytalkصفحه اختصاصي در هارموني تاك <http://www.harmonytalk.com/mohajeri>

Shaahin Mohajeri in Wikipedia شاهين مهاجري دردائره المعارف ويكي پديا <http://en.wikipedia.org/wiki/Shaahin_mohajeri> <http://www.harmonytalk.com/id/908>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of monz
Sent: Thursday, November 23, 2006 3:14 AM
To: tuning@yahoogroups.com
Subject: [tuning] Re: what EDO to simulate pythagorean scale?

Hi Mohajeri,

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> , "Mohajeri Shahin" <shahinm@...> wrote:
>
> Hi monzo
>
>
>
> All of these have PC=o
>
> >
> > 117 degrees of 200-edo = 702 cents
> > 148 degrees of 253-edo = ~701.9762846 cents
> > 179 degrees of 306-edo = ~701.9607843 cents
> >
> > The next closer series of convergents gives the same
> > approximations with double the cardinality (i.e., 400-,
> > 506-, and 612-edo), with other EDOs between and then
> > continuing beyond 612-edo with 665-edo, which is
> > extremely close to pythagorean tuning.

Not quite sure what you mean ... it seems like you're
saying that all of these temper out the pythagorean-comma,
but actually none of them do.

The size of steps in the pythagorean-comma for
200-, 253-, 306-, and 665-edo is respectively
4, 5, 6, and 13 steps.

-monz
http://tonalsoft.com <http://tonalsoft.com> <http://tonalsoft.com <http://tonalsoft.com> >
Tonescape microtonal music software

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:

> If we have a fifth as 700+((X)/12) , so it is a good approximation
for pythagoean scale or chains of 3/2.
>
> For example :
>
> Is 154-EDO a good approximation for pythagoean scale?
>
> In 154-EDO we have third degree equal to 23.37662 cent as comma ,
so our fifth must be 701.94805 but it isn't , fifth is 701.2987 cent.

Are you asking when does the best fifth of an edo lead to the best
Pythagorean comma? In the case of 154, the best fifth gives a
Pythagorean comma of 1/77 octave, or 15.584 cents, whereas the closest
we get to a Pythagorean comma is actually 3/154 octave, or 23.377 cents.

Here's a list of edo where it does, going up to 306. Note the
denominators of convergents to log2(3) I talked about are always on
this list for n>5. The same is not true for semiconvergents;
semiconvergents can be not on the list, and things on the list might
not be semiconvergents.

12, 24, 29, 41, 53, 65, 82, 94, 106, 118, 135, 147, 159, 171, 188,
200, 212, 224, 241, 253, 265, 277, 282, 294, 306

🔗Mohajeri Shahin <shahinm@kayson-ir.com>

Hi gene

No , considering relation between 3/2 and pyth.comma , it is very clear from edo structure what cardinality show the best pythagoren approximation by FJC and PC but using the degrees of the circle of 3/2 which nearly close at 1200 is very useful to get the final result :

Degree...........FJC....................PC...........Closing octave at degree (related to number of fifth cycles)

282...........0.999909229...........0........... -48.69

29...........0.999215490........... 0 ........... -43.305

82...........0.999745577...........0........... -39.69

135...........0.999859521...........0........... -36.075

188...........0.999909229...........0........... -32.46

241...........0.999937075...........0........... -28.845

294...........0.999954882...........0........... -25.23

41...........0.999745577...........0........... -19.845

94...........0.999909229...........0........... -16.23

147..........0.999954882...........0........... -12.615

200...........0.999976341...........0........... -9

253...........0.99998881...........0........... -5.385

306...........0.999996959...........0........... -1.77<<

665...........1.00000006...........0........... 0.075<<

53...........1.000035864...........0........... 3.615<<

106...........1.000035864...........0........... 7.23<<<<<

159...........1.000035864...........0........... 10.845<<<<<

212...........1.000035864...........0........... 14.46<<<<<

265...........1.000035864...........0........... 18.075<<<<<

12...........1.001028948...........0........... 23.46

65...........1.000219054...........0........... 27.075

118...........1.000136765...........0........... 30.69

171...........1.000105489...........0........... 34.305

224...........1.000089015...........0........... 37.92

277...........1.000078844...........0........... 41.535

24...........1.001028948...........0........... 46.92

( 53 is better than 106 and so on )

My FJC <http://www.esnips.com/web/Shaahinsmicrotonaldocumentfolder/> ( Fifth justness coefficient = N * Log3(2)/(M+N)) is inspired by the denominator of a rational approximation of log2(3) as seen in : The Mathematics of Musical Scales <http://home.comcast.net/~olga.radko/UCLA-fiat-lux-small.ppt_files/UCLA-fiat-lux-small.ppt.ppt>

Shaahin Mohajeri

Tombak Player & Researcher , Microtonal Composer

My web siteوب سايت شاهين مهاجري <http://240edo.tripod.com/>

My farsi page in Harmonytalkصفحه اختصاصي در هارموني تاك <http://www.harmonytalk.com/mohajeri>

Shaahin Mohajeri in Wikipedia شاهين مهاجري دردائره المعارف ويكي پديا <http://en.wikipedia.org/wiki/Shaahin_mohajeri> <http://www.harmonytalk.com/id/908>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Gene Ward Smith
Sent: Saturday, November 25, 2006 1:37 AM
To: tuning@yahoogroups.com
Subject: to monzo and petr : RE: [tuning] Re: what EDO to simulate pythagorean scale? and

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> , "Mohajeri Shahin" <shahinm@...> wrote:

12, 24, 29, 41, 53, 65, 82, 94, 106, 118, 135, 147, 159, 171, 188,
200, 212, 224, 241, 253, 265, 277, 282, 294, 306

to monzo and petr : RE: [tuning] Re: what EDO to simulate pythagorean scale? and about 530-edo

🔗Mohajeri Shahin <shahinm@kayson-ir.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Mohajeri Shahin <shahinm@kayson-ir.com>

🔗monz <monz@tonalsoft.com>