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what EDO to simulate pythagorean scale?

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

11/19/2006 4:57:03 AM

Hi all

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.

For example :

Is 154-EDO a good simulation 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.

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>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

11/19/2006 10:55:31 AM

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

> 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.

If we take denominators of the convergents of log2(3), we get
the equal divisions which give the best pythagorean scales, relatively
speaking. These go 2, 5, 12, 41, 53, 306, 665... The corresponding
size of the Pythagorean comma is -1200, 240, 0, 29.268, 22.642,
23.529, 23.459..., as compared to the actual value of 23.460 cents.

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

11/19/2006 11:32:45 PM

Hi gene

By considering (M = Cardinality of EDO) and (N = Degree of fifth in EDO) I define two coefficient :

1- FJC = Fifth justness coefficient as N * Log3(2) /(M+N)

2- PC = pythagoreanism coefficient as Fifth of EDO-(700+(Comma in EDO/12))

According to these two coefficient, 665-EDO is the best cardinality to simulate Pythagorean scale with these coefficients :.

- FGC = 1.00000006

- PC = 0

The best cardinality must show FGC of 1 and PC of 0.

For those EDOs which have PC of 0 if FGC ------> 1 we have better simulation for Pythagorean scale.

see the result for edos with PC of 0 , here but for more you can see a related excel spread sheet here :

http://www.esnips.com/web/Shaahinsmicrotonaldocumentfolder

EDO ....……. ……. FJC..... ..………. PC

12 ………. 1.001028948 ………. 0

24 ………. 1.001028948 ………. 0

65 ………. 1.000219054 ………. 0

118 ………. 1.000136765 ………. 0

171 ………. 1.000105489 ………. 0

224 ………. 1.000089015 ………. 0

277 ………. 1.000078844 ………. 0

330 ………. 1.000071941 ………. 0

383 ………. 1.000066949 ………. 0

53 ………. 1.000035864 ………. 0

106 ………. 1.000035864 ………. 0

159 ………. 1.000035864 ………. 0

212 ………. 1.000035864 ………. 0

265 ………. 1.000035864 ………. 0

318 ………. 1.000035864 ………. 0

371 ………. 1.000035864 ………. 0

424 ………. 1.000035864 ………. 0

477 ………. 1.000035864 ………. 0

530 ………. 1.000035864 ………. 0

583 ………. 1.000035864 ………. 0

636 ………. 1.000035864 ………. 0

689 ………. 1.000035864 ………. 0

1048 ………. 1.000024504 ………. 0

995 ………. 1.000023899 ………. 0

942 ………. 1.000023226 ………. 0

889 ………. 1.000022472 ………. 0

836 ………. 1.000021623 ………. 0

783 ………. 1.000020659 ………. 0

730 ………. 1.000019555 ………. 0

677 ………. 1.000018279 ………. 0

624 ………. 1.000016785 ………. 0

1195 ………. 1.000015939 ………. 0

571 ………. 1.000015014 ………. 0

1142 ………. 1.000015014 ………. 0

1089 ………. 1.000014000 ............ 0

518 ………. 1.000012881 ………. 0

1036 ………. 1.000012881 ………. 0

983 ………. 1.000011642 ………. 0

465 ………. 1.000010262 ………. 0

930 ………. 1.000010262 ………. 0

877 ………. 1.000008714 ………. 0

412 ………. 1.000006968 ………. 0

824 ………. 1.000006968 ………. 0

1183 ………. 1.000005674 ………. 0

771 ………. 1.000004982 ………. 0

1130 ………. 1.000004258 ………. 0

359 ………. 1.000002703 ………. 0

718 ………. 1.000002703 ………. 0

1077 ………. 1.000002703 ………. 0

1024 ………. 1.000000986 ………. 0

665 ………. 1.000000060 ………. 0

971 ………. 0.999999083 ………. 0

306 ………. 0.999996959 ………. 0

612 ………. 0.999996959 ………. 0

918 ………. 0.999996959 ………. 0

1171 ………. 0.999995198 ………. 0

865 ………. 0.999994576 ………. 0

559 ………. 0.999993271 ………. 0

1118 ………. 0.999993271 ………. 0

812 ………. 0.999991881 ………. 0

1065 ………. 0.999991151 ………. 0

759 ………. 0.999988810 ………. 0

253 ………. 0.999988810 ………. 0

506 ………. 0.999988810 ………. 0

1012 ………. 0.999988810 ………. 0

959 ………. 0.999986209 ………. 0

706 ………. 0.999985277 ………. 0

1159 ………. 0.999984506 ………. 0

453 ………. 0.999983305 ………. 0

906 ………. 0.999983305 ………. 0

1106 ………. 0.999982046 ………. 0

653 ………. 0.999981172 ………. 0

853 ………. 0.999980039 ………. 0

1053 ………. 0.999979337 ………. 0

200 ………. 0.999976341 ………. 0

400 ………. 0.999976341 ………. 0

600 ………. 0.999976341 ………. 0

800 ………. 0.999976341 ………. 0

1000 ………. 0.999976341 ………. 0

947 ………. 0.999973010 ………. 0

747 ………. 0.999972118 ………. 0

547 ………. 0.999970574 ………. 0

347 ………. 0.999967250 ………. 0

694 ………. 0.999967250 ………. 0

494 ………. 0.999963570 ………. 0

641 ………. 0.999961578 ………. 0

441 ………. 0.999954882 ………. 0

147 ………. 0.999954882 ………. 0

294 ………. 0.999954882 ………. 0

388 ………. 0.999943822 ………. 0

241 ………. 0.999937075 ………. 0

335 ………. 0.999929261 ………. 0

94 ………. 0.999909229 ………. 0

188 ………. 0.999909229 ………. 0

282 ………. 0.999909229 ………. 0

135 ………. 0.999859521 ………. 0

41 ………. 0.999745577 ………. 0

82 ………. 0.999745577 ………. 0

29 ………. 0.999215490 ………. 0

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: Sunday, November 19, 2006 10:26 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: what EDO to simulate pythagorean scale?

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

> 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.

🔗Danny <dawiertx@sbcglobal.net>

11/20/2006 2:33:25 AM

Mohajeri,

Here's a list of ETs/EDOs that have the most precise fifths. The numbers on the left are the number of equal steps in an octave; on the right, how much the fifths in each EDO deviate from the true fifth in percent of a step in the EDO. Positive percentages mean the EDO fifth is sharp; negative means flat. They're ordered from worst to best, and I listed only tunings as good or better than 12-EDO and no scales larger than 1200-EDO.

Note that one of the best tunings is 1024-EDO, and 2^10 = 1024. And sorry about the bad text formatting.

12 -1.954985
983 -1.812494
318 -1.807094
624 -1.659203
359 -1.532912
930 -1.511312
265 -1.505911
571 -1.358020
877 -1.210129
212 -1.204729
1183 -1.062238
518 -1.056838
824 -0.908947
159 -0.9035468
1130 -0.7610559
465 -0.7556558
771 -0.6077647
106 -0.6023645
1077 -0.4598737
412 -0.4544735
718 -0.3065825
53 -0.3011823
1024 -0.1586914
665 -0.005400181

653 1.949584
347 1.801693
1012 1.796293
41 1.653802
706 1.648402
400 1.500511
1065 1.495111
94 1.352620
759 1.347220
453 1.199329
1118 1.193929
147 1.051438
812 1.046038
506 0.8981466
1171 0.8927464
200 0.7502556
865 0.7448554
559 0.5969644
253 0.4490733
918 0.4436731
612 0.2957821
306 0.1478910
971 0.1424909

~D.

----- Original Message ----- From: Mohajeri Shahin
To: tuning@yahoogroups.com ; tuning-math@yahoogroups.com
Sent: Sunday, 19 November, 2006 06:57
Subject: [tuning] what EDO to simulate pythagorean scale?

Hi all

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.
For example :
Is 154-EDO a good simulation 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.

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

11/20/2006 5:01:39 AM

Hi danny

According to :

/tuning/topicId_68056.html#68060

and my spread sheet in : http://www.esnips.com/web/Shaahinsmicrotonaldocumentfolder <http://www.esnips.com/web/Shaahinsmicrotonaldocumentfolder>

665 is the best cardinality to simulate Pythagorean scale . we can also see edos with fifth and commas greater than it like 306 and smaller than it like 1025.

I believe that The size of comma and relation between (comma/12) and size of fifth is very important , so we see that 53-EDO is very far from 665-EDO.

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 Danny
Sent: Monday, November 20, 2006 2:03 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] what EDO to simulate pythagorean scale?

Mohajeri,

Here's a list of ETs/EDOs that have the most precise fifths. The numbers on
the left are the number of equal steps in an octave; on the right, how much
the fifths in each EDO deviate from the true fifth in percent of a step in
the EDO. Positive percentages mean the EDO fifth is sharp; negative means
flat. They're ordered from worst to best, and I listed only tunings as good
or better than 12-EDO and no scales larger than 1200-EDO.

Note that one of the best tunings is 1024-EDO, and 2^10 = 1024. And sorry
about the bad text formatting.

~D.

----- Original Message -----
From: Mohajeri Shahin
To: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ; tuning-math@yahoogroups.com <mailto:tuning-math%40yahoogroups.com>
Sent: Sunday, 19 November, 2006 06:57
Subject: [tuning] what EDO to simulate pythagorean scale?

Hi all

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.
For example :
Is 154-EDO a good simulation 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.

🔗a_sparschuh <a_sparschuh@yahoo.com>

11/20/2006 6:19:09 AM

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> If we take denominators of the convergents of log2(3), we get
> the equal divisions which give the best pythagorean scales, relatively
> speaking. These go 2, 5, 12, 41, 53, 306, 665... The corresponding
> size of the Pythagorean comma is -1200, 240, 0, 29.268, 22.642,
> 23.529, 23.459..., as compared to the actual value of 23.460 cents.
>
further appoximations of that series in:
http://www.research.att.com/~njas/sequences/A005664
"
1, 1, 2, 5, 12, 41, 53, 306, 665, 15601, 31867, 79335, 111202, 190537,
10590737, 10781274, 53715833, 171928773, 225644606, 397573379,
6189245291, 6586818670, 65470613321, 137528045312, 753110839881,
5409303924479, 6162414764360"EDO... whoever needs/wants such ETs?
should visit:
http://www.eddaardvark.co.uk/t3a1/int_log32.htm

Discussions:
http://mathforum.org/kb/message.jspa?messageID=16668&tstart=134565
http://groups.google.com.my/group/sci.math/browse_thread/thread/a5a932563ec4e56e/feb8aa721a6aaeae?hl=en

also appearing in Collatz "3x+1 cycles" problems
http://www.ericr.nl/wondrous/cycles.html
http://www.geocities.com/mattiksinisalo/collatz.doc

http://www.research.att.com/~njas/sequences/DUNNE/TEMPERAMENT.HTML

in german:
http://wissen-welt.de/posting_700_1135181746_elemenarer-algorithmus-zur-berechnung-des-zweierlogarithmus-gesucht-.htm#
Enzyklopädisch:
http://www.riat-serra.org/musik-3.pdf

but I do prefer simply the fraction of an just pure 5th:
3/2=1.5
instead bareley irrational approximations of that rational number,
alike Werckmeister stayed within them also too.
A.S.

🔗Petr Pařízek <p.parizek@chello.cz>

11/20/2006 2:46:01 AM

Danny wrote:

> Here's a list of ETs/EDOs that have the most precise fifths. The numbers
on
> the left are the number of equal steps in an octave; on the right, how
much
> the fifths in each EDO deviate from the true fifth in percent of a step in
> the EDO. Positive percentages mean the EDO fifth is sharp; negative means
> flat. They're ordered from worst to best, and I listed only tunings as
good
> or better than 12-EDO and no scales larger than 1200-EDO.

Isn't 212, 159 and 106 all the same as 53?

Petr

🔗Danny <dawiertx@sbcglobal.net>

11/20/2006 8:22:39 AM

At most, 53-tET deviates from Pythagorean by 1.773 cents (at degrees 11 and 42, or 26 fifths up or down), so it's still quite accurate, unless you really want dead-on precision which you get with 665-EDO. The fifth in 53-tET is flat of the true fifth by only 0.0682 cents.

But don't forget that 612-EDO is great not only for high-precision Pythagorean), but for 11-limit JI. It's been talked about a lot on the list.

~D.

----- Original Message ----- From: Mohajeri Shahin
To: tuning@yahoogroups.com
Sent: Monday, 20 November, 2006 07:01
Subject: RE: [tuning] what EDO to simulate pythagorean scale?

Hi danny

According to :
/tuning/topicId_68056.html#68060
and my spread sheet in : http://www.esnips.com/web/Shaahinsmicrotonaldocumentfolder
665 is the best cardinality to simulate Pythagorean scale . we can also see edos with fifth and commas greater than it like 306 and smaller than it like 1025.
I believe that The size of comma and relation between (comma/12) and size of fifth is very important , so we see that 53-EDO is very far from 665-EDO.

🔗a_sparschuh <a_sparschuh@yahoo.com>

11/20/2006 10:56:44 AM

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:
>
> I believe that The size of comma and relation between (comma/12) and
size of fifth is very important , so we see that 53-EDO is very far
from 665-EDO.
>
>>>> 665 -0.005400181
>
It appears that the 665 division was probably used already long ago:

http://www.xs4all.nl/~huygensf/doc/measures.html
"Delfi unit: 1/665 part of an octave
Used in Byzantine music theory? Approximately 1/12 part of the
syntonic comma and 1/13 part of the Pythagorean comma. "

that's today's greek:
http://www.musicheaven.gr/html/modules.php?name=News&file=print&sid=432
12 .................N#ON3 54;N5ON1ON<N-N=N? NOOO ;ON1O
N:O N7N<N9OON=N9ο
53 .................N:ON<N<N1 ON?O NN5ON:N,ON?ON1
68 .................NON1N2N9N:N. N<N?N=N,N4α, N2ON6N1N=ON9N=O N7OοN<OON9N?
72 .................NON6αN=ON9N=O N7ON?N<OO 53;N?
301 ...............Savart
665 ...............Delfi unit
1200 ..............cent
http://www.musicheaven.gr/html/modules.php?name=News&file=print&sid=432
http://www.mathsforyou.gr/plugins/preview/index.php?option=com_content&task=view&id=600&Itemid=275

http://www.phys.uoa.gr/~nektar/arts/music/hellenic_byzantine_music.htm

Prime factor decompostion:
665:=19*7*5
hence 665ET contains also the ETs: 5,7,19,35,95,133 within itself.
due to that, the
http://www.google.de/search?hl=de&q=satanic-comma+665&btnG=Suche&meta=
can be expressed in terms of limmata and apotomata:

limma:= 256/243 = 2^8/3^5
using (256/243)^133 = 1 023.9553
yields 1024/((256/243)^133) = 1.00004366 ~0.0755754826...Cents

apotome:= 2187/2048 = 3^7/2^11
using (2187/2048)^95 = 512.022351
yields ((2187/2048)^95)/512 = 1.00004366 ~1/13 Cents

an remarkable well approximation of the prime interval 1/1.
hence I do see no need for ETs bigger than 665edo.
even i.m.o. the classical Byzantine DU:=2^(1/665)
is even more apt fitting than the less natural chosen Cent:=2^(1/1200).
What about to change from Cents DUs?
A.S.

🔗Danny <dawiertx@sbcglobal.net>

11/20/2006 3:25:53 PM

Petr Pařízek wrote:

> Danny wrote:
>
>> Here's a list of ETs/EDOs that have the most precise fifths. The numbers
> on
>> the left are the number of equal steps in an octave; on the right, how
> much
>> the fifths in each EDO deviate from the true fifth in percent of a step >> in
>> the EDO. Positive percentages mean the EDO fifth is sharp; negative means
>> flat. They're ordered from worst to best, and I listed only tunings as
> good
>> or better than 12-EDO and no scales larger than 1200-EDO.
>
> Isn't 212, 159 and 106 all the same as 53?

They would be if you're calculating the error of the fifth in cents, but I was measuring error as percentage of a step in the particular EDO, so it would be twice as much for 106 than it is for 53.

~D.

🔗Danny <dawiertx@sbcglobal.net>

11/20/2006 3:28:13 PM

a_sparschuh wrote:

> --- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:
>>
>> I believe that The size of comma and relation between (comma/12) and
> size of fifth is very important , so we see that 53-EDO is very far
> from 665-EDO.
>>
>>>>> 665 -0.005400181
>>
> It appears that the 665 division was probably used already long ago:

I was hoping there was a better name than "Satanic comma". But why is it called "delfi" or "delphi"?

~D.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

11/20/2006 4:15:48 PM

--- In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@...> wrote:

> an remarkable well approximation of the prime interval 1/1.
> hence I do see no need for ETs bigger than 665edo.
> even i.m.o. the classical Byzantine DU:=2^(1/665)
> is even more apt fitting than the less natural chosen Cent:=2^(1/1200).
> What about to change from Cents DUs?
> A.S.

I think 665 is serious overkill in the 3-limit at the expense of
higher limits. I would choose 612 in preference; in fact I did.

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

11/20/2006 11:44:35 PM

All these edos have the same FJC and PC :

106

159

212

265

318

371

424

477

530

583

636

689

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 Petr Pa??zek
Sent: Monday, November 20, 2006 2:16 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] what EDO to simulate pythagorean scale?

Danny wrote:

Isn't 212, 159 and 106 all the same as 53?

Petr

🔗Petr Pařízek <p.parizek@chello.cz>

11/21/2006 12:50:28 AM

Shaahin wrote:

> All these edos have the same FJC and PC :

This is because all these EDOs use 31/53-octave as the period. And that's
why I was asking: What is it good for to use 530-EDO if you always choose
multiples of 10 steps anyway? Isn't it better to use 53-EDO in the first
place then?

Petr

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

11/21/2006 2:09:56 AM

665-EDO is also good for 19-limmit with a high precision , but 612-EDO is slightely better :

Consider 11-ADO:

1/1….….….0

12/11….150.6370585

13/11….289.2097194

14/11….417.5079641

15/11….536.9507724

16/11….648.6820576

17/11….753.6374671

18/11….852.5920594

19/11….946.1950738

20/11….1034.995772

21/11….1119.462965

22/11….1200.000

Cent difference with 665-EDO:

-0.8626

-0.4879

-0.6659

0.7936

-0.8625

0.6482

-0.8627

-0.6312

0.7937

-0.6660

0.0000

Cent difference with 612-EDO:

0.3433

-0.9744

0.1391

0.3041

0.3376

-0.6963

0.3491

0.8637

0.2983

0.1449

0.0000

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 Danny
Sent: Monday, November 20, 2006 7:53 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] what EDO to simulate pythagorean scale?

At most, 53-tET deviates from Pythagorean by 1.773 cents (at degrees 11 and
42, or 26 fifths up or down), so it's still quite accurate, unless you
really want dead-on precision which you get with 665-EDO. The fifth in
53-tET is flat of the true fifth by only 0.0682 cents.

But don't forget that 612-EDO is great not only for high-precision
Pythagorean), but for 11-limit JI. It's been talked about a lot on the list.

~D.

----- Original Message -----
From: Mohajeri Shahin
To: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
Sent: Monday, 20 November, 2006 07:01
Subject: RE: [tuning] what EDO to simulate pythagorean scale?

Hi danny

According to :
/tuning/topicId_68056.html#68060 </tuning/topicId_68056.html#68060>
and my spread sheet in :
http://www.esnips.com/web/Shaahinsmicrotonaldocumentfolder <http://www.esnips.com/web/Shaahinsmicrotonaldocumentfolder>
665 is the best cardinality to simulate Pythagorean scale . we can also see
edos with fifth and commas greater than it like 306 and smaller than it
like 1025.
I believe that The size of comma and relation between (comma/12) and size of
fifth is very important , so we see that 53-EDO is very far from 665-EDO.

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

11/21/2006 3:16:51 AM

If I understood your question correctly because of my bad English ;-):

The more the degrees of edo , the better we have intervals resulted from fifth chains. so 530-EDO is better than 53 to simulate chains of fifth.

now another thing:

in these edos we may have different choices for comma but only one of them must be considered to have PC=0 , like 530edo or in 1060-edo we have different choices for comma but only the common comma of n*53 (22.64151 cent) fit with PC=0.

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 Petr Pa??zek
Sent: Tuesday, November 21, 2006 12:20 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] what EDO to simulate pythagorean scale?

Shaahin wrote:

> All these edos have the same FJC and PC :

Petr

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

11/21/2006 4:13:06 AM

Attachments

Unfortunately I cant read greek words.

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 a_sparschuh
Sent: Monday, November 20, 2006 10:27 PM
To: tuning@yahoogroups.com
Subject: [tuning] 665 division; "satanic"-comma, was Re: what EDO to simulate pythagorean scale?

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> , "Mohajeri Shahin" <shahinm@...> wrote:
>
> I believe that The size of comma and relation between (comma/12) and
size of fifth is very important , so we see that 53-EDO is very far
from 665-EDO.
>
>>>> 665 -0.005400181
>
It appears that the 665 division was probably used already long ago:

http://www.xs4all.nl/~huygensf/doc/measures.html <http://www.xs4all.nl/~huygensf/doc/measures.html>
"Delfi unit: 1/665 part of an octave
Used in Byzantine music theory? Approximately 1/12 part of the
syntonic comma and 1/13 part of the Pythagorean comma. "

that's today's greek:
http://www.musicheaven.gr/html/modules.php?name=News&file=print&sid=432 <http://www.musicheaven.gr/html/modules.php?name=News&file=print&sid=432>
12 .................Συγκερασμένο Ευρωπαϊκό ημιτόνιο
53 .................κόμμα του Μερκάτορα
68 .................Αραβική μονάδα, βυζαντινό ηχομόριο
72 .................Βυζαντινό ηχομόριο
301 ...............Savart
665 ...............Delfi unit
1200 ..............cent
http://www.musicheaven.gr/html/modules.php?name=News&file=print&sid=432 <http://www.musicheaven.gr/html/modules.php?name=News&file=print&sid=432>
http://www.mathsforyou.gr/plugins/preview/index.php?option=com_content&task=view&id=600&Itemid=275 <http://www.mathsforyou.gr/plugins/preview/index.php?option=com_content&task=view&id=600&Itemid=275>

http://www.phys.uoa.gr/~nektar/arts/music/hellenic_byzantine_music.htm <http://www.phys.uoa.gr/~nektar/arts/music/hellenic_byzantine_music.htm>

Prime factor decompostion:
665:=19*7*5
hence 665ET contains also the ETs: 5,7,19,35,95,133 within itself.
due to that, the
http://www.google.de/search?hl=de&q=satanic-comma+665&btnG=Suche&meta= <http://www.google.de/search?hl=de&q=satanic-comma+665&btnG=Suche&meta=>
can be expressed in terms of limmata and apotomata:

limma:= 256/243 = 2^8/3^5
using (256/243)^133 = 1 023.9553
yields 1024/((256/243)^133) = 1.00004366 ~0.0755754826...Cents

apotome:= 2187/2048 = 3^7/2^11
using (2187/2048)^95 = 512.022351
yields ((2187/2048)^95)/512 = 1.00004366 ~1/13 Cents

🔗Petr Pařízek <p.parizek@chello.cz>

11/21/2006 9:45:18 AM

Shaahin wrote:

> The more the degrees of edo , the better we have intervals resulted from
fifth chains. so 530-EDO is better than 53 to simulate chains of fifth.

But 530 is not better than 53. In both of them, the nearest to 3/2 is the
31/53-octave. That means that the approximation of Pythagorean tuning is
"equally good" in both of these tunings. So I don't know why it should be
important to make difference between 31/53-octave and 310/530-octave if this
is actually the very same interval, only written in different step sizes.

> in these edos we may have different choices for comma but only one of them
must be considered to have PC=0 , like 530edo or in 1060-edo we have
different choices for comma but only the common comma of n*53 (22.64151
cent) fit with PC=0.

Now I'm not sure if I've understood you correctly. You can't approximate
Pythagorean tuning by finding the nearest comma but by finding the nearest
fifth approximating 3/2. And if, for example, the nearest fifth in 530-EDO
takes 310 steps, you realize that 310/530 is the same as 31/53. That means
530-EDO is no better than 53-EDO for approximating Pythagorean tuning
because the fifth is just the same in both of these.

Petr

🔗monz <monz@tonalsoft.com>

11/21/2006 10:26:47 AM

If your concern is to get a better approximation of 3/2
from an EDO, you can find several which are better than
530-edo which have far lower cardinality. One convergent
series gives these three candidates:

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.

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

--- In tuning@yahoogroups.com, Petr PaÅÃzek <p.parizek@...> wrote:
>
> Shaahin wrote:
>
> > The more the degrees of edo , the better we have
> > intervals resulted from fifth chains. so 530-EDO is
> > better than 53 to simulate chains of fifth.
>
> But 530 is not better than 53. In both of them, the nearest
> to 3/2 is the 31/53-octave. That means that the approximation
> of Pythagorean tuning is "equally good" in both of these
> tunings. So I don't know why it should be important to
> make difference between 31/53-octave and 310/530-octave if
> this is actually the very same interval, only written in
> different step sizes.
>
> > in these edos we may have different choices for comma
> > but only one of them must be considered to have PC=0 ,
> > like 530edo or in 1060-edo we have different choices
> > for comma but only the common comma of n*53 (22.64151
> > cent) fit with PC=0.
>
> Now I'm not sure if I've understood you correctly. You
> can't approximate Pythagorean tuning by finding the nearest
> comma but by finding the nearest fifth approximating 3/2.
> And if, for example, the nearest fifth in 530-EDO takes
> 310 steps, you realize that 310/530 is the same as 31/53.
> That means 530-EDO is no better than 53-EDO for approximating
> Pythagorean tuning because the fifth is just the same in
> both of these.
>
> Petr

🔗monz <monz@tonalsoft.com>

11/21/2006 10:52:57 AM

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> If your concern is to get a better approximation of 3/2
> from an EDO, you can find several which are better than
> 530-edo which have far lower cardinality. One convergent
> series gives these three candidates:
>
> 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.

I also meant to say that there is no better approximation
to the 3/2 ratio between 53-edo and 200-edo.

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

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

11/21/2006 10:19:05 PM

Hi petr

1- Can you find an interval approximating (3^72)/( 2^114) =140.760 cent in 53-edo with high precision . but I have it in 530-EDO as 140.377358

And in 665-EDO as 140.75188.the nearest in 53-EDO is 135.849057 which is between (3^19) and (3^ -33) . So 530 , 665 , .... approximate higher degrees of fifth chain very and very better than 53 , although their relation is like a tree with its branches.

2- can you tell me what is the difference of 142-EDO and 253-EDO to approximate chain of fifths according to importance of only fifth?

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 Petr Pa??zek
Sent: Tuesday, November 21, 2006 9:15 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] what EDO to simulate pythagorean scale?

Shaahin wrote:

> The more the degrees of edo , the better we have intervals resulted from
fifth chains. so 530-EDO is better than 53 to simulate chains of fifth.

Petr

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

11/21/2006 10:35:09 PM

Hi monzo

All of these have PC=o

And also :

EDO……..……..FJC……..……..size of fifth……..……..PC

665……..1.00000006……..701.9548872……..23.45864662

971……..0.999999083……..701.9567456……..3.48094748

306……..0.999996959……..701.9607843……..23.52941176

612……..0.999996959……..701.9607843……..23.52941176

918……..0.999996959……..701.9607843……..23.52941176

1171……..0.999995198……..701.9641332……..23.56959863

865……..0.999994576……..701.9653179……..23.58381503

559……..0.999993271……..701.9677996……..23.61359571

1118……..0.999993271……..701.9677996……..23.61359571

812……..0.999991881……..701.9704433……..23.6453202

1065……..0.999991151……..701.971831……..23.66197183

759……..0.99998881……..701.9762846……..23.71541502

253……..0.99998881……..701.9762846……..23.71541502

For FJC and PC : http://www.esnips.com/web/Shaahinsmicrotonaldocumentfolder <http://www.esnips.com/web/Shaahinsmicrotonaldocumentfolder>

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: Tuesday, November 21, 2006 9:57 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: what EDO to simulate pythagorean scale?

117 degrees of 200-edo = 702 cents
148 degrees of 253-edo = ~701.9762846 cents
179 degrees of 306-edo = ~701.9607843 cents

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

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> , Petr PaÅ™Ãzek <p.parizek@...> wrote:
>
> Shaahin wrote:
>
> > The more the degrees of edo , the better we have
> > intervals resulted from fifth chains. so 530-EDO is
> > better than 53 to simulate chains of fifth.
>
> But 530 is not better than 53. In both of them, the nearest
> to 3/2 is the 31/53-octave. That means that the approximation
> of Pythagorean tuning is "equally good" in both of these
> tunings. So I don't know why it should be important to
> make difference between 31/53-octave and 310/530-octave if
> this is actually the very same interval, only written in
> different step sizes.
>
> > in these edos we may have different choices for comma
> > but only one of them must be considered to have PC=0 ,
> > like 530edo or in 1060-edo we have different choices
> > for comma but only the common comma of n*53 (22.64151
> > cent) fit with PC=0.
>
> Now I'm not sure if I've understood you correctly. You
> can't approximate Pythagorean tuning by finding the nearest
> comma but by finding the nearest fifth approximating 3/2.
> And if, for example, the nearest fifth in 530-EDO takes
> 310 steps, you realize that 310/530 is the same as 31/53.
> That means 530-EDO is no better than 53-EDO for approximating
> Pythagorean tuning because the fifth is just the same in
> both of these.
>
> Petr

🔗Petr Pařízek <p.parizek@chello.cz>

11/21/2006 10:51:07 PM

Shaahin wrote:

> 1- Can you find an interval approximating (3^72)/( 2^114) =140.760 cent
in 53-edo with high precision . but I have it in 530-EDO as 140.377358
>
> And in 665-EDO as 140.75188.the nearest in 53-EDO is 135.849057 which is
between (3^19) and (3^ -33) . So 530 , 665 , .... approximate higher degrees
of fifth chain very and very better than 53 , although their relation is
like a tree with its branches.

For 665, I say "yes". But for 530, I say "no". What you're saying here seems
like your way of approximating Pythagorean tuning in 530-EDO is
inconsistent. If you had based your approximations on the fifths, you'd
realize that you couldn't reach this interval at all then.

> 2- can you tell me what is the difference of 142-EDO and 253-EDO to
approximate chain of fifths according to importance of only fifth?

I don't understand your question.

Petr

🔗ciarán maher <ciaran@rhizomecowboy.com>

11/22/2006 10:10:56 AM

hi all

sorry to bug you about this, but i was really hoping someone could recommend a retunable softsynth/vst plugin i could use to make demos of new variations of tenney's spectral canon. i'm hoping to record the finished pieces on a midi grand piano which we can properly retune, but for now, i just need to test.

i have the midi files generated and running in logic. just need a reasonable piano sound i can retune (to the 1st 24 partials of a harmonic series).

can anyone help

🔗monz <monz@tonalsoft.com>

11/22/2006 3:44:12 PM

Hi Mohajeri,

--- In tuning@yahoogroups.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>
Tonescape microtonal music software