Yahoo Tuning Groups Ultimate Backup tuning Pythagorean basis of 24-edo

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:
>
>
> Hi all
> Is there any pyth.basis for 24-edo.what about chain of (3/2)^(1/2)?

Chains of (3/2)^(1/2) are quite interesting. It splits the difference
between the two septimal neutral thirds of 49/40 and 60/49, and so it
is almost inevitable that you temper out 2401/2400--which is quite
small. If you allow yourself to take multiple chains of neutral
thirds, then 10/7 times 49/40 is 7/4, and 7/4 times another 10/7 is
5/4. So three chains, rather than two, of (3/2)^(1/2) is preferable.
Three chains of size eight each for a total of 24 could be considered
a scale of the breed temperament, but because of all the 225/224
relationshops, it could also be considered a highly irregualar
miracle I suppose. There are various other temperamrnts and scales I
could mention in this connection.

Anyway, for whoever might find it interesting here is this Shahin-
inspired scale:

! sha.scl
Three chains of sqrt(3/2) separated by 10/7
24
!
34.975615
119.442808
147.067499
182.043113
203.910002
238.885617
266.510307
350.977500
385.953115
470.420309
498.044999
533.020614
617.487807
701.955001
736.930616
764.555306
821.397809
849.022500
883.998114
968.465308
1052.932501
1087.908116
1115.532807
1200.000000

🔗monz <monz@tonalsoft.com>

🔗Cameron Bobro <misterbobro@yahoo.com>

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

1-if considering "grad" as (23.46/12)or 12th root of pyth.comma , we can consider something like "semigrad" as (23.46/24)or 24th root of pyth.comma.
2-Considering chains of (3/2)^(1/2) we can have this result:

....................................Cent ..............After temp.
Degree in chain ..............0 .............. 0
-23 .............. 33.382 .............. 50
7 .............. 56.842 .............. 50
-22 .............. 90.2249 .............. 100
14 .............. 113.685 .............. 100
-21 .............. 147.0675.............. 150
21 .............. 170.5275.............. 150
-20 .............. 180.45 .............. 200
4 .............. 203.91 .............. 200
-19 .............. 237.2925.............. 250
11 .............. 260.7525.............. 250
-18 .............. 294.135 .............. 300
18 .............. 317.595 .............. 300
-17 .............. 327.5175.............. 350
1 .............. 350.9775.............. 350
-16 .............. 384.36 ............. 400
8 .............. 407.82 .............. 400
-15 .............. 441.2025.............. 450
15 .............. 464.6625.............. 450
-14 .............. 498.045 .............. 500
22 .............. 521.505 .............. 500
-13 .............. 531.4275.............. 550
5 .............. 554.8875.............. 550
-12 .............. 588.27 .............. 600
12 .............. 611.73 .............. 600
-11 .............. 645.1125.............. 650
19 .............. 668.5725.............. 650
-10 .............. 678.495 .............. 700
2 .............. 701.955 .............. 700
-9 .............. 735.3375.............. 750
9 .............. 758.7975.............. 750
-8 .............. 792.18 .............. 800
16 .............. 815.64 .............. 800
-7 .............. 849.0225.............. 850
23 .............. 872.4825.............. 850
-6 .............. 882.405 .............. 900
6 .............. 905.865 .............. 900
-5 .............. 939.2475.............. 950
13 .............. 962.7075.............. 950
-4 .............. 996.09 .............. 1000
20 .............. 1019.55 .............. 1000
-3 .............. 1029.4725.............. 1050
3 .............. 1052.9325.............. 1050
-2 .............. 1086.315.............. 1100
10 .............. 1109.775.............. 1100
-1 .............. 1143.1575.............. 1150
17 .............. 1166.6175.............. 1150
24 .............. 1223.46 .............. 1200

So we see that in this result we have all intervals in chain of 3/2 and two size for quarter tones , (like lima and appotom).
We have also here "schisma of philolaus".and this (3/2)^(1/2) is not a new thing , in http://198.66.217.172/monzo/aristoxenus/318tet.htm we have " mese - hemiolic chromatic lichanos" as "3 semitones + enharmonic diesis" measured 350.978 cent or :
(3/4)*(256/243)*((2187/2048)^(1/2)) 0.816497 ~-350.978 hemiolic chromatic lichanos

Shaahin Mohajeri

Tombak Player & Researcher , Microtonal Composer

My web site?? ???? ????? ?????? <http://240edo.googlepages.com/>

My farsi page in Harmonytalk ???? ??????? ?? ??????? ??? <http://www.harmonytalk.com/mohajeri>

Shaahin Mohajeri in Wikipedia ????? ?????? ??????? ??????? ???? ???? <http://en.wikipedia.org/wiki/Shaahin_mohajeri>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of monz
Sent: Wednesday, January 10, 2007 12:52 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: Pythagorean basis of 24-edo

Hi Mohajeri,

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> , "Mohajeri Shahin" <shahinm@...> wrote:
>
>
> Hi all
> Is there any pyth.basis for 24-edo.what about chain of (3/2)^(1/2)?

Very unlikely that pythagorean tuning would historically
have ever been considered as a basis for 24-edo, because
one finds a pythagorean interval resembling a quartertone
only at (3/2)^24 = ~46.9 cents.

The first few points at which a pythagorean cycle create
a set of pitches resembling an EDO are 12, 41, and 53.
53-edo is close enough to pythagorean for most practical
purposes that it's usually not necessary to find a higher
cardinality EDO approximation other than for experimental
reasons.

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

🔗yahya_melb <yahya@melbpc.org.au>

🔗Cameron Bobro <misterbobro@yahoo.com>

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
>
> Shaahin Mohajeri asked:
> > Is there any pyth.basis for 24-edo.what about chain of (3/2)^
(1/2)?
>
>
> Cameron Bobro replied:
> > It seems to me that, historically speaking, divisions of the
> tetrachord are where to look first. In other words, the basic
> structure was established in a pythogorean manner, then people
> starting filling in between frets and nudging frets around, within
> the tetrachord. Establish the minor and major third via pyth.,
then
> drop a fret inbetween, etc. When you step back afterward and look
at
> the resulting octave as a whole, 24-EDO would be a fair grid to
> describe the result. Just an idea.
>
>
> Hi Cameron,
>
> Intriguing! Could you flesh this idea out a little, with some
> actual, historically-appropriate numbers?
>
> Regards,
> Yahya

One minute after typing that post I remembered that someone had
mentioned John Chalmer's book "Divisions of the Tetrachord" being on
the Net, so I started printing it out and reading it for the first
time. And there it all is, wonderful. I'll have to read through
and find the specific quotes that much better describe what I meant,
then post them.

-Cameron Bobro

🔗Cameron Bobro <misterbobro@yahoo.com>

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

> Intriguing! Could you flesh this idea out a little, with some
> actual, historically-appropriate numbers?

Yahya,

Take a look at page 96 of Divisions of the Tetrachord, there is
exactly what I meant, with a specific example.

And below it looks like Shaahin has found what he's looking for?
That's very nice, man!

-Cameron Bobro

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:
>
> Hi
>
> 1-if considering "grad" as (23.46/12)or 12th root of pyth.comma ,
we can consider something like "semigrad" as (23.46/24)or 24th root
of pyth.comma.
> 2-Considering chains of (3/2)^(1/2) we can have this result:
>
> ....................................Cent ..............After
temp.
> Degree in chain ..............0 .............. 0
> -23 .............. 33.382 .............. 50
> 7 .............. 56.842 .............. 50
> -22 .............. 90.2249 .............. 100
> 14 .............. 113.685 .............. 100
> -21 .............. 147.0675.............. 150
> 21 .............. 170.5275.............. 150
> -20 .............. 180.45 .............. 200
> 4 .............. 203.91 .............. 200
> -19 .............. 237.2925.............. 250
> 11 .............. 260.7525.............. 250
> -18 .............. 294.135 .............. 300
> 18 .............. 317.595 .............. 300
> -17 .............. 327.5175.............. 350
> 1 .............. 350.9775.............. 350
> -16 .............. 384.36 ............. 400
> 8 .............. 407.82 .............. 400
> -15 .............. 441.2025.............. 450
> 15 .............. 464.6625.............. 450
> -14 .............. 498.045 .............. 500
> 22 .............. 521.505 .............. 500
> -13 .............. 531.4275.............. 550
> 5 .............. 554.8875.............. 550
> -12 .............. 588.27 .............. 600
> 12 .............. 611.73 .............. 600
> -11 .............. 645.1125.............. 650
> 19 .............. 668.5725.............. 650
> -10 .............. 678.495 .............. 700
> 2 .............. 701.955 .............. 700
> -9 .............. 735.3375.............. 750
> 9 .............. 758.7975.............. 750
> -8 .............. 792.18 .............. 800
> 16 .............. 815.64 .............. 800
> -7 .............. 849.0225.............. 850
> 23 .............. 872.4825.............. 850
> -6 .............. 882.405 .............. 900
> 6 .............. 905.865 .............. 900
> -5 .............. 939.2475.............. 950
> 13 .............. 962.7075.............. 950
> -4 .............. 996.09 .............. 1000
> 20 .............. 1019.55 .............. 1000
> -3 .............. 1029.4725.............. 1050
> 3 .............. 1052.9325.............. 1050
> -2 .............. 1086.315.............. 1100
> 10 .............. 1109.775.............. 1100
> -1 .............. 1143.1575.............. 1150
> 17 .............. 1166.6175.............. 1150
> 24 .............. 1223.46 .............. 1200
>
> So we see that in this result we have all intervals in chain of
3/2 and two size for quarter tones , (like lima and appotom).
> We have also here "schisma of philolaus".and this (3/2)^(1/2) is
not a new thing , in
http://198.66.217.172/monzo/aristoxenus/318tet.htm we have " mese -
hemiolic chromatic lichanos" as "3 semitones + enharmonic diesis"
measured 350.978 cent or :
> (3/4)*(256/243)*((2187/2048)^(1/2)) 0.816497 ~-
350.978 hemiolic chromatic lichanos
>
>
> Shaahin Mohajeri
>
> Tombak Player & Researcher , Microtonal Composer
>
> My web site?? ???? ????? ?????? <http://240edo.googlepages.com/>
>
> My farsi page in Harmonytalk ???? ??????? ?? ??????? ???
<http://www.harmonytalk.com/mohajeri>
>
> Shaahin Mohajeri in
Wikipedia ????? ?????? ??????? ??????? ???? ????
<http://en.wikipedia.org/wiki/Shaahin_mohajeri>
>
>
>
> ________________________________
>
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On
Behalf Of monz
> Sent: Wednesday, January 10, 2007 12:52 PM
> To: tuning@yahoogroups.com
> Subject: [tuning] Re: Pythagorean basis of 24-edo
>
>
>
> Hi Mohajeri,
>
> --- In tuning@yahoogroups.com <mailto:tuning%
40yahoogroups.com> , "Mohajeri Shahin" <shahinm@> wrote:
> >
> >
> > Hi all
> > Is there any pyth.basis for 24-edo.what about chain of (3/2)^
(1/2)?
>
> Very unlikely that pythagorean tuning would historically
> have ever been considered as a basis for 24-edo, because
> one finds a pythagorean interval resembling a quartertone
> only at (3/2)^24 = ~46.9 cents.
>
> The first few points at which a pythagorean cycle create
> a set of pitches resembling an EDO are 12, 41, and 53.
> 53-edo is close enough to pythagorean for most practical
> purposes that it's usually not necessary to find a higher
> cardinality EDO approximation other than for experimental
> reasons.
>
> -monz
> http://tonalsoft.com <http://tonalsoft.com>
> Tonescape microtonal music software
>

Pythagorean basis of 24-edo

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

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗monz <monz@tonalsoft.com>

🔗Cameron Bobro <misterbobro@yahoo.com>

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

🔗yahya_melb <yahya@melbpc.org.au>

🔗Cameron Bobro <misterbobro@yahoo.com>

🔗Cameron Bobro <misterbobro@yahoo.com>