Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7.

Hi all.

For quite a long time, we've been discussing the fact that there are
actually two "well-known" ways to approximate 1:2:3:5 in a 2/1-periodic
tuning, one of them being meantones (tempering out 81/80) and the other one
being schismatic tunings (tempering out 32805/32768). And although either
uses different methods of interval grouping to achieve this 1:2:3:5, they
both can make a "two-intervals-per-class" system in a 12-tone version (in
contrast to other cardinalities which are unique to either system) and, also
in both cases, the generator is the fifth. Interestingly, a striking
similarity appears in the 3/1-periodic tunings approximating 1:3:5:7. There
are also two major ways to do this. One is the classic method which I
believe BP comes from and which makes the mentioned "BP diatonic" of two
disjunct pentachords adding up to 3/1. Though I haven't seen a word about
generators for this, I've learned that the generator comes out as something
slightly smaller than 7/3 and the most important approximant then is the
5/3. I could describe this as something like a "meanthird" system because
the "BP third" is made wider than 9/7 to get closer to "sqrt(5/3)" and
therefore temper out 245/243. But then there's a totally different
possibility yet. You can also decide to temper out 16875/16807 (that's what
you get if you take a 27/5 and subtract five 7/5s). The generator then comes
out as something a bit smaller than 15/7. And, to prove the point, when
comparing this with the classic BP, both of these methods can make a
"two-intervals-per-class" system in a 13-tone version! Again, other
cardinalities are unique to either system. Even the "quasi-diatonic" mode
comes out as a 10-tone scale, not a 9-tone one as in the regular BP. This is
because the "symmetric interval pattern" for regular BP is "slsslssslssls",
while for this different method the symmetric pattern is "llslllslllsll".

Okay, what about some examples? Well, for the 1:2:3:5, some very good ones
can be quarter-comma meantone for one method and the 1/9-schisma temperament
for the other one. As far as 1:3:5:7 is concerned, one example of the
classic "BP style" tempering can be the scale I made in April (you can find
it in the message called "BP as a linear temperament"). And to have an
example of the other method of approaching this, I'm going to show you one
now. The period is 3/1 and the generator is actually a "fifth-kleisma"
smaller than 15/7 (by "kleisma" I mean the 16875/16807). The interesting
thing is that if you approximate 1:3:5:7 in this tuning, the 1:3:5 is pure
and the 7 is just a "fifth-kleisma" higher than in JI. Unfortunately,
similarly to the "schismatic-versus-meantone" comparison, there are less
places where this is true in this scale as compared to how often the regular
BP method does its best approximations of this.

! bptemp2.scl
!
BP tempered using a different generator
13
!
150.22423
300.44846
433.68603
583.91026
734.13449
5/3
9/5
1167.82052
1318.04474
1468.26897
1601.50655
1751.73077
3/1

Petr

And what about this one :

0
125.0044
250.0088
400.6458
525.6502
650.6546
775.659
926.296
1051.3004
1176.3048
1301.3092
1451.9462
1576.9506
1701.955

With this Intervallic symmetry pattern of aabaaabaaabaa:

125.0044
125.0044
12/11
125.0044
125.0044
125.0044
12/11
125.0044
125.0044
125.0044
12/11
125.0044
125.0044

Shaahin Mohaajeri

Tombak Player & Researcher , Microtonal Composer

My web site

My page in Harmonytalk

My tombak musics in Rhythmweb

My article in DrumDojo

My musics in Wikipedia, the free encyclopedia :

- A composition based on a folk melody of Shiraz region, in shur-dastgah by Mohajeri Shahin

- An experiment in Iranian homayun and chahargah modes by Mohajeri Shahin

-----Original Message-----
From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Petr Pa??zek
Sent: Sunday, October 01, 2006 9:51 AM
To: Tuning List
Subject: [tuning] Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7.

Hi all.

For quite a long time, we've been discussing the fact that there are
actually two "well-known" ways to approximate 1:2:3:5 in a 2/1-periodic
tuning, one of them being meantones (tempering out 81/80) and the other one
being schismatic tunings (tempering out 32805/32768). And although either
uses different methods of interval grouping to achieve this 1:2:3:5, they
both can make a "two-intervals-per-class" system in a 12-tone version (in
contrast to other cardinalities which are unique to either system) and, also
in both cases, the generator is the fifth. Interestingly, a striking
similarity appears in the 3/1-periodic tunings approximating 1:3:5:7. There
are also two major ways to do this. One is the classic method which I
believe BP comes from and which makes the mentioned "BP diatonic" of two
disjunct pentachords adding up to 3/1. Though I haven't seen a word about
generators for this, I've learned that the generator comes out as something
slightly smaller than 7/3 and the most important approximant then is the
5/3. I could describe this as something like a "meanthird" system because
the "BP third" is made wider than 9/7 to get closer to "sqrt(5/3)" and
therefore temper out 245/243. But then there's a totally different
possibility yet. You can also decide to temper out 16875/16807 (that's what
you get if you take a 27/5 and subtract five 7/5s). The generator then comes
out as something a bit smaller than 15/7. And, to prove the point, when
comparing this with the classic BP, both of these methods can make a
"two-intervals-per-class" system in a 13-tone version! Again, other
cardinalities are unique to either system. Even the "quasi-diatonic" mode
comes out as a 10-tone scale, not a 9-tone one as in the regular BP. This is
because the "symmetric interval pattern" for regular BP is "slsslssslssls",
while for this different method the symmetric pattern is "llslllslllsll".

Okay, what about some examples? Well, for the 1:2:3:5, some very good ones
can be quarter-comma meantone for one method and the 1/9-schisma temperament
for the other one. As far as 1:3:5:7 is concerned, one example of the
classic "BP style" tempering can be the scale I made in April (you can find
it in the message called "BP as a linear temperament"). And to have an
example of the other method of approaching this, I'm going to show you one
now. The period is 3/1 and the generator is actually a "fifth-kleisma"
smaller than 15/7 (by "kleisma" I mean the 16875/16807). The interesting
thing is that if you approximate 1:3:5:7 in this tuning, the 1:3:5 is pure
and the 7 is just a "fifth-kleisma" higher than in JI. Unfortunately,
similarly to the "schismatic-versus-meantone" comparison, there are less
places where this is true in this scale as compared to how often the regular
BP method does its best approximations of this.

! bptemp2.scl
!
BP tempered using a different generator
13
!
150.22423
300.44846
433.68603
583.91026
734.13449
5/3
9/5
1167.82052
1318.04474
1468.26897
1601.50655
1751.73077
3/1

Petr

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Hi Shaahin.

I'm afraid I don't quite understand. What is your approximant? What are you
tempering out with this? Where did you get the 1701.955 cents?

Petr

Hi Petr

For a non-octavic scale with 13 unequal , unsorted hybrid divisions with a symmetric free pattern and with 2 diffrent sizes of intervals, I got this scale as below :

1701.955= 10a + 3b

choosing b as 12/11 will result a=125.0044

You can construct scales by choosing size of intervals (a,b,c,…)and their rhythmic patterns within octave or other intervals.

Shaahin Mohaajeri

Tombak Player & Researcher , Microtonal Composer

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

My page in Harmonytalk <http://www.harmonytalk.com/id/908>

My tombak musics in Rhythmweb <http://www.rhythmweb.com/gdg>

My article in DrumDojo <http://www.drumdojo.com/world/persia/tonbak_acoustics.htm>

My musics in Wikipedia, the free encyclopedia :

- A composition based on a folk melody of Shiraz region, in shur-dastgah by Mohajeri Shahin <http://www.xenharmony.org/mp3/shaahin/shur.mp3>

- An experiment in Iranian homayun and chahargah modes by Mohajeri Shahin <http://www.xenharmony.org/mp3/shaahin/homayun.mp3>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Petr Pa??zek
Sent: Sunday, October 01, 2006 11:31 AM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7.

Hi Shaahin.

I'm afraid I don't quite understand. What is your approximant? What are you
tempering out with this? Where did you get the 1701.955 cents?

Petr

Okay, but why a period of right 1701.955 cents and not something else? Where
did you get that interval?

Petr

Can't choose What ever we want (-: for period of a scale to repeat its intervallic structure?

Another one for a non-octavic scale with 7 unsorted hybrid divisions with a symmetric free pattern and with 2 diffrent unequal sizes of intervals:

0

b 282.5 282.5

a 90 372.5

b 282.5 655

a 90 745

b 282.5 1027.5

a 90 1117.5

b 282.5 1400

when repeated we have :

0

b 282.5 282.5

a 90 372.5

b 282.5 655

a 90 745

b 282.5 1027.5

a 90 1117.5

b 282.5 1400

b 282.5 1682.5

a 90 1772.5

b 282.5 2055

a 90 2145

b 282.5 2427.5

a 90 2517.5

b 282.5 2800

but if we consider a pattern of babababa…….. we have a period of 1490 cent , the next period is 1862.5 and the other one is 2235 and so on ….

0

b 282.5 282.5

a 90 372.5

b 282.5 655

a 90 745

b 282.5 1027.5

a 90 1117.5

b 282.5 1400

a 90 1490

b 282.5 1772.5

a 90 1862.5

b 282.5 2145

a 90 2235

b 282.5 2517.5

a 90 2607.5

b 282.5 2890

a 90 2980

b 282.5 3262.5

a 90 3352.5

b 282.5 3635

a 90 3725

b 282.5 4007.5

a 90 4097.5

b 282.5 4380

a 90 4470

b 282.5 4752.5

Shaahin Mohaajeri

Tombak Player & Researcher , Microtonal Composer

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

My page in Harmonytalk <http://www.harmonytalk.com/id/908>

My tombak musics in Rhythmweb <http://www.rhythmweb.com/gdg>

My article in DrumDojo <http://www.drumdojo.com/world/persia/tonbak_acoustics.htm>

My musics in Wikipedia, the free encyclopedia :

- A composition based on a folk melody of Shiraz region, in shur-dastgah by Mohajeri Shahin <http://www.xenharmony.org/mp3/shaahin/shur.mp3>

- An experiment in Iranian homayun and chahargah modes by Mohajeri Shahin <http://www.xenharmony.org/mp3/shaahin/homayun.mp3>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Petr Pa??zek
Sent: Sunday, October 01, 2006 3:11 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7.

Okay, but why a period of right 1701.955 cents and not something else? Where
did you get that interval?

Petr

Right. I do understand all of your concepts but it's not clear to me if you
got such periods like 1400 or 1701.955 cents by approximating some other
intervals or by simply "picking up" an interval of your choice. That's why I
was asking.

Petr

Hi petr

You can choose the period simply by your choice.

Shaahin Mohaajeri

Tombak Player & Researcher , Microtonal Composer

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

My page in Harmonytalk <http://www.harmonytalk.com/id/908>

My tombak musics in Rhythmweb <http://www.rhythmweb.com/gdg>

My article in DrumDojo <http://www.drumdojo.com/world/persia/tonbak_acoustics.htm>

My musics in Wikipedia, the free encyclopedia :

- A composition based on a folk melody of Shiraz region, in shur-dastgah by Mohajeri Shahin <http://www.xenharmony.org/mp3/shaahin/shur.mp3>

- An experiment in Iranian homayun and chahargah modes by Mohajeri Shahin <http://www.xenharmony.org/mp3/shaahin/homayun.mp3>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Petr Pa??zek
Sent: Sunday, October 01, 2006 6:58 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7.

Right. I do understand all of your concepts but it's not clear to me if you
got such periods like 1400 or 1701.955 cents by approximating some other
intervals or by simply "picking up" an interval of your choice. That's why I
was asking.

Petr