Topic: Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7.
1 scales
| File | Description | Notes | Period (¢) |
|---|---|---|---|
| bptemp2 | BP tempered using a different generator | 13 | 1902.0 |
Thread (8 messages)
From: Petr Pařízek (2006-10-01) Subject: 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
From: Mohajeri Shahin (2006-10-01) Subject: RE: [tuning] Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7. 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 You can configure your subscription by sending an empty email to one of these addresses (from the address at which you receive the list): tuning-subscribe@yahoogroups.com - join the tuning group. tuning-unsubscribe@yahoogroups.com - leave the group. tuning-nomail@yahoogroups.com - turn off mail from the group. tuning-digest@yahoogroups.com - set group to send daily digests. tuning-normal@yahoogroups.com - set group to send individual emails. tuning-help@yahoogroups.com - receive general help information. Yahoo! Groups Links
From: Petr Pařízek (2006-10-01) 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
From: Mohajeri Shahin (2006-10-01) Subject: RE: [tuning] Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7. 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
From: Petr Pařízek (2006-10-01) 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
From: Mohajeri Shahin (2006-10-01)
Subject: RE: [tuning] Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7.
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
From: Petr Pařízek (2006-10-01) 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
From: Mohajeri Shahin (2006-10-02) Subject: RE: [tuning] Another similarity! Another reason why to use 12 for 1:2:3:5 and 13 for 1:3:5:7. 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