Topic: Pythagorean basis of 24-edo
1 scales
| File | Description | Notes | Period (¢) |
|---|---|---|---|
| sha | Three chains of sqrt(3/2) separated by 10/7 | 24 | 1200.0 |
Thread (8 messages)
From: Mohajeri Shahin (2007-01-09) Subject: Pythagorean basis of 24-edo Hi all Is there any pyth.basis for 24-edo.what about chain of (3/2)^(1/2)? Shaahin Mohajeri Tombak Player & Researcher , Microtonal Composer My web siteوب سايت شاهين مهاجري <http://240edo.googlepages.com/> My farsi page in Harmonytalk صفحه 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>
From: Gene Ward Smith (2007-01-10) Subject: Re: Pythagorean basis of 24-edo --- In [email protected], "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
From: monz (2007-01-10) Subject: Re: Pythagorean basis of 24-edo Hi Mohajeri, --- In [email protected], "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 Tonescape microtonal music software
From: Cameron Bobro (2007-01-10) Subject: Re: Pythagorean basis of 24-edo --- In [email protected], "Mohajeri Shahin" <shahinm@...> wrote: > > > Hi all > Is there any pyth.basis for 24-edo.what about chain of (3/2)^(1/2)? > > Shaahin Mohajeri 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. -Cameron Bobro
From: Mohajeri Shahin (2007-01-10) Subject: RE: [tuning] Re: Pythagorean basis of 24-edo 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: [email protected] [mailto:[email protected]] On Behalf Of monz Sent: Wednesday, January 10, 2007 12:52 PM To: [email protected] Subject: [tuning] Re: Pythagorean basis of 24-edo Hi Mohajeri, --- In [email protected] <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
From: yahya_melb (2007-01-11) Subject: Re: Pythagorean basis of 24-edo 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
From: Cameron Bobro (2007-01-11) Subject: Re: Pythagorean basis of 24-edo --- In [email protected], "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
From: Cameron Bobro (2007-01-14) Subject: Re: Pythagorean basis of 24-edo --- In [email protected], "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 [email protected], "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: [email protected] [mailto:[email protected]] On Behalf Of monz > Sent: Wednesday, January 10, 2007 12:52 PM > To: [email protected] > Subject: [tuning] Re: Pythagorean basis of 24-edo > > > > Hi Mohajeri, > > --- In [email protected] <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 >