User talk:Xenwolf: Difference between revisions

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:::: In the case of [[9/7]], we have two prime factors to consider- the 3-limit, and the 7-limit. Now, I can already assure you that you can only use one tempered 7-limit interval max before the absolute error exceeds 3.5 cents, as after [[7/4]], the next interval in the 7-limit chain is [[49/32]], and the difference between the JI version and the 159edo-tempered version exceeds 3.5 cents, and this is also true for the 205edo-tempered version of 49/32. However, when two different EDOs have the same number of intervals of a given p-limit that can be stacked before the absolute error exceeds 3.5 cents, it is the absolute error in cents of the tempered stack relative to the JI equvalent that determines which EDO is superior for representing that p-limit, with the better EDO for representation having the smaller absolute error in cents. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 20:05, 8 October 2020 (UTC)

:::: Using [http://musictheory.zentral.zone/huntsystemcalc.html the same calculator] that you linked, I tested how 159edo and 205edo each represent this set of intervals- [[3/2]], [[9/8]], [[27/16]], [[81/64]], [[243/128]], [[729/512]], [[5/4]], [[25/16]], [[125/64]], [[625/512]], [[7/4]], [[49/32]], [[343/256]], [[11/8]], [[121/64]], [[13/8]], [[169/128]], [[17/16]] and [[289/256]]- the ~~stipulation being~~ that the odd-limit of any interval in the set has to be less than 1024. I looked for the number of "P" ratings given by both 205edo and 159edo, as "P" ratings are the only ratings I'm really interested in at this point, and I also looked at their distribution. Both 205edo and 159edo give 7 "P" ratings total out of this set, and are surpassed in this respect by [[147edo]], which has 8. However, one of the P-rated intervals for 147edo is for 343/326, the best approximation of which cannot be reached by stacking three of 147edo's best tempered version of 7/4 and octave reducing, thus resulting in this interval's disqualification. Furthermore, the 147edo-tempered versions of six of the other 7 intervals in the starting interval set given a "P" rating in Hunt's system have absolute errors in cents that are ''greater'' than those of their 159edo-tempered counterparts which are also P-rated- a decisive loss for 147edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:13, 9 October 2020 (UTC)

:::: Using [http://musictheory.zentral.zone/huntsystemcalc.html the same calculator] that you linked, I tested how 159edo and 205edo each represent this set of intervals- [[3/2]], [[9/8]], [[27/16]], [[81/64]], [[243/128]], [[729/512]], [[5/4]], [[25/16]], [[125/64]], [[625/512]], [[7/4]], [[49/32]], [[343/256]], [[11/8]], [[121/64]], [[13/8]], [[169/128]], [[17/16]] and [[289/256]]- and as I'm forced to limit the interval set somehow, I've decided that the odd-limit of any interval in the set has to be less than 1024. I looked for the number of "P" ratings given by both 205edo and 159edo, as "P" ratings are the only ratings I'm really interested in at this point, and I also looked at their distribution. Both 205edo and 159edo give 7 "P" ratings total out of this set, and are surpassed in this respect by [[147edo]], which has 8. However, one of the P-rated intervals for 147edo is for 343/326, the best approximation of which cannot be reached by stacking three of 147edo's best tempered version of 7/4 and octave reducing, thus resulting in this interval's disqualification. Furthermore, the 147edo-tempered versions of six of the other 7 intervals in the starting interval set given a "P" rating in Hunt's system have absolute errors in cents that are ''greater'' than those of their 159edo-tempered counterparts which are also P-rated- a decisive loss for 147edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:13, 9 October 2020 (UTC)

:::: As for the remaining head to head comparison of 159edo and 205edo in terms of how good their representation is, one of the P-rated intervals for 205edo is 49/32, the best approximation of which cannot be reached by stacking two of 205edo's best tempered version of 7/4 and octave reducing, resulting in this interval's disqualification. Four of 205edo's remaining P-rated intervals are all solidly in the in the 5-limit chain, and another P-rated interval is 17/16- the 205edo-tempered version of which is better than the 159edo-tempered version- the final P-rated interval for 205edo is 3/2, and 159edo-tempered version of this interval has ''less'' absolute error in cents than the 205edo-tempered version, leaving 205edo with only 5 P-rated intervals that are not outperformed by their 159edo counterparts. This means a decisive loss for 205edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:40, 9 October 2020 (UTC)

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 :::: In the case of [[9/7]], we have two prime factors to consider- the 3-limit, and the 7-limit.  Now, I can already assure you that you can only use one tempered 7-limit interval max before the absolute error exceeds 3.5 cents, as after [[7/4]], the next interval in the 7-limit chain is [[49/32]], and the difference between the JI version and the 159edo-tempered version exceeds 3.5 cents, and this is also true for the 205edo-tempered version of 49/32.  However, when two different EDOs have the same number of intervals of a given p-limit that can be stacked before the absolute error exceeds 3.5 cents, it is the absolute error in cents of the tempered stack relative to the JI equvalent that determines which EDO is superior for representing that p-limit, with the better EDO for representation having the smaller absolute error in cents. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 20:05, 8 October 2020 (UTC)
-:::: Using [http://musictheory.zentral.zone/huntsystemcalc.html the same calculator] that you linked, I tested how 159edo and 205edo each represent this set of intervals- [[3/2]], [[9/8]], [[27/16]], [[81/64]], [[243/128]], [[729/512]], [[5/4]], [[25/16]], [[125/64]], [[625/512]], [[7/4]], [[49/32]], [[343/256]], [[11/8]], [[121/64]], [[13/8]], [[169/128]], [[17/16]] and [[289/256]]- the stipulation being that the odd-limit of any interval in the set has to be less than 1024.  I looked for the number of "P" ratings given by both 205edo and 159edo, as "P" ratings are the only ratings I'm really interested in at this point, and I also looked at their distribution.  Both 205edo and 159edo give 7 "P" ratings total out of this set, and are surpassed in this respect by [[147edo]], which has 8.  However, one of the P-rated intervals for 147edo is for 343/326, the best approximation of which cannot be reached by stacking three of 147edo's best tempered version of 7/4 and octave reducing, thus resulting in this interval's disqualification.  Furthermore, the 147edo-tempered versions of six of the other 7 intervals in the starting interval set given a "P" rating in Hunt's system have absolute errors in cents that are ''greater'' than those of their 159edo-tempered counterparts which are also P-rated- a decisive loss for 147edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:13, 9 October 2020 (UTC)
+:::: Using [http://musictheory.zentral.zone/huntsystemcalc.html the same calculator] that you linked, I tested how 159edo and 205edo each represent this set of intervals- [[3/2]], [[9/8]], [[27/16]], [[81/64]], [[243/128]], [[729/512]], [[5/4]], [[25/16]], [[125/64]], [[625/512]], [[7/4]], [[49/32]], [[343/256]], [[11/8]], [[121/64]], [[13/8]], [[169/128]], [[17/16]] and [[289/256]]- and as I'm forced to limit the interval set somehow, I've decided that the odd-limit of any interval in the set has to be less than 1024.  I looked for the number of "P" ratings given by both 205edo and 159edo, as "P" ratings are the only ratings I'm really interested in at this point, and I also looked at their distribution.  Both 205edo and 159edo give 7 "P" ratings total out of this set, and are surpassed in this respect by [[147edo]], which has 8.  However, one of the P-rated intervals for 147edo is for 343/326, the best approximation of which cannot be reached by stacking three of 147edo's best tempered version of 7/4 and octave reducing, thus resulting in this interval's disqualification.  Furthermore, the 147edo-tempered versions of six of the other 7 intervals in the starting interval set given a "P" rating in Hunt's system have absolute errors in cents that are ''greater'' than those of their 159edo-tempered counterparts which are also P-rated- a decisive loss for 147edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:13, 9 October 2020 (UTC)
 :::: As for the remaining head to head comparison of 159edo and 205edo in terms of how good their representation is, one of the P-rated intervals for 205edo is 49/32, the best approximation of which cannot be reached by stacking two of 205edo's best tempered version of 7/4 and octave reducing, resulting in this interval's disqualification.  Four of 205edo's remaining P-rated intervals are all solidly in the in the 5-limit chain, and another P-rated interval is 17/16- the 205edo-tempered version of which is better than the 159edo-tempered version- the final P-rated interval for 205edo is 3/2, and 159edo-tempered version of this interval has ''less'' absolute error in cents than the 205edo-tempered version, leaving 205edo with only 5 P-rated intervals that are not outperformed by their 159edo counterparts.  This means a decisive loss for 205edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:40, 9 October 2020 (UTC)