171edo: Difference between revisions
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171edo is a remarkable edo which serves as a [[7-limit]] [[microtemperament]], approximating the [[9-odd-limit]] [[tonality diamond]] within about 2/5 of a cent. The excellence of its 7-limit approximations is good enough to make it the eleventh [[zeta integral edo]] but not enough to make it a [[zeta gap edo|zeta gap]]. It is also almost consistent in the 17-odd-limit, only missing [[15/11]] and [[22/15]]. | 171edo is a remarkable edo which serves as a [[7-limit]] [[microtemperament]], approximating the [[9-odd-limit]] [[tonality diamond]] within about 2/5 of a cent. The excellence of its 7-limit approximations is good enough to make it the eleventh [[zeta integral edo]] but not enough to make it a [[zeta gap edo|zeta gap]]. It is also almost consistent in the 17-odd-limit, only missing [[15/11]] and [[22/15]]. | ||
Remarkable 5-limit commas 171et [[tempering out|tempers out]] are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -29 -11 20 }} ([[gammic comma]]), and remarkable 7-limit commas 171et tempers out are 2401/2400 ([[breedsma]]), 4375/4374 ([[ragisma]]), 65625/65536 ([[horwell comma]]), 250047/250000 ([[landscape comma]]), 420175/419904 ([[wizma]]), and 703125/702464 ([[meter]]). Therefore, 171et [[support]]s a number of notable 7-limit rank-2 temperaments: [[pontiac]], [[sesquiquartififths]], [[term]], [[ennealimmal]], [[tertiaseptal]], [[supermajor (temperament)|supermajor]], [[enneadecal]], [[neptune]], [[mitonic]], and [[mutt]]. It notably provides the [[optimal patent val]] for the rank-3 [[horwell]] temperament, and is also an excellent tuning for the 5-limit [[Helmholtz (temperament)|helmholtz]] temperament, tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }}. | Remarkable 5-limit commas 171et [[tempering out|tempers out]] are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -29 -11 20 }} ([[gammic comma]]), and remarkable 7-limit commas 171et tempers out are 2401/2400 ([[breedsma]]), 4375/4374 ([[ragisma]]), 65625/65536 ([[horwell comma]]), 250047/250000 ([[landscape comma]]), 420175/419904 ([[wizma]]), and 703125/702464 ([[meter]]). Therefore, 171et [[support]]s a number of notable 7-limit rank-2 temperaments: [[pontiac]], [[sesquiquartififths]], [[term]], [[ennealimmal]], [[tertiaseptal]], [[supermajor (temperament)|supermajor]], [[enneadecal]], [[[[Neptune (temperament)|neptune]]]], [[mitonic]], and [[mutt]]. It notably provides the [[optimal patent val]] for the rank-3 [[horwell]] temperament, and is also an excellent tuning for the 5-limit [[Helmholtz (temperament)|helmholtz]] temperament, tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }}. | ||
171edo is notably [[Consistency#Consistency to distance d|consistent to distance ''6'']] in the [[7-limit|7-prime]] [[15-odd-limit]], and to distance ''8'' in the [[9-odd-limit]]. No other edo is so consistent until [[3125edo]]. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify 7-limit commas by size. For example, one step represents [[225/224]], two steps [[126/125]], three steps [[81/80]], and four steps [[64/63]]. | 171edo is notably [[Consistency#Consistency to distance d|consistent to distance ''6'']] in the [[7-limit|7-prime]] [[15-odd-limit]], and to distance ''8'' in the [[9-odd-limit]]. No other edo is so consistent until [[3125edo]]. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify 7-limit commas by size. For example, one step represents [[225/224]], two steps [[126/125]], three steps [[81/80]], and four steps [[64/63]]. | ||
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* 171et is lower in relative error than any previous equal temperaments in the 7-limit | * 171et is lower in relative error than any previous equal temperaments in the 7-limit. Not until [[441edo|441]] do we find a better equal temperaments in terms of absolute error, and not until [[3125edo|3125]] do we find one in terms of relative error. | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
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| 182.46 | | 182.46 | ||
| 10/9 | | 10/9 | ||
| [[ | | [[Mitonic]] / mineral (171) / ore (171e) / goldmine (171ef) | ||
|- | |- | ||
| 1 | | 1 | ||
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| 498.25 | | 498.25 | ||
| 4/3 | | 4/3 | ||
| [[ | | [[Pontiac]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 328: | Line 328: | ||
| 582.46 | | 582.46 | ||
| 7/5 | | 7/5 | ||
| [[Neptune]] | | [[Neptune (temperament)|Neptune]] | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 346: | Line 346: | ||
| 182.46 | | 182.46 | ||
| 10/9 | | 10/9 | ||
| [[ | | [[Domain (temperament)|Domain]] | ||
|- | |- | ||
| 3 | | 3 | ||
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| 315.79<br>(49.12) | | 315.79<br>(49.12) | ||
| 6/5<br>(36/35) | | 6/5<br>(36/35) | ||
| [[Ennealimmal]] ( | | [[Ennealimmal]] / enneabiotic (171ef) / ennealympic (171) / ennealimnic (171) / ennealiminal (171ef) | ||
|- | |- | ||
| 9 | | 9 | ||