53edo: Difference between revisions

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== Theory ==

53edo is notable as a [[5-limit]] system, a fact apparently first noted by Isaac Newton, notably tempering out [[Mercator's comma]] (353/284), the [[schisma]] (32805/32768), and the [[kleisma]] (15625/15552). More complex 5-limit commas tempered out include the [[amity comma]] (1600000/1594323), the [[semicomma]] (2109375/2097152), and the [[vulture comma]] ({{monzo| 24 -21 4 }}). In the 7-limit it tempers out the ~~Marvel~~ comma ([[225/224]]), ~~Orwell comma (~~[[1728/1715]]), and ~~Gariboh~~ comma ([[3125/3087]]). In the 11-limit, it tempers out [[99/98]] and [[121/120]] (in addition to their difference, [[540/539]]), and is the [[optimal patent val]] for [[big brother]] temperament, which tempers out both, as well as 11-limit [[orwell]] temperament, which also tempers out the 11-limit commas [[176/175]] and [[385/384]]. In the 13-limit, it tempers out [[169/168]], [[275/273]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[2080/2079]], and [[4096/4095]], and gives the optimal patent val for [[Marvel family #Athene|athene]] temperament. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. It can also be treated as a no-elevens, no-seventeens tuning, on which it is [[consistent]] all the way up to the 23-odd-limit.

53edo is notable as a [[5-limit]] system, a fact apparently first noted by Isaac Newton, notably tempering out [[Mercator's comma]] (353/284), the [[schisma|schisma (32805/32768)]], and the [[15625/15552|kleisma (15625/15552)]]. More complex 5-limit commas tempered out include the [[amity comma|amity comma (1600000/1594323)]], the [[semicomma|semicomma (2109375/2097152)]], and the [[vulture comma|vulture comma ({{monzo| 24 -21 4 }}]]). In the 7-limit it tempers out the [[225/224|marvel comma (225/224)]], [[1728/1715|orwellisma (1728/1715)]], and [[3125/3087|gariboh comma (3125/3087)]]. In the 11-limit, it tempers out [[99/98]] and [[121/120]] (in addition to their difference, [[540/539]]), and is the [[optimal patent val]] for [[big brother]] temperament, which tempers out both, as well as 11-limit [[orwell]] temperament, which also tempers out the 11-limit commas [[176/175]] and [[385/384]]. In the 13-limit, it tempers out [[169/168]], [[275/273]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[2080/2079]], and [[4096/4095]], and gives the optimal patent val for [[Marvel family #Athene|athene]] temperament. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. It can also be treated as a no-elevens, no-seventeens tuning, on which it is [[consistent]] all the way up to the 23-odd-limit.

53edo has also found a certain dissemination as an edo tuning for [[Arabic, Turkish, Persian music]]. Due to its fifths being almost indistinguishable from just, it can also be used as an extended [[3-limit|Pythagorean tuning]].

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53edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].

Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and [[954edo]] have good consistency ~~limit~~ and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.

Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.

== Intervals ==

@@ Line 10: / Line 10: @@
 == Theory ==
-edo is notable as a [[5-limit]] system, a fact apparently first noted by Isaac Newton, notably tempering out [[Mercator's comma]] (3<sup>53</sup>/2<sup>84</sup>), the [[schisma]] (32805/32768), and the [[kleisma]] (15625/15552). More complex 5-limit commas tempered out include the [[amity comma]] (1600000/1594323), the [[semicomma]] (2109375/2097152), and the [[vulture comma]] ({{monzo| 24 -21 4 }}). In the 7-limit it tempers out the Marvel comma ([[225/224]]), Orwell comma ([[1728/1715]]), and Gariboh comma ([[3125/3087]]). In the 11-limit, it tempers out [[99/98]] and [[121/120]] (in addition to their difference, [[540/539]]), and is the [[optimal patent val]] for [[big brother]] temperament, which tempers out both, as well as 11-limit [[orwell]] temperament, which also tempers out the 11-limit commas [[176/175]] and [[385/384]]. In the 13-limit, it tempers out [[169/168]], [[275/273]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[2080/2079]], and [[4096/4095]], and gives the optimal patent val for [[Marvel family #Athene|athene]] temperament. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. It can also be treated as a no-elevens, no-seventeens tuning, on which it is [[consistent]] all the way up to the 23-odd-limit.
+edo is notable as a [[5-limit]] system, a fact apparently first noted by Isaac Newton, notably tempering out [[Mercator's comma]] (3<sup>53</sup>/2<sup>84</sup>), the [[schisma|schisma (32805/32768)]], and the [[15625/15552|kleisma (15625/15552)]]. More complex 5-limit commas tempered out include the [[amity comma|amity comma (1600000/1594323)]], the [[semicomma|semicomma (2109375/2097152)]], and the [[vulture comma|vulture comma ({{monzo| 24 -21 4 }}]]). In the 7-limit it tempers out the [[225/224|marvel comma (225/224)]], [[1728/1715|orwellisma (1728/1715)]], and [[3125/3087|gariboh comma (3125/3087)]]. In the 11-limit, it tempers out [[99/98]] and [[121/120]] (in addition to their difference, [[540/539]]), and is the [[optimal patent val]] for [[big brother]] temperament, which tempers out both, as well as 11-limit [[orwell]] temperament, which also tempers out the 11-limit commas [[176/175]] and [[385/384]]. In the 13-limit, it tempers out [[169/168]], [[275/273]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[2080/2079]], and [[4096/4095]], and gives the optimal patent val for [[Marvel family #Athene|athene]] temperament. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. It can also be treated as a no-elevens, no-seventeens tuning, on which it is [[consistent]] all the way up to the 23-odd-limit.
 edo has also found a certain dissemination as an edo tuning for [[Arabic, Turkish, Persian music]]. Due to its fifths being almost indistinguishable from just, it can also be used as an extended [[3-limit|Pythagorean tuning]].
@@ Line 22: / Line 22: @@
 edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].
-Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and [[954edo]] have good consistency limit and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.
+Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.
 == Intervals ==