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|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 is notable as an excellent [[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)]]. 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]].

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

53edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma ~~(and is almost exactly equal to the geometric mean of the two)~~.

53edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma.

=== Prime harmonics ===

Line 894:

| Tribilawa

| 53-comma, [[Mercator's comma]]

|-

| 5

| [[2109375/2097152|(14 digits)]]

| {{monzo| -21 3 7 }}

| 10.061

| Lasepyo

| [[Semicomma]]

|-

| 5

Line 901:

Line 908:

| Tribiyo

| Kleisma

|-

| 5

| <abbr title="1600000/1594323">(14 digits)</abbr>

| {{monzo| 9 -13 5 }}

| 6.15

| Saquinyo

| [[Amity comma]]

|-

| 5

| <abbr title="10485760000/10460353203">(22 digits)</abbr>

| {{monzo| 24 -21 4 }}

| 4.20

| Sasaquadyo

| [[Vulture comma]]

|-

| 5

Line 908:

Line 929:

| Layo

| Schisma

|-

| 7

| [[3125/3087]]

| 21.18

| {{monzo| 0 -2 5 -3 }}

| Triru-aquinyo

| Gariboh comma

|-

| 7

| [[1728/1715]]

| {{monzo| 6 3 -1 -3 }}

| 13.07

| Triru-agu

| Orwellisma

|-

| 7

| [[225/224]]

| {{monzo| -5 2 2 -1 }}

| 7.7115

| Ruyoyo

| Marvel comma, septimal kleisma

|}

Line 1,014:

Line 1,056:

|-

|1

|23\53

| 23\53

|520.75

| 520.75

|4/3

| 4/3

|[[Pelogic family|Mavila]]

| [[Pelogic family|Mavila]]

|-

| 1

@@ 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|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 is notable as an excellent [[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)]]. 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]].
+edo has also found a certain dissemination as an edo tuning for [[Arabic, Turkish, Persian|Arabic, Turkish, and Persian music]]. It can also be used as an extended [[3-limit|Pythagorean tuning]], since its fifths are almost indistinguishable from just.
-edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma (and is almost exactly equal to the geometric mean of the two).
+edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma.
 === Prime harmonics ===
@@ Line 894: / Line 894: @@
 | Tribilawa
 | 53-comma, [[Mercator's comma]]
+|-
+| 5
+| [[2109375/2097152|(14 digits)]]
+| {{monzo| -21 3 7 }}
+| 10.061
+| Lasepyo
+| [[Semicomma]]
 |-
 | 5
@@ Line 901: / Line 908: @@
 | Tribiyo
 | Kleisma
+|-
+| 5
+| <abbr title="1600000/1594323">(14 digits)</abbr>
+| {{monzo| 9 -13 5 }}
+| 6.15
+| Saquinyo
+| [[Amity comma]]
+|-
+| 5
+| <abbr title="10485760000/10460353203">(22 digits)</abbr>
+| {{monzo| 24 -21 4 }}
+| 4.20
+| Sasaquadyo
+| [[Vulture comma]]
 |-
 | 5
@@ Line 908: / Line 929: @@
 | Layo
 | Schisma
+|-
+| 7
+| [[3125/3087]]
+| 21.18
+| {{monzo| 0 -2 5 -3 }}
+| Triru-aquinyo
+| Gariboh comma
+|-
+| 7
+| [[1728/1715]]
+| {{monzo| 6 3 -1 -3 }}
+| 13.07
+| Triru-agu
+| Orwellisma
+|-
+| 7
+| [[225/224]]
+| {{monzo| -5 2 2 -1 }}
+| 7.7115
+| Ruyoyo
+| Marvel comma, septimal kleisma
 |}
@@ Line 1,014: / Line 1,056: @@
 |-
 |1
-|23\53
+| 23\53
-|520.75
+| 520.75
-|4/3
+| 4/3
-|[[Pelogic family|Mavila]]
+| [[Pelogic family|Mavila]]
 |-
 | 1