53edo: Difference between revisions

Line 11:

== Theory ==

53edo is notable as an excellent [[5-limit]] system, a fact apparently first noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (~~3-4~~):~~223-48~~. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. In the opinion of some, 53edo is the first equal division to deal adequately with the [[13-limit]], while others award that distinction to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]] (and if we exclude the most damaged interval pair, 7/5 and 10/7, is [[consistent to distance]] 2), but among them, 53 is the first that finds the [[interseptimal interval]]s [[15/13]] and [[13/10]] distinctly from adjacent [[7-limit|septimal]] intervals [[8/7]] and [[7/6]], and [[9/7]] and [[21/16]], respectively, which is essential to its 13-limit credibility. It also avoids equating [[11/9]] with [[16/13]], so that the former is tuned very flat to equate it with a slightly flat [[~]][[39/32]] – a feature shared by 46edo. It is almost consistent to the entire [[15-odd-limit]], with the only inconsistency occurring at [[14/11]] (and its octave complement), which is mapped inconsistently sharp and equated with [[9/7]], but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the [[27-odd-limit]].

53edo is notable as an excellent [[5-limit]] system, a fact apparently first noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. In the opinion of some, 53edo is the first equal division to deal adequately with the [[13-limit]], while others award that distinction to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]] (and if we exclude the most damaged interval pair, 7/5 and 10/7, is [[consistent to distance]] 2), but among them, 53 is the first that finds the [[interseptimal interval]]s [[15/13]] and [[13/10]] distinctly from adjacent [[7-limit|septimal]] intervals [[8/7]] and [[7/6]], and [[9/7]] and [[21/16]], respectively, which is essential to its 13-limit credibility. It also avoids equating [[11/9]] with [[16/13]], so that the former is tuned very flat to equate it with a slightly flat [[~]][[39/32]] – a feature shared by 46edo. It is almost consistent to the entire [[15-odd-limit]], with the only inconsistency occurring at [[14/11]] (and its octave complement), which is mapped inconsistently sharp and equated with [[9/7]], but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the [[27-odd-limit]].

As an equal temperament, it notably [[tempering out|tempers out]] [[Mercator's comma]] (3<sup>53</sup>/2<sup>84</sup>), the [[schisma|schisma (32805/32768)]], [[15625/15552|kleisma (15625/15552)]], and [[amity comma|amity comma (1600000/1594323)]]. In the 7-limit it tempers out the [[225/224|marvel comma (225/224)]] for which it is a [[Marvel#Tunings|relatively efficient tuning]], [[1728/1715|orwellisma (1728/1715)]], [[3125/3087|gariboh comma (3125/3087)]], and [[4375/4374|ragisma (4375/4374)]]. 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.

Line 538:

| downminor

| zo

| (a, b, 0, 1)

| {{nowrap|(a, b, 0, 1)}}

| 7/6, 7/4

|-

| minor

| fourthward wa

| (a, b) with b < -1

| {{nowrap|(a, b)}} with {{nowrap|b < −1}}

| 32/27, 16/9

|-

| upminor

| gu

| (a, b, -1)

| {{nowrap|(a, b, −1)}}

| 6/5, 9/5

|-

| dupminor

| ilo

| (a, b, 0, 0, 1)

| {{nowrap|(a, b, 0, 0, 1)}}

| 11/9, 11/6

|-

| dudmajor

| lu

| (a, b, 0, 0, -1)

| {{nowrap|(a, b, 0, 0, −1)}}

| 12/11, 18/11

|-

| downmajor

| yo

| (a, b, 1)

| {{nowrap|(a, b, 1)}}

| 5/4, 5/3

|-

| major

| fifthward wa

| (a, b) with b > 1

| {{nowrap|(a, b)}} with {{nowrap|b > 1}}

| 9/8, 27/16

|-

| upmajor

| ru

| (a, b, 0, -1)

| {{nowrap|(a, b, 0, −1)}}

| 9/7, 12/7

|}

All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a ~~stacked-3rds~~ chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are.

All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked−3rds chord {{nowrap|{{dash|6, 1, 3, 5, 7, 9, 11, 13}}}}). Alterations are always enclosed in parentheses, additions never are.

Here are the zo, gu, ilo, lu, yo and ru triads:

Line 590:

| zo

| 6:7:9

| ~~0-12-31~~

| 0–12–31

| C vEb G

| Cvm

Line 597:

| gu

| 10:12:15

| ~~0-14-31~~

| 0–14–31

| C ^Eb G

| C^m

Line 604:

| ilo

| 18:22:27

| ~~0-15-31~~

| 0–15–31

| C ^^Eb G

| C^^m

Line 611:

| lu

| 22:27:33

| ~~0-16-31~~

| 0–16–31

| C vvE G

| Cvv

Line 618:

| yo

| 4:5:6

| ~~0-17-31~~

| 0–17–31

| C vE G

| Cv

Line 625:

| ru

| 14:18:21

| ~~0-19-31~~

| 0–19–31

| C ^E G

| C^

@@ Line 11: / Line 11: @@
 == Theory ==
-edo is notable as an excellent [[5-limit]] system, a fact apparently first noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. In the opinion of some, 53edo is the first equal division to deal adequately with the [[13-limit]], while others award that distinction to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]] (and if we exclude the most damaged interval pair, 7/5 and 10/7, is [[consistent to distance]] 2), but among them, 53 is the first that finds the [[interseptimal interval]]s [[15/13]] and [[13/10]] distinctly from adjacent [[7-limit|septimal]] intervals [[8/7]] and [[7/6]], and [[9/7]] and [[21/16]], respectively, which is essential to its 13-limit credibility. It also avoids equating [[11/9]] with [[16/13]], so that the former is tuned very flat to equate it with a slightly flat [[~]][[39/32]] – a feature shared by 46edo. It is almost consistent to the entire [[15-odd-limit]], with the only inconsistency occurring at [[14/11]] (and its octave complement), which is mapped inconsistently sharp and equated with [[9/7]], but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the [[27-odd-limit]].
+edo is notable as an excellent [[5-limit]] system, a fact apparently first noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. In the opinion of some, 53edo is the first equal division to deal adequately with the [[13-limit]], while others award that distinction to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]] (and if we exclude the most damaged interval pair, 7/5 and 10/7, is [[consistent to distance]] 2), but among them, 53 is the first that finds the [[interseptimal interval]]s [[15/13]] and [[13/10]] distinctly from adjacent [[7-limit|septimal]] intervals [[8/7]] and [[7/6]], and [[9/7]] and [[21/16]], respectively, which is essential to its 13-limit credibility. It also avoids equating [[11/9]] with [[16/13]], so that the former is tuned very flat to equate it with a slightly flat [[~]][[39/32]] – a feature shared by 46edo. It is almost consistent to the entire [[15-odd-limit]], with the only inconsistency occurring at [[14/11]] (and its octave complement), which is mapped inconsistently sharp and equated with [[9/7]], but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the [[27-odd-limit]].
 As an equal temperament, it notably [[tempering out|tempers out]] [[Mercator's comma]] (3<sup>53</sup>/2<sup>84</sup>), the [[schisma|schisma (32805/32768)]], [[15625/15552|kleisma (15625/15552)]], and [[amity comma|amity comma (1600000/1594323)]]. In the 7-limit it tempers out the [[225/224|marvel comma (225/224)]] for which it is a [[Marvel#Tunings|relatively efficient tuning]], [[1728/1715|orwellisma (1728/1715)]], [[3125/3087|gariboh comma (3125/3087)]], and [[4375/4374|ragisma (4375/4374)]]. 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.
@@ Line 538: / Line 538: @@
 | downminor
 | zo
-| (a, b, 0, 1)
+| {{nowrap|(a, b, 0, 1)}}
 | 7/6, 7/4
 |-
 | minor
 | fourthward wa
-| (a, b) with b &lt; -1
+| {{nowrap|(a, b)}} with {{nowrap|b &lt; −1}}
 | 32/27, 16/9
 |-
 | upminor
 | gu
-| (a, b, -1)
+| {{nowrap|(a, b, −1)}}
 | 6/5, 9/5
 |-
 | dupminor
 | ilo
-| (a, b, 0, 0, 1)
+| {{nowrap|(a, b, 0, 0, 1)}}
 | 11/9, 11/6
 |-
 | dudmajor
 | lu
-| (a, b, 0, 0, -1)
+| {{nowrap|(a, b, 0, 0, −1)}}
 | 12/11, 18/11
 |-
 | downmajor
 | yo
-| (a, b, 1)
+| {{nowrap|(a, b, 1)}}
 | 5/4, 5/3
 |-
 | major
 | fifthward wa
-| (a, b) with b &gt; 1
+| {{nowrap|(a, b)}} with {{nowrap|b &gt; 1}}
 | 9/8, 27/16
 |-
 | upmajor
 | ru
-| (a, b, 0, -1)
+| {{nowrap|(a, b, 0, −1)}}
 | 9/7, 12/7
 |}
-All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are.
+All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked−3rds chord {{nowrap|{{dash|6, 1, 3, 5, 7, 9, 11, 13}}}}). Alterations are always enclosed in parentheses, additions never are.
 Here are the zo, gu, ilo, lu, yo and ru triads:
@@ Line 590: / Line 590: @@
 | zo
 | 6:7:9
-| 0-12-31
+| 0–12–31
 | C vEb G
 | Cvm
@@ Line 597: / Line 597: @@
 | gu
 | 10:12:15
-| 0-14-31
+| 0–14–31
 | C ^Eb G
 | C^m
@@ Line 604: / Line 604: @@
 | ilo
 | 18:22:27
-| 0-15-31
+| 0–15–31
 | C ^^Eb G
 | C^^m
@@ Line 611: / Line 611: @@
 | lu
 | 22:27:33
-| 0-16-31
+| 0–16–31
 | C vvE G
 | Cvv
@@ Line 618: / Line 618: @@
 | yo
 | 4:5:6
-| 0-17-31
+| 0–17–31
 | C vE G
 | Cv
@@ Line 625: / Line 625: @@
 | ru
 | 14:18:21
-| 0-19-31
+| 0–19–31
 | C ^E G
 | C^