53edo: Difference between revisions

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== 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]]. It shines however in the 2.3.5.19 and [[2.3.5.13 subgroup|2.3.5.13]] subgroups, where it offers excellent approximations with decent complexity.

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 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 indistinguishable from just in most contexts.

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.

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 ===

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 == 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]]. It shines however in the 2.3.5.19 and [[2.3.5.13 subgroup|2.3.5.13]] subgroups, where it offers excellent approximations with decent complexity.
-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 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 indistinguishable from just in most contexts.
-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.
+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 ===