53edo: Difference between revisions

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 distintion 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]].  

| (a, b, 0, 1)

| (a, b) with b &lt; -1

| (a, b, -1)

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

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

| (a, b, 1)

| (a, b) with b &gt; 1

| (a, b, 0, -1)

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.

| 0-12-31

| 0-14-31

| 0-15-31

| 0-16-31

| 0-17-31

| 0-19-31

53edo's [[circle of fifths|circle of 53 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This is possible because 31\53 is on the 7\12 kite in the [[scale tree]]. Stated another way, it is possible because the absolute value of 53edo's [[sharpness#dodeca-sharpness|dodeca-sharpness]] (edosteps per [[Pythagorean comma]]) is 1.  

53edo has only 5 pairs of inconsistent intervals in the full 27-odd-limit: {11/7, 14/11}, {[[17/11]], [[22/17]]}, {[[19/17]], [[34/19]]}, {[[21/11]], [[22/21]]}, and {[[23/22]], [[44/23]]}. This is perhaps remarkable compared to 8 pairs in 46edo and 11 pairs in 41edo, because the smallest edo after 53edo to get 5 or less inconsistencies in the 27-odd-limit is [[99edo]] (using the 99[[wart|ef]] [[val]]), followed by [[111edo]] ([[patent val]]), [[130edo]] (patent val) and [[159edo]] (also patent); all of these get 5 inconsistencies as well except 159edo which gets 1 and which is itself a superset of 53edo. However, most interpret the approximation of prime 17 in 53edo as too off for all but the most opportunistic harmonies, and some question the 23 and possibly also 11, so the practical significance of this is debatable.

| {{monzo| -84 53 }}

| {{mapping| 53 84 }}

| {{mapping| 53 84 123 }}

| {{mapping| 53 84 123 149 }}

| {{mapping| 53 84 123 149 183 }}

| {{mapping| 53 84 123 149 183 196 }}

| {{mapping| 53 84 123 149 183 196 225 }}

| {{monzo| -84 53 }}

| {{monzo| -21 3 7 }}

| {{monzo| -6 -5 6 }}

| {{monzo| 9 -13 5 }}

| {{monzo| 24 -21 4 }}

| {{monzo| -15 8 1 }}

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

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

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

| {{monzo| -1 -7 4 1 }}

| {{monzo| -1 2 0 -2 1 }}

| {{monzo| -3 -1 -1 0 2 }}

| {{monzo| 4 0 -2 -1 1 }}

| {{monzo| 33 -23 0 0 1 }}

| {{monzo| -7 -1 1 1 1 }}

| {{monzo| 2 3 1 -2 -1 }}

| {{monzo| 0 -1 2 -1 1 -1 }}

| {{monzo| -3 -1 0 -1 0 2 }}

| {{monzo| -4 -1 4 0 0 -1 }}

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

| {{monzo| -3 0 -3 1 1 1 }}

| {{monzo| 5 -3 1 -1 -1 1 }}

| {{monzo| 12 -2 -1 -1 0 -1 }}

| [[Tricot]]

=== MOS scales ===

* Palace (approximated from [[Porky]] in [[29edo]]): 7 7 8 9 7 7 8

* "It's a Good Idea to Have a Good Time." from ''Random Sentences'' (2025) – [https://open.spotify.com/track/3rYiNMcOQ5Oxz7F6mQZsfw Spotify] | [https://francium223.bandcamp.com/track/its-a-good-idea-to-have-a-good-time Bandcamp] | [https://www.youtube.com/watch?v=D-i-4Sv-vqE YouTube] – in Ellic, 53edo tuning