53edo: Difference between revisions
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| Prime factorization = 53 | | Prime factorization = 53 | ||
| Subgroup = 2.3.5.7.13.19.23 | | Subgroup = 2.3.5.7.13.19.23 | ||
| Step size = 22. | | Step size = 22.642¢ | ||
| Fifth type = [[schismic]] 31\53 701.886¢ | | Fifth type = [[schismic]] 31\53 = 701.886¢ | ||
| Major 2nd = 9\53 = 204¢ | |||
| Minor 2nd = 4\53 = 91¢ | |||
| Augmented 1sn = 5\53 = 113¢ | |||
| Common uses = Extended Pythagorean system<br/>Turkish music | | Common uses = Extended Pythagorean system<br/>Turkish music | ||
| Important MOS = | | Important MOS = | ||
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== Theory == | == Theory == | ||
53edo is notable as a [[5-limit]] system, a fact apparently first noted by Isaac Newton, tempering out the [[schisma]], 32805/32768, the [[kleisma]], 15625/15552, the [[amity comma]], 1600000/1594323 and the [[semicomma]], 2109375/2097152. In the 7-limit it tempers out [[225/224]], 1728/1715 and [[3125/3087]], the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out [[99/98]] and [[121/120]], and is the [[optimal patent val]] for [[Big Brother]] temperament, which tempers out both, as well as [[Semicomma family #Orwell|11-limit orwell temperament]], which also tempers out the 11-limit comma [[176/175]]. In the 13-limit, it tempers out [[169/168]], [[275/273]] and [[676/675]], and gives the optimal patent val for [[Marvel family #Athene|athene temperament]]. It is the eighth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]] and the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]]. | {| class="wikitable center-all" | ||
! colspan="2" | | |||
! prime 2 | |||
! prime 3 | |||
! prime 5 | |||
! prime 7 | |||
! prime 11 | |||
! prime 13 | |||
! prime 17 | |||
! prime 19 | |||
! prime 23 | |||
|- | |||
! rowspan="2" |Error | |||
! absolute (¢) | |||
| 0 | |||
| -0.07 | |||
| -1.41 | |||
| +4.76 | |||
| -7.92 | |||
| -2.79 | |||
| +8.26 | |||
| -3.17 | |||
| +5.69 | |||
|- | |||
! relative (%) | |||
| 0 | |||
| -0.3 | |||
| -6 | |||
| +21 | |||
| -35 | |||
| -12 | |||
| +36 | |||
| -14 | |||
| +25 | |||
|- | |||
! colspan="2" |nearest edomapping | |||
|53 | |||
|31 | |||
|17 | |||
|43 | |||
|24 | |||
|37 | |||
|5 | |||
|13 | |||
|28 | |||
|- | |||
! colspan="2" |[[fifthspan]] | |||
| 0 | |||
| +1 | |||
| -8 | |||
| -14 | |||
| +23 | |||
| +20 | |||
| +7 | |||
| -3 | |||
| +18 | |||
|} | |||
53edo is notable as a [[5-limit]] system, a fact apparently first noted by Isaac Newton, tempering out the [[schisma]], 32805/32768, the [[kleisma]], 15625/15552, the [[amity comma]], 1600000/1594323 and the [[semicomma]], 2109375/2097152. In the 7-limit it tempers out [[225/224]], 1728/1715 and [[3125/3087]], the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out [[99/98]] and [[121/120]], and is the [[optimal patent val]] for [[Big Brother]] temperament, which tempers out both, as well as [[Semicomma family #Orwell|11-limit orwell temperament]], which also tempers out the 11-limit comma [[176/175]]. In the 13-limit, it tempers out [[169/168]], [[275/273]] and [[676/675]], and gives the optimal patent val for [[Marvel family #Athene|athene temperament]]. It is the eighth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]] and the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]]. | |||
53edo has also found a certain dissemination as an EDO tuning for [[Arabic, Turkish, Persian]] music. | 53edo has also found a certain dissemination as an EDO tuning for [[Arabic, Turkish, Persian]] music. | ||
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=== Selected just intervals by error === | === Selected just intervals by error === | ||
The following table shows how [[15-odd-limit intervals]] are represented in 53edo. Octave-reduced prime harmonics are '''bolded'''; inconsistent intervals are in ''italic''. | |||
The following table shows how [[15-odd-limit intervals]] are represented in 53edo. Octave-reduced prime harmonics are '''bolded'''; inconsistent intervals are in ''italic''. | |||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||