49edo: Difference between revisions
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== Theory == | == Theory == | ||
49edo is very much on the sharp side of things, with sharp tunings of [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]]. It is the [[optimal patent val]] for [[superpyth]] temperament in the 7- and 11-limit, [[Archytas family #Archytas|archytas]] ([[7-limit]]) and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments, being almost exactly equal to {{frac|3|10}}-comma superpyth and the {{w|e (mathematical constant)|e-based}} analog of [[Lucy tuning]]. It [[tempering out|tempers out]] [[64/63]], [[245/243]], and [[3125/3087]] in the 7-limit, and [[100/99]] and [[1375/1372]] in the 11-limit. | 49edo is very much on the sharp side of things, with sharp tunings of [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]]. It is the [[optimal patent val]] for [[superpyth]] temperament in the 7- and 11-limit, [[Archytas family #Archytas|archytas]] ([[7-limit]]), and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments, being almost exactly equal to {{frac|3|10}}-comma superpyth and the {{w|e (mathematical constant)|''e''-based}} analog of [[Lucy tuning]]. It [[tempering out|tempers out]] [[64/63]], [[245/243]], and [[3125/3087]] in the 7-limit, and [[100/99]] and [[1375/1372]] in the 11-limit. | ||
=== Harmonics === | === Harmonics === | ||
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== Intervals == | == Intervals == | ||
{| class="wikitable center-all right-2 left-3" | {| class="wikitable center-all right-2 left-3" | ||
! # | |- | ||
! # | |||
! Cents | ! Cents | ||
! Approximate | ! Approximate ratios* | ||
! [[Ups and downs notation | ! [[Ups and downs notation]] | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 272: | Line 273: | ||
== Notation == | == Notation == | ||
=== Ups and downs notation === | === Ups and downs notation === | ||
Using [[Helmholtz–Ellis | Using [[Helmholtz–Ellis]] accidentals, 49edo can be notated using [[ups and downs notation|ups and downs]]: | ||
{{Sharpness-sharp7}} | {{Sharpness-sharp7}} | ||
===Sagittal notation=== | === Sagittal notation === | ||
====Evo flavor==== | ==== Evo flavor ==== | ||
<imagemap> | <imagemap> | ||
File:49-EDO_Evo_Sagittal.svg | File:49-EDO_Evo_Sagittal.svg | ||
| Line 289: | Line 289: | ||
</imagemap> | </imagemap> | ||
====Revo flavor==== | ==== Revo flavor ==== | ||
<imagemap> | <imagemap> | ||
File:49-EDO_Revo_Sagittal.svg | File:49-EDO_Revo_Sagittal.svg | ||
| Line 304: | Line 303: | ||
== Approximation to JI == | == Approximation to JI == | ||
[[File:49ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 49edo]] | [[File:49ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 49edo]] | ||
=== Interval mappings === | === Interval mappings === | ||
{{Q-odd-limit intervals|49}} | {{Q-odd-limit intervals|49}} | ||
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|- | |- | ||
| ϕ | | ϕ | ||
| | | −0.437 | ||
| 34 | | 34 | ||
|- | |- | ||
| ϕ<sup>ϕ<sup>-1</sup></sup> | | ϕ<sup>ϕ<sup>-1</sup></sup> | ||
| | | −0.592 | ||
| 21 | | 21 | ||
|} | |} | ||
| Line 339: | Line 339: | ||
=== Music === | === Music === | ||
* [https://www.youtube.com/watch?v=vZyAm-D3nlk&ab_channel=Sevish Sevish - Star Nursery] uses a scale based on [[acoustic phi]] and ϕ<sup>ϕ<sup> | * [https://www.youtube.com/watch?v=vZyAm-D3nlk&ab_channel=Sevish Sevish - Star Nursery] uses a scale based on [[acoustic phi]] and ϕ<sup>ϕ<sup>−1</sup></sup>. 49edo provides a suitable approximation for this scale with the mode: 5 3 5 5 3 5 3 5 | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| {{monzo| 78 -49 }} | | {{monzo| 78 -49 }} | ||
| {{mapping| 49 78 }} | | {{mapping| 49 78 }} | ||
| | | −2.60 | ||
| 2.60 | | 2.60 | ||
| 10.62 | | 10.62 | ||
| Line 363: | Line 363: | ||
| 15625/15552, 20480/19683 | | 15625/15552, 20480/19683 | ||
| {{mapping| 49 78 114 }} | | {{mapping| 49 78 114 }} | ||
| | | −2.53 | ||
| 2.12 | | 2.12 | ||
| 8.69 | | 8.69 | ||
| Line 370: | Line 370: | ||
| 64/63, 245/243, 3125/3087 | | 64/63, 245/243, 3125/3087 | ||
| {{mapping| 49 78 114 138 }} | | {{mapping| 49 78 114 138 }} | ||
| | | −2.85 | ||
| 1.92 | | 1.92 | ||
| 7.87 | | 7.87 | ||
| Line 377: | Line 377: | ||
| 64/63, 100/99, 245/243, 1331/1323 | | 64/63, 100/99, 245/243, 1331/1323 | ||
| {{mapping| 49 78 114 138 170 }} | | {{mapping| 49 78 114 138 170 }} | ||
| | | −2.97 | ||
| 1.74 | | 1.74 | ||
| 7.11 | | 7.11 | ||
| Line 476: | Line 476: | ||
== Zeta properties == | == Zeta properties == | ||
===Zeta peak index=== | === Zeta peak index === | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | ! colspan="3" | Tuning | ||
! | ! colspan="3" | Strength | ||
! colspan="2" | Closest EDO | |||
! colspan="2" | Integer limit | |||
!EDO | |||
! | |||
|- | |- | ||
|[[233zpi]] | ! ZPI | ||
|49.1412057629230 | ! Steps per octave | ||
|24.4194252332612 | ! Step size (cents) | ||
|5.691547 | ! Height | ||
|0.920677 | ! Integral | ||
|15.624024 | ! Gap | ||
|49edo | ! EDO | ||
|1196.55183642980 | ! Octave (cents) | ||
|7 | ! Consistent | ||
|7 | ! Distinct | ||
|- | |||
| [[233zpi]] | |||
| 49.1412057629230 | |||
| 24.4194252332612 | |||
| 5.691547 | |||
| 0.920677 | |||
| 15.624024 | |||
| 49edo | |||
| 1196.55183642980 | |||
| 7 | |||
| 7 | |||
|} | |} | ||
== Scales == | == Scales == | ||
=== MOS scales === | === MOS scales === | ||
| Line 519: | Line 522: | ||
; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune | * [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune – ''Man'' (cover)] (2023) | ||
[[Category:Archytas]] | [[Category:Archytas]] | ||