10edo: Difference between revisions
No need to remind readers of what a regular temperament is everywhere Tag: Undo |
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== Theory == | == Theory == | ||
10edo can be thought of as two circles of [[5edo]] separated by 120 cents | 10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. | ||
Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L 4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. | Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L 4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. | ||
It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all. | |||
This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15. | |||
By treating 360c as 11/9, we arrive at 11/8 = 600c (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup. | |||
10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]]. | |||
Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]]. | Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]]. | ||
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[[Enharmonic unison]]: d2 | [[Enharmonic unison]]: d2 | ||
See below: 3L 4s | See below: 3L 4s mosh notation | ||
=== 3L 4s (mosh) notation === | === 3L 4s (mosh) notation === | ||
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{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed" | {| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed" | ||
! | ! # | ||
! Cents | ! Cents | ||
! Note | ! Note | ||
! Name | ! Name | ||
! Associated | ! Associated ratios | ||
|- | |- | ||
| 0 | | 0 | ||
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=== Sagittal notation === | === Sagittal notation === | ||
This notation is a subset of the notations for | This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]]. | ||
==== Evo and Revo flavors ==== | ==== Evo and Revo flavors ==== | ||
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</imagemap> | </imagemap> | ||
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is | Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation. | ||
== Approximation to JI == | == Approximation to JI == | ||
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==== Selected 13-limit intervals ==== | ==== Selected 13-limit intervals ==== | ||
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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|} | |} | ||
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]] | [[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]] | ||
=== Lumatone === | |||
''See [[Lumatone mapping for 10edo]]''. | |||
== Music == | == Music == | ||
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[[Category:10-tone scales]] | [[Category:10-tone scales]] | ||