10edo: Difference between revisions
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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 | 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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! Degree | ! Degree | ||
! Cents | ! Cents | ||
! Approximate ratios<ref> | ! Approximate ratios<ref group="note">{{sg|limit=2.15.7.13-subgroup}}</ref> | ||
! Additional ratios <br> of 3, 5 and 9<ref> | ! Additional ratios<br />of 3, 5, and 9<ref group="note">Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals</ref> | ||
! Interval names | ! Interval names | ||
! colspan="3" | [[Ups and downs notation]]<br />([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2) | ! colspan="3" | [[Ups and downs notation]]<br />([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2) | ||
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| [[File:0-1200 octave.mp3|frameless]] | | [[File:0-1200 octave.mp3|frameless]] | ||
|} | |} | ||
== Notation == | == Notation == | ||
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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 === | ||
See above: Heptatonic 3rd-generated notation. | See above: Heptatonic 3rd-generated 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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! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve stretch (¢) | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
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=== Uniform maps === | === Uniform maps === | ||
{{Uniform map| | {{Uniform map|edo=10}} | ||
=== Commas === | === Commas === | ||
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{| class="commatable wikitable center-1 center-2 right-4 center-5" | {| class="commatable wikitable center-1 center-2 right-4 center-5" | ||
|- | |- | ||
! [[Harmonic limit|Prime<br>limit]] | ! [[Harmonic limit|Prime<br />limit]] | ||
! [[Ratio]]<ref group="note">{{rd}}</ref> | ! [[Ratio]]<ref group="note">{{rd}}</ref> | ||
! [[Monzo]] | ! [[Monzo]] | ||
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| Island comma, parizeksma | | Island comma, parizeksma | ||
|} | |} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|- | |- | ||
! Periods <br> per 8ve | ! Periods<br />per 8ve | ||
! Generator | ! Generator | ||
! Temperament(s) | ! Temperament(s) | ||
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== Scales == | == Scales == | ||
=== MOS scales === | === MOS scales === | ||
* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1 | * Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1 | ||
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1 | * Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1 | ||
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1 | * Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1 | ||
=== Other scales === | === Other scales === | ||
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=== Horagrams === | === Horagrams === | ||
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]] | [[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]] | ||
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]] | [[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]] | ||
== Diagrams == | == Diagrams == | ||
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== Instruments == | == Instruments == | ||
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 | 10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple. | ||
Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features. | Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features. | ||
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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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<references group="note" /> | <references group="note" /> | ||
[[Category: | [[Category:10-tone scales]] | ||