m Text replacement - "Ups and Downs Notation" to "Ups and downs notation"
(107 intermediate revisions by 37 users not shown)
Line 1:
Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:phylingual|phylingual]] and made on <tt>2012-05-04 11:24:21 UTC</tt>.<br>
: The original revision id was <tt>330017132</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">35-tET or 35-[[xenharmonic/edo|EDO]] refers to a tuning system which divides the octave into 35 steps of approximately [[xenharmonic/cent|34.29¢]] each.
As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[xenharmonic/macrotonal edos|macrotonal edos]]: [[xenharmonic/5edo|5edo]] and [[xenharmonic/7edo|7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71¢ and 5edo's wide fifth of 720¢. 35edo can also represent the 2.3.5.7.11.17 [[xenharmonic/Just intonation subgroups|subgroup]] and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of all other subgroup generators. Therefore among whitewood tunings it is very versatile, you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9, and if you ignore [[xenharmonic/22edo|22edo]]'s consistent representation of both subgroups. 35edo has the optimal patent val for [[xenharmonic/Greenwoodmic temperaments|greenwood]] and [[xenharmonic/Greenwoodmic temperaments#Secund|secund]] temperaments.
== Theory ==
As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[macrotonal edos]]: [[5edo]] and [[7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71{{c}} and 5edo's wide fifth of 720{{c}}. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1{{c}} of error.
A good beggining for start to play 35-EDO is with the Sub-diatonic scale, that is a [[xenharmonic/MOS|MOS]] of 3L2s: 9 4 9 9 4.
35edo can also represent the 2.3.5.7.11.17 [[subgroup]] and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of all other subgroup generators (7/5 and 17/11 stand out, having less than 1 cent error). Therefore among [[whitewood]] tunings it is very versatile; you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9 (a characteristic of whitewood tunings), and if you ignore [[22edo]]'s more in-tune versions of 35edo MOS's and consistent representation of both subgroups.
=Intervals=
35edo has the optimal [[patent val]] for [[greenwood]] and [[secund]] temperaments, as well as 11-limit [[muggles]], and the 35f val is an excellent tuning for 13-limit muggles. 35edo is the largest edo with a lack of a [[diatonic scale]] (unless 7edo is considered a diatonic scale).
=== Odd harmonics ===
{{Harmonics in equal|35}}
|| Degrees of 35-EDO || Cents value || Ratios in 2.5.7.11.17 subgroup || Ratios with flat 3 || Ratios with 9 ||
== Notation ==
|| 0 || 0 || 1/1 || || ||
The 7edo fifth is preferred as the diatonic generator for ups and downs notation due to being much easier to notate than the 5edo fifth (which involves E and F being enharmonic), as well as being closer to 3/2.
|| 1 || 34.29 || || || ||
{| class="wikitable"
|| 2 || 68.57 || || || ||
|-
|| 3 || 102.86 || 17/16 || || 18/17 ||
! Degrees
|| 4 || 137.14 || || 12/11 || ||
! Cents
|| 5 || 171.43 || 11/10 || || 10/9 ||
! colspan="3" | [[Ups and downs notation]]
|| 6 || 205.71 || || || 9/8 ||
! [[Dual-fifth tuning|Dual-fifth]] notation
|| 7 || 240 || 8/7 || || ||
<small>based on closest 12edo interval</small>
|| 8 || 274.29 || 20/17 || 7/6 || ||
|-
|| 9 || 308.57 || || 6/5 || ||
| 0
|| 10 || 342.86 || 17/14 || || 11/9 ||
| 0.000
|| 11 || 377.14 || 5/4 || || ||
| unison
|| 12 || 411.43 || 14/11 || || 14/11 ||
| 1
|| 13 || 445.71 || 22/17 || || 9/7 ||
| D
|| 14 || 480 || || || ||
| 1sn, prime
|| 15 || 514.29 || || 4/3 || ||
|-
|| 16 || 548.57 || 11/8 || || ||
| 1
|| 17 || 582.86 || 7/5 || 24/17 || ||
| 34.286
|| 18 || 617.14 || 10/7 || 17/12 || ||
| up unison
|| 19 || 651.43 || 16/11 || || ||
| ^1
|| 20 || 685.71 || || 3/2 || ||
| ^D
|| 21 || 720 || || || ||
| augmented 1sn
|| 22 || 754.29 || 17/11 || || 14/9 ||
|-
|| 23 || 788.57 || 11/7 || || ||
| 2
|| 24 || 822.86 || 8/5 || || ||
| 68.571
|| 25 || 857.15 || || || 18/11 ||
| dup unison
|| 26 || 891.43 || || 5/3 || ||
| ^^1
|| 27 || 925.71 || 17/10 || 12/7 || ||
| ^^D
|| 28 || 960 || 7/4 || || ||
| diminished 2nd
|| 29 || 994.29 || || || 16/9 ||
|-
|| 30 || 1028.57 || 20/11 || || 9/5 ||
| 3
|| 31 || 1062.86 || || 11/6 || ||
| 102.857
|| 32 || 1097.14 || 32/17 || || 17/9 ||
| dud 2nd
|| 33 || 1131.43 || || || ||
| vv2
|| 34 || 1165.71 || || || ||
| vvE
=Rank two temperaments=
| minor 2nd
|-
| 4
| 137.143
| down 2nd
| v2
| vE
| neutral 2nd
|-
| 5
| 171.429
| 2nd
| 2
| E
| submajor 2nd
|-
| 6
| 205.714
| up 2nd
| ^2
| ^E
| major 2nd
|-
| 7
| 240
| dup 2nd
| ^^2
| ^^E
| supermajor 2nd
|-
| 8
| 274.286
| dud 3rd
| vv3
| vvF
| diminished 3rd
|-
| 9
| 308.571
| down 3rd
| v3
| vF
| minor 3rd
|-
| 10
| 342.857
| 3rd
| 3
| F
| neutral 3rd
|-
| 11
| 377.143
| up 3rd
| ^3
| ^F
| major 3rd
|-
| 12
| 411.429
| dup 3rd
| ^^3
| ^^F
| augmented 3rd
|-
| 13
| 445.714
| dud 4th
| vv4
| vvG
| diminished 4th
|-
| 14
| 480
| down 4th
| v4
| vG
| minor 4th
|-
| 15
| 514.286
| 4th
| 4
| G
| major 4th
|-
| 16
| 548.571
| up 4th
| ^4
| ^G
| augmented 4th
|-
| 17
| 582.857
| dup 4th
| ^^4
| ^^G
| minor tritone
|-
| 18
| 617.143
| dud 5th
| vv5
| vvA
| major tritone
|-
| 19
| 651.429
| down 5th
| v5
| vA
| diminished 5th
|-
| 20
| 685.714
| 5th
| 5
| A
| minor 5th
|-
| 21
| 720
| up 5th
| ^5
| ^A
| major 5th
|-
| 22
| 754.286
| dup 5th
| ^^5
| ^^A
| augmented 5th
|-
| 23
| 788.571
| dud 6th
| vv6
| vvB
| diminished 6th
|-
| 24
| 822.857
| down 6th
| v6
| vB
| minor 6th
|-
| 25
| 857.143
| 6th
| 6
| B
| neutral 6th
|-
| 26
| 891.429
| up 6th
| ^6
| ^B
| major 6th
|-
| 27
| 925.714
| dup 6th
| ^^6
| ^^B
| augmented 6th
|-
| 28
| 960
| dud 7th
| vv7
| vvC
| diminished 7th
|-
| 29
| 994.286
| down 7th
| v7
| vC
| minor 7th
|-
| 30
| 1028.571
| 7th
| 7
| C
| superminor 7th
|-
| 31
| 1062.857
| up 7th
| ^7
| ^C
| neutral 7th
|-
| 32
| 1097.143
| dup 7th
| ^^7
| ^^C
| major 7th
|-
| 33
| 1131.429
| dud 8ve
| vv8
| vvD
| augmented 7th
|-
| 34
| 1165.714
| down 8ve
| v8
| vD
| diminished 8ve
|-
| 35
| 1200
| 8ve
| 8
| D
| 8ve
|}
===Sagittal notation===
====Best fifth notation====
This notation uses the same sagittal sequence as EDOs [[30edo#Second-best fifth notation|30b]] and [[40edo#Sagittal notation|40]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].
This notation uses the same sagittal sequence as [[42edo#Sagittal notation|42-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>35edo</title></head><body>35-tET or 35-<a class="wiki_link" href="http://xenharmonic.wikispaces.com/edo">EDO</a> refers to a tuning system which divides the octave into 35 steps of approximately <a class="wiki_link" href="http://xenharmonic.wikispaces.com/cent">34.29¢</a> each.<br />
desc none
<br />
rect 80 0 300 50 [[Sagittal_notation]]
As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic <a class="wiki_link" href="http://xenharmonic.wikispaces.com/macrotonal%20edos">macrotonal edos</a>: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5edo">5edo</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7edo">7edo</a>. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71¢ and 5edo's wide fifth of 720¢. 35edo can also represent the 2.3.5.7.11.17 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Just%20intonation%20subgroups">subgroup</a> and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of all other subgroup generators. Therefore among whitewood tunings it is very versatile, you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9, and if you ignore <a class="wiki_link" href="http://xenharmonic.wikispaces.com/22edo">22edo</a>'s consistent representation of both subgroups. 35edo has the optimal patent val for <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Greenwoodmic%20temperaments">greenwood</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Greenwoodmic%20temperaments#Secund">secund</a> temperaments.<br />
rect 391 0 551 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
<br />
rect 20 80 391 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
A good beggining for start to play 35-EDO is with the Sub-diatonic scale, that is a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS">MOS</a> of 3L2s: 9 4 9 9 4.<br />
Ups and downs can be used to name 35edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used. An up or down immediately 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.
<table class="wiki_table">
0-10-20 = C E G = C = C or C perfect
<tr>
<td>Degrees of 35-EDO<br />
</td>
<td>Cents value<br />
</td>
<td>Ratios in 2.5.7.11.17 subgroup<br />
</td>
<td>Ratios with flat 3<br />
</td>
<td>Ratios with 9<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td>1/1<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>34.29<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>68.57<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>102.86<br />
</td>
<td>17/16<br />
</td>
<td><br />
</td>
<td>18/17<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>137.14<br />
</td>
<td><br />
</td>
<td>12/11<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>171.43<br />
</td>
<td>11/10<br />
</td>
<td><br />
</td>
<td>10/9<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>205.71<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>9/8<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>240<br />
</td>
<td>8/7<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>274.29<br />
</td>
<td>20/17<br />
</td>
<td>7/6<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>308.57<br />
</td>
<td><br />
</td>
<td>6/5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>342.86<br />
</td>
<td>17/14<br />
</td>
<td><br />
</td>
<td>11/9<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>377.14<br />
</td>
<td>5/4<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>411.43<br />
</td>
<td>14/11<br />
</td>
<td><br />
</td>
<td>14/11<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>445.71<br />
</td>
<td>22/17<br />
</td>
<td><br />
</td>
<td>9/7<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>480<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>514.29<br />
</td>
<td><br />
</td>
<td>4/3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>548.57<br />
</td>
<td>11/8<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>582.86<br />
</td>
<td>7/5<br />
</td>
<td>24/17<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>617.14<br />
</td>
<td>10/7<br />
</td>
<td>17/12<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>651.43<br />
</td>
<td>16/11<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>685.71<br />
</td>
<td><br />
</td>
<td>3/2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>720<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>754.29<br />
</td>
<td>17/11<br />
</td>
<td><br />
</td>
<td>14/9<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>788.57<br />
</td>
<td>11/7<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>822.86<br />
</td>
<td>8/5<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>857.15<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>18/11<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>891.43<br />
</td>
<td><br />
</td>
<td>5/3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>925.71<br />
</td>
<td>17/10<br />
</td>
<td>12/7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>960<br />
</td>
<td>7/4<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>994.29<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>16/9<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>1028.57<br />
</td>
<td>20/11<br />
</td>
<td><br />
</td>
<td>9/5<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>1062.86<br />
</td>
<td><br />
</td>
<td>11/6<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>1097.14<br />
</td>
<td>32/17<br />
</td>
<td><br />
</td>
<td>17/9<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>1131.43<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>1165.71<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank two temperaments</h1>
35 equal divisions of the octave (abbreviated 35edo or 35ed2), also called 35-tone equal temperament (35tet) or 35 equal temperament (35et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 35 equal parts of about 34.3 ¢ each. Each step represents a frequency ratio of 21/35, or the 35th root of 2.
As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic macrotonal edos: 5edo and 7edo. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71 ¢ and 5edo's wide fifth of 720 ¢. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1 ¢ of error.
35edo can also represent the 2.3.5.7.11.17 subgroup and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of all other subgroup generators (7/5 and 17/11 stand out, having less than 1 cent error). Therefore among whitewood tunings it is very versatile; you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9 (a characteristic of whitewood tunings), and if you ignore 22edo's more in-tune versions of 35edo MOS's and consistent representation of both subgroups.
35edo has the optimal patent val for greenwood and secund temperaments, as well as 11-limit muggles, and the 35f val is an excellent tuning for 13-limit muggles. 35edo is the largest edo with a lack of a diatonic scale (unless 7edo is considered a diatonic scale).
The 7edo fifth is preferred as the diatonic generator for ups and downs notation due to being much easier to notate than the 5edo fifth (which involves E and F being enharmonic), as well as being closer to 3/2.
This notation uses the same sagittal sequence as EDOs 30b and 40, and is a superset of the notation for 7-EDO.
Second-best fifth notation
This notation uses the same sagittal sequence as 42-EDO, and is a superset of the notation for 5-EDO.
Chord Names
Ups and downs can be used to name 35edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used. An up or down immediately 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.
