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{{interwiki
This page is about of a [[MOSScales|MOSScale]] with 7 large steps and 2 small steps arranged LLLsLLLLs (or any rotation of that, such as LLsLLLsLL).
| de = Mavila
| en =
| es =
| ja =
}}
{{Infobox MOS
| Name = mavila, superdiatonic
| Periods = 1
| nLargeSteps = 7
| nSmallSteps = 2
| Equalized = 5
| Paucitonic = 4
| Pattern = LLLsLLLLs
| Neutral = 5L 4s
}}
'''7L 2s''', '''mavila''' (/ˈmɑːvɪlə/ or /ˈmævɪlə/ ''MA(H)-vil-ə''), or '''superdiatonic''' refers to the structure of octave-equivalent [[MOS]] scales with generators ranging from 4\7 (four degrees of [[7edo]] = 685.71¢) to 5\9 (five degrees of [[9edo]] = 666.67¢) and its associated harmonic framework. In the case of 9edo, L and s are the same size; in the case of 7edo, s becomes so small it disappears (and all that remains are the seven equal L's). Mavila was first discovered by [[Erv Wilson]] after studying the tuning of the "Timbila" music of the Chopi tribe in Mozambique. It is also closely related to the "pelog" scale in Indonesian and Balinese Gamelan music.
== Introduction ==
In mavila, the fifths are so flat that they are even flatter than 7-EDO. As a result, stacking 7 of these fifths gives you an "[[2L 5s|anti-diatonic]]" MOS scale, where in a certain sense, major and minor intervals get "reversed." For example, stacking four fifths and octave-reducing now gets you a *minor* third, whereas stacking three fourths and octave-reducing now gets you a *major* third. (Note that since we have a heptatonic scale, terms like "fifths," "thirds," etc make perfect sense and really are five, three, etc steps in the anti-diatonic scale.)


This has some very strange implications for music. The mavila diatonic scale is similar to the normal diatonic scale - except interval classes are flipped. Wherever there was a major third, you'll find a minor third, and vice versa. Half steps become whole steps and whole steps become half steps (closer to neutral second range, however). When you sharpen the leading tone in minor, you end up sharpening it down instead, meaning you flatten it. Also, minor is now major - you end up with three parallel natural/harmonic/melodic major scales, and only one minor scale. Instead of a diminished triad in the major scale, there is now an augmented triad.
If you're looking for highly accurate scales (that is, ones that approximate JI closely), there are much better scale patterns to look at. However, if your harmonic entropy is coarse enough (that is, if 678 cents is an acceptable '3/2' to you), then [[Pelogic_family|mávila]] is an important harmonic entropy minimum here. So a general name for this MOS pattern could be "Mávila Superdiatonic" or simply 'Superdiatonic'.


As an example, the anti-Ionian scale has steps of ssLsssL, which looks like the regular Ionian scale except the "L" intervals are now "s" and vice versa.
These scales are strongly associated with the [[Armodue|Armodue]] project/system applied too on Septimal-mávila and Hornbostel temperaments.


In addition to the 7-note anti-diatonic scale described, Mavila also has a 9 note "superdiatonic" MOS, the "super-Ionian" mode of which is LLLsLLLLs. This is the basis for [[Armodue theory]].
Optional types of 'JI [[Blown_Fifth|Blown Fifth]]' Generators: 31/21, 34/23, 65/44, 71/48, 99/67, 105/71, 108/73, 133/90, 145/98, 176/119 & 250/169.


== Notation ==
In this article we use JKLMNOPQRJ = the symmetric Olympian mode (LLSLLLSLL), J = 261.6255653. &/@ = raise/lower by one chroma. So the fifth chain becomes ... P@ L@ Q M R N J O K P L Q& M& ...
A possible alternative is to keep the 7-note fifth chain FCGDAEB from diatonic and extend it to a 9-note one, YFCGDAEBX, and to use #/b for the 7L 2s chroma. So the extended chain becomes ... Bb Xb Y F C G D A E B X Y# F# ...
== Tunings ==
Much like [[5L 2s|5L 2s diatonic]], mavila is supported by several low-numbered EDOs, which will basically be the same size as the MOS's listed above.
7edo can be thought of as a degenerate tuning, yielding a totally equal heptatonic scale that is equally diatonic and anti-diatonic.
The next EDO supporting Mavila is 9edo, which can be thought of as the first mavila EDO (and the first EDO in general) differentiating between major and minor chords. This is fairly interesting, as there is no real equivalent in meantone terms. It is larger than the "diatonic" sized MOS, but smaller than the 16-tone "chromatic" MOS. It is best thought of as a "superdiatonic" scale. The fifth is 667 cents.
It is also supported by 16edo, which is probably the most common tuning for mavila. This can be thought of as the first EDO offering the potential for chromatic mavila harmony, similar to 12edo for meantone. This is also the usual setting for the aforementioned Armodue theory, although the Armodue theory can easily be extended to larger mavila scales such as mavila[23]. The fifth is 675 cents.
The next EDO supporting mavila is 23edo, which is the second-most common tuning for mavila, used frequently by Igliashon Jones in his Cryptic Ruse albums. The fifth is 678 cents, and as a result the harmonic properties are slightly better than 16edo, although still fairly inharmonic compared to meantone. The anti-diatonic scale is more "quasi-equal" in this tuning than in 16edo.
25edo also contains mavila, although the tuning is 672 cents and hence very flat, even flatter than 16edo. The major third is 384¢, close to a just [[5/4]] and the minor third is 288¢, close to a just [[13/11]].
Mavila defines a tuning "spectrum", similarly to the meantone spectrum. The fifth of 7edo (~686 cents) is often thought of as a dividing line between diatonic and mavila: if the fifth is flatter than this, it will generate anti-diatonic scales but ''not'' 7L 2s superdiatonic scales (it will generate [[2L 7s]] instead), and if it is sharper than this, it will generate diatonic scales. The fifth of 9edo is also often thought of as the other tuning endpoint on the mavila spectrum.
== Intervals ==
== Modes ==
:''See also [[Mavila modal harmony]]
From brightest to darkest, the superdiatonic modes are:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | LLLLsLLLs - (Super)Lydian
! colspan="3" | Generator
! | <span style="display: block; text-align: center;">'''Generator size (cents)'''</span>
! | Pentachord steps
! | Comments
|-
| | 4\[[7edo|7]]
| |
| |
| | 685.714
| | 1 1 1 0
| |
|-
| |
| |  
| |  
| | 102\[[179edo|179]]
| | 683.798
| | 25 25 25 2
| | Approximately 0.03 cents away from [[95/64]]
|-
|-
| | LLLsLLLLs - (Super)Ionian
| | 33\[[58edo|58]]
| |
| |  
| |  
| | 682.758
| | 8 8 8 1
| | 2 generators equal 11/10, 6 equal 4/3, creating a hybrid Mávila/Porcupine scale with three perfect 5ths as well as the flat ones.
|-
|-
| | LLLsLLLsL - (Super)Mixolydian
| | 21\37
| |
| |
| | 681.081
| | 5 5 5 1
| |  
| |  
|-
|-
| | LLsLLLLsL - (Super)Corinthian
| | 17\30
| |
| |
| | 680
| | 4 4 4 1
| | L/s = 4
|-
| |
| | 30\53
| |
| | 679.245
| | 7 7 7 2
| |  
| |  
|-
|-
| | LLsLLLsLL - (Super)Olympian
| |  
| | [[File:MavilaOlympian16edo.mp3]]
| | 43\76
| |
| | 678.947
| | 10 10 10 3
| |
|-
|-
| | LsLLLLsLL - (Super)Dorian
| |  
| | 56\99
| |
| | 678.788
| | 13 13 13 4
| |  
| |  
|-
|-
| | LsLLLsLLL - (Super)Aeolian
| |  
| | 69\122
| |
| | 678.6885
| | 16 16 16 5
| |  
| |  
|-
|-
| | sLLLLsLLL - (Super)Phrygian
| |  
| | 82\145
| |
| | 678.621
| | 19 19 19 6
| |  
| |  
|-
|-
| | sLLLsLLLL - (Super)Locrian
| |  
| | [[File:MavilaSuperdiatonic16edo.mp3]]
| | 95\168
|}
| |
 
| | 678.571
The modes of the antidiatonic scale are simply named after the existing diatonic scale modes, but with the "Anti-" prefix (e.g. Anti-ionian, Anti-aeolian, etc). The modes of the superdiatonic scale are also named after the existing modes, but contain the "Super" prefix (e.g. Superionian, Superaeolian, etc.). The "anti" and "super" prefixes can be left in to explicitly distinguish which MOS's modes you're talking about, or can be omitted for convention.
| | 22 22 22 7
 
| |
Each superdiatonic mode contains its corresponding mode in the antidiatonic scale. Additionally, the superdiatonic modes also resemble the "shape" of their meantone diatonic counterparts. This leads to a pattern: LLLsLLLLs both resembles the meantone LLsLLLs Ionian mode, and contains the mavila ssLsssL anti-Ionian mode as well. Additionally, sLLLsLLLL resembles the diatonic sLLsLLL Locrian mode, and also contains the mavila LssLsss anti-Ionian mode. Furthermore, every mavila mode contains a tonic triad of the -opposite- quality as the corresponding diatonic mode, so that Superionian and Anti-ionian contain a minor triad, and Superphrygian and Antiphrygian contain a major triad.
 
Since there are only seven diatonic modes, two of the superdiatonic modes need additional names and cannot reference any mode of the diatonic scale. These two modes present themselves as "mixed" modes, which begin with the LLs tetrachord, and so contain both the ~300 cent minor third and the ~375 cent major third (and hence both minor and major triads). These are the only two modes to exhibit this behavior. They're interspersed on the rotational continuum between Ionian and Dorian, and Mixolydian and Aeolian.
 
As were the original modes named after regions of ancient Greece, so are these new superdiatonic extensions. The one between Ionian and Dorian is called Corinthian, after the Greek island of Corinth, set up so that the Ionian -&gt; Corinthian -&gt; Dorian cyclic sequence will resemble the columns of ancient Greek architecture. The mode cyclically placed between Mixolydian and Aeolian, which is the symmetrical LLsLLLsLL scale, has a number of noteworthy theoretical properties, in that it contains every modal rotation of mavila[5], and hence the entire mavila-tempered 5-limit tonality diamond; it was given the name of Olympian to match its unique status in this regard.
=== Table of modes ===
 
== Chords and extended harmony ==
 
<!--== Primodal theory ==
=== Neji versions of superdiatonic modes ===
* 40:48:52:54:59:64:70:77:80 Pental Superionian
=== 16nejis ===
=== 23nejis ===
=== 25nejis ===
-->
 
== Scale tree ==
Mavila generates a 16 tone chromatic MOS. In a certain sense, much of mavila makes sense if viewed within the lens of a 16-tone chromatic gamut, similarly to how much of meantone is thought of in the setting of a 12-tone chromatic gamut.
 
After the 16 tone chromatic scale is the 23 tone enharmonic MOS, which can be thought of as an "extended mavila" analogous to the "extended meantone" 19-tone enharmonic scale. If the mavila fifth is flatter than that of 16-EDO (675 cents), it will instead generate an MOS at 25 notes. This is similar to how if the meantone fifth is tuned sharper than 12-EDO, it will instead generate a 17-tone MOS rather than a 19-tone one.
 
{| class="wikitable"
|-
|-
! colspan="5" | Generator
| |
! | Generator size (cents)
| |
! | L/s
| |
! | Comments
| | 678.569
| | π π π 1
| | L/s = π
|-
|-
| | 4\[[7edo|7]]
| |  
| |  
| | 108\191
| |  
| |  
| |
| | 678.534
| |
| | 25 25 25 8
| | 685.714
| | 1/0
| |  
| |  
|-
|-
| |
| |  
| |
| | 121\214
| |
| |  
| |
| | 678.505
| | 21\37
| | 28 28 28 9
| | 681.08
| | 28;9 Superdiatonic 1/28-tone <span style="font-size: 12.8000001907349px;">(a slight exceeded representation of the ratio 262144/177147, the Pythagorean wolf Fifth)</span>
| | 5/1
| |
|-
|-
| |
| |  
| |
| | 134\237
| |
| |  
| | 17\30
| | 678.481
| |
| | 31 31 31 10
| | 680
| | HORNBOSTEL TEMPERAMENT <span style="font-size: 12.8000001907349px;">(1/31-tone; Optimum high size of Hornbostel '6th')</span>
| | 4/1
| |
|-
|-
| |
| |
| | 13\23
| | 13\23
| |
| |  
| |
| |  
| | 678.261
| | 678.261
| | 3/1
| | 3 3 3 1
| |
| | HORNBOSTEL TEMPERAMENT <span style="font-size: 12.8000001907349px;">(Armodue 1/3-tone)</span>
|-
| |
| | 126\223
| |
| | 678.027
| | 29 29 29 10
| | HORNBOSTEL TEMPERAMENT
 
<span style="font-size: 12.8000001907349px;">(Armodue 1/29-tone)</span>
|-
| |
| | 113\200
| |
| | 678
| | 26 26 26 9
| | HORNBOSTEL (&amp; [[Alexei_Stepanovich_Ogolevets|OGOLEVETS]]) TEMPERAMENT <span style="font-size: 12.8000001907349px;">(Armodue 1/26-tone; Best equillibrium between 6/5, Phi (833.1 Cent) and Square root of Pi (990.9 Cent), the notes '3', '7' &amp; '8')</span>
|-
| |
| | 100\177
| |
| | 677.966
| | 23 23 23 8
| |
|-
| |
| | 87\154
| |
| | 677.922
| | 20 20 20 7
| |  
|-
|-
| |
| |  
| |  
| |
| | 74\131
| |
| | 677.863
| | 17 17 17 6
| | Armodue-Hornbostel 1/17-tone <span style="font-size: 12.8000001907349px;">(the Golden Tone System of Thorvald Kornerup and a temperament of the Alexei Ogolevets's list of temperaments)</span>
|-
| |
| | 61\108
| |
| | 677.778
| | 14 14 14 5
| | Armodue-Hornbostel 1/14-tone
|-
| |
| | 109\193
| |
| | 677.720
| | 25 25 25 9
| | Armodue-Hornbostel 1/25-tone
|-
| |
| | 48\85
| |
| | 677.647
| | 11 11 11 4
| | Armodue-Hornbostel 1/11-tone <span style="font-size: 12.8000001907349px;">(Optimum accuracy of Phi interval, the note '7')</span>
|-
| |
| |
| |
| | 677.562
| | e e e 1
| | L/s = e
|-
| |
| | 35\62
| |
| | 677.419
| | 8 8 8 3
| | Armodue-Hornbostel 1/8-tone
|-
| |
| | 92\163
| |
| | 677.301
| | 21 21 21 8
| | 21;8 Superdiatonic 1/21-tone
|-
| |
| |
| |
| | 677.28
| | <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ+1 φ+1 φ+1 1</span>
| | Split φ superdiatonic relation (34;13 - 55;21 - 89;34 - 144;55 - 233;89 - 377;144 - 610;233..)
|-
| |
| | 57\101
| |
| | 677.228
| | 13 13 13 5
| | 13;5 Superdiatonic 1/13-tone
|-
| | 22\39
| | 22\39
| |
| |
| |  
| | 676.923
| | 676.923
| | 5/2
| | 5 5 5 2
| | Armodue-Hornbostel 1/5-tone <span style="font-size: 12.8000001907349px;">(Optimum low size of Hornbostel '6th')</span>
|-
| |
| | 75\133
| |
| | 676.692
| | 17 17 17 7
| | 17;7 Superdiatonic 1/17-tone <span style="font-size: 12.8000001907349px;">(Note the very accuracy of the step 75 with the ratio 34/23 with an error of +0.011 Cents)</span>
|-
| |
| | 53\94
| |
| | 676.596
| | 12 12 12 5
| |
|-
| |
| | 31\55
| |
| | 676.364
| | 7 7 7 3
| | 7;3 Superdiatonic 1/7-tone
|-
| |
| | 40\71
| |
| | 676.056
| | 9 9 9 4
| | 9;4 Superdiatonic 1/9-tone
|-
| |
| | 49\87
| |
| | 675.862
| | 11 11 11 5
| | 11;5 Superdiatonic 1/11-tone
|-
| |
| | 58\103
| |
| | 675.728
| | 13 13 13 6
| | 13;6 Superdiatonic 1/13-tone
|-
|-
| |
| | 9\16
| | 9\16
| |  
| |  
| |  
| |  
| |
| | 675
| | 675
| | 2/1
| | 2 2 2 1
| | Boundary of propriety; smaller generators are strictly proper
| | <span style="display: block; text-align: left;">'''[BOUNDARY OF PROPRIETY: smaller generators are strictly proper]'''</span>ARMODUE ESADECAFONIA (or Goldsmith Temperament)
|-
| |
| | 59\105
| |
| | 674.286
| | 13 13 13 7
| | Armodue-Mávila 1/13-tone
|-
| |
| | 50\89
| |
| | 674.157
| | 11 11 11 6
| | Armodue-Mávila 1/11-tone
|-
| |
| | 41\73
| |
| | 673.973
| | 9 9 9 5
| | Armodue-Mávila 1/9-tone <span style="font-size: 12.8000001907349px;">(with an approximation of the Perfect Fifth + 1/5 Pyth.Comma [706.65 Cents]: 43\73 is 706.85 Cents)</span>
|-
| |
| | 32\57
| |
| | 673.684
| | 7 7 7 4
| | Armodue-Mávila 1/7-tone <span style="font-size: 12.8000001907349px;">(the 'Commatic' version of Armodue, because its high accuracy of the [[7/4|7/4]] interval, the note '8')</span>
|-
| |
| |
| |
| | 673.577
| | <span style="background-color: #ffffff;">√3 √3 √3 1</span>
| |
|-
| |
| | 55\98
| |
| | 673.469
| | 12 12 12 7
| |
|-
| |
| | 78\139
| |
| | 673.381
| | 17 17 17 10
| | Armodue-Mávila 1/17-tone
|-
| |
| | 101\180
| |
| | 673.333
| | 22 22 22 13
| |
|-
|-
| |
| |
| | 23\41
| | 23\41
| |  
| |  
| |  
| |  
| | 673.171
| | 673.171
| | 5/3
| | 5 5 5 3
| | 5;3 Golden Armodue-Mávila 1/5-tone
|-
| |  
| |  
| | 60\107
| |
| | 672.897
| | 13 13 13 8
| | 13;8 Golden Mávila 1/13-tone
|-
|-
| |  
| |  
| |  
| |  
| |  
| |  
| |
| |
| | 672.85
| | 672.85
| | φ/1
| | <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ φ φ 1</span>
| | Golden mavila
| | GOLDEN MÁVILA (L/s = <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ)</span>
|-
| |
| |
| | 97\173
| | 672.832
| | 21 21 21 13
| | 21;13 Golden Mávila 1/21-tone <span style="font-size: 12.8000001907349px;">(Phi is the step 120\173)</span>
|-
| |
| | 37\66
| |
| | 672.727
| | 8 8 8 5
| | 8;5 Golden Mávila 1/8-tone
|-
| |
| | 51\91
| |
| | 672.527
| | 11 11 11 7
| | 11;7 Superdiatonic 1/11-tone
|-
| |
| |
| |
| | 672.523
| | π π π 2
| |
|-
| |
| |
| | 116\207
| | 672.464
| | 25 25 25 16
| | 25;16 Superdiatonic 1/25-tone
|-
| |
| | 65\116
| |
| | 672.414
| | 14 14 14 9
| | 14;9 Superdiatonic 1/14-tone
|-
| |
| | 79\141
| |
| | 672.340
| | 17 17 17 11
| | 17;11 Superdiatonic 1/17-tone
|-
| |
| | 93\166
| |
| | 672.289
| | 20 20 20 13
| |
|-
| |
| | 107\191
| |
| | 672.251
| | 23 23 23 15
| |
|-
| |
| | 121\216
| |
| | 672.222
| | 26 26 26 17
| | 26;17 Superdiatonic 1/26-tone
|-
| |
| | 135\241
| |
| | 672.199
| | 29 29 29 19
| | 29;19 Superdiatonic 1/29-tone
|-
|-
| |
| |
| | 14\25
| | 14\25
| |  
| |  
| |
| |  
| | 672
| | 672
| | 3/2
| | 3 3 3 2
| | 3;2 Golden Armodue-Mávila 1/3-tone
|-
| |
| | 145\259
| |
| | 671.815
| | 31 31 31 21
| | 31;21 Superdiatonic 1/31-tone
|-
| |
| | 131\234
| |
| | 671.795
| | 28 28 28 19
| | 28;19 Superdiatonic 1/28-tone
|-
| |
| | 117\209
| |
| | 671.770
| | 25 25 25 17
| |
|-
| |
| | 103\184
| |
| | 671.739
| | 22 22 22 15
| |
|-
| |
| | 89\159
| |
| | 671.698
| | 19 19 19 13
| |
|-
| |
| | 75\134
| |
| | 671.642
| | 16 16 16 11
| |
|-
| |
| | 61\109
| |
| | 671.560
| | 13 13 13 9
| |
|-
| | 47\84
| |
| |
| | 671.429
| | 10 10 10 7
| |
|-
| | 33\59
| |
| |
| | 671.186
| | 7 7 7 5
| |  
| |  
|-
|-
| |
| |
| |
| | 19\34
| | 19\34
| |
| |  
| |  
| | 670.588
| | 670.588
| | 4/3
| | 4 4 4 3
| |  
| |  
|-
|-
| |
| |
| |
| |
| | 24\43
| | 24\43
| |
| |
| | 669.767
| | 669.767
| | 5/4
| | 5 5 5 4
| |  
| |  
|-
|-
Line 212: Line 516:
| |  
| |  
| |  
| |  
| |
| |
| | 666.667
| | 666.667
| | 1/1
| | 1 1 1 1
| |  
| |  
|}
|}
 
[[Category:diatonic]]
== Pieces ==
[[Category:mavila]]
* [https://soundcloud.com/starshine99/undercity-ft-hatsune-miku (Undercity ft. Hatsune Miku)]
* [https://www.youtube.com/watch?v=bgnGZCQr5yE ks26 - Wallowing in Madness]
* [https://www.youtube.com/watch?v=2p3z9YEpW1k Sevish - Sea Poem]
* [https://www.youtube.com/watch?v=1tdHPqKPOWc Marooned at Home (mavila tuning)]
* [https://www.youtube.com/watch?v=QzZw-KCn2ig "Netbeans" (in a tuning of Mavila)]
* [https://en.xen.wiki/w/File:Mavila_Jazz_Rhodes_1.mp3 Mavila Jazz Rhodes 1]
* [https://cityoftheasleep.bandcamp.com/track/illegible-red-ink Illegible Red Ink]
* [https://cityoftheasleep.bandcamp.com/track/run-run-red-robot Run Run Red Robot]
 
[[Category:Mavila| ]] <!-- main article -->
[[Category:Theory]]
[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]
[[Category:Abstract MOS patterns]]
[[Category:Superdiatonic]]
[[Category:scale]]
[[Category:scales]]
[[Category:superdiatonic]]
[[Category:theory]]

Revision as of 04:22, 14 April 2021

This page is about of a MOSScale with 7 large steps and 2 small steps arranged LLLsLLLLs (or any rotation of that, such as LLsLLLsLL).

If you're looking for highly accurate scales (that is, ones that approximate JI closely), there are much better scale patterns to look at. However, if your harmonic entropy is coarse enough (that is, if 678 cents is an acceptable '3/2' to you), then mávila is an important harmonic entropy minimum here. So a general name for this MOS pattern could be "Mávila Superdiatonic" or simply 'Superdiatonic'.

These scales are strongly associated with the Armodue project/system applied too on Septimal-mávila and Hornbostel temperaments.

Optional types of 'JI Blown Fifth' Generators: 31/21, 34/23, 65/44, 71/48, 99/67, 105/71, 108/73, 133/90, 145/98, 176/119 & 250/169.

Generator Generator size (cents) Pentachord steps Comments
4\7 685.714 1 1 1 0
102\179 683.798 25 25 25 2 Approximately 0.03 cents away from 95/64
33\58 682.758 8 8 8 1 2 generators equal 11/10, 6 equal 4/3, creating a hybrid Mávila/Porcupine scale with three perfect 5ths as well as the flat ones.
21\37 681.081 5 5 5 1
17\30 680 4 4 4 1 L/s = 4
30\53 679.245 7 7 7 2
43\76 678.947 10 10 10 3
56\99 678.788 13 13 13 4
69\122 678.6885 16 16 16 5
82\145 678.621 19 19 19 6
95\168 678.571 22 22 22 7
678.569 π π π 1 L/s = π
108\191 678.534 25 25 25 8
121\214 678.505 28 28 28 9 28;9 Superdiatonic 1/28-tone (a slight exceeded representation of the ratio 262144/177147, the Pythagorean wolf Fifth)
134\237 678.481 31 31 31 10 HORNBOSTEL TEMPERAMENT (1/31-tone; Optimum high size of Hornbostel '6th')
13\23 678.261 3 3 3 1 HORNBOSTEL TEMPERAMENT (Armodue 1/3-tone)
126\223 678.027 29 29 29 10 HORNBOSTEL TEMPERAMENT

(Armodue 1/29-tone)

113\200 678 26 26 26 9 HORNBOSTEL (& OGOLEVETS) TEMPERAMENT (Armodue 1/26-tone; Best equillibrium between 6/5, Phi (833.1 Cent) and Square root of Pi (990.9 Cent), the notes '3', '7' & '8')
100\177 677.966 23 23 23 8
87\154 677.922 20 20 20 7
74\131 677.863 17 17 17 6 Armodue-Hornbostel 1/17-tone (the Golden Tone System of Thorvald Kornerup and a temperament of the Alexei Ogolevets's list of temperaments)
61\108 677.778 14 14 14 5 Armodue-Hornbostel 1/14-tone
109\193 677.720 25 25 25 9 Armodue-Hornbostel 1/25-tone
48\85 677.647 11 11 11 4 Armodue-Hornbostel 1/11-tone (Optimum accuracy of Phi interval, the note '7')
677.562 e e e 1 L/s = e
35\62 677.419 8 8 8 3 Armodue-Hornbostel 1/8-tone
92\163 677.301 21 21 21 8 21;8 Superdiatonic 1/21-tone
677.28 φ+1 φ+1 φ+1 1 Split φ superdiatonic relation (34;13 - 55;21 - 89;34 - 144;55 - 233;89 - 377;144 - 610;233..)
57\101 677.228 13 13 13 5 13;5 Superdiatonic 1/13-tone
22\39 676.923 5 5 5 2 Armodue-Hornbostel 1/5-tone (Optimum low size of Hornbostel '6th')
75\133 676.692 17 17 17 7 17;7 Superdiatonic 1/17-tone (Note the very accuracy of the step 75 with the ratio 34/23 with an error of +0.011 Cents)
53\94 676.596 12 12 12 5
31\55 676.364 7 7 7 3 7;3 Superdiatonic 1/7-tone
40\71 676.056 9 9 9 4 9;4 Superdiatonic 1/9-tone
49\87 675.862 11 11 11 5 11;5 Superdiatonic 1/11-tone
58\103 675.728 13 13 13 6 13;6 Superdiatonic 1/13-tone
9\16 675 2 2 2 1 [BOUNDARY OF PROPRIETY: smaller generators are strictly proper]ARMODUE ESADECAFONIA (or Goldsmith Temperament)
59\105 674.286 13 13 13 7 Armodue-Mávila 1/13-tone
50\89 674.157 11 11 11 6 Armodue-Mávila 1/11-tone
41\73 673.973 9 9 9 5 Armodue-Mávila 1/9-tone (with an approximation of the Perfect Fifth + 1/5 Pyth.Comma [706.65 Cents]: 43\73 is 706.85 Cents)
32\57 673.684 7 7 7 4 Armodue-Mávila 1/7-tone (the 'Commatic' version of Armodue, because its high accuracy of the 7/4 interval, the note '8')
673.577 √3 √3 √3 1
55\98 673.469 12 12 12 7
78\139 673.381 17 17 17 10 Armodue-Mávila 1/17-tone
101\180 673.333 22 22 22 13
23\41 673.171 5 5 5 3 5;3 Golden Armodue-Mávila 1/5-tone
60\107 672.897 13 13 13 8 13;8 Golden Mávila 1/13-tone
672.85 φ φ φ 1 GOLDEN MÁVILA (L/s = φ)
97\173 672.832 21 21 21 13 21;13 Golden Mávila 1/21-tone (Phi is the step 120\173)
37\66 672.727 8 8 8 5 8;5 Golden Mávila 1/8-tone
51\91 672.527 11 11 11 7 11;7 Superdiatonic 1/11-tone
672.523 π π π 2
116\207 672.464 25 25 25 16 25;16 Superdiatonic 1/25-tone
65\116 672.414 14 14 14 9 14;9 Superdiatonic 1/14-tone
79\141 672.340 17 17 17 11 17;11 Superdiatonic 1/17-tone
93\166 672.289 20 20 20 13
107\191 672.251 23 23 23 15
121\216 672.222 26 26 26 17 26;17 Superdiatonic 1/26-tone
135\241 672.199 29 29 29 19 29;19 Superdiatonic 1/29-tone
14\25 672 3 3 3 2 3;2 Golden Armodue-Mávila 1/3-tone
145\259 671.815 31 31 31 21 31;21 Superdiatonic 1/31-tone
131\234 671.795 28 28 28 19 28;19 Superdiatonic 1/28-tone
117\209 671.770 25 25 25 17
103\184 671.739 22 22 22 15
89\159 671.698 19 19 19 13
75\134 671.642 16 16 16 11
61\109 671.560 13 13 13 9
47\84 671.429 10 10 10 7
33\59 671.186 7 7 7 5
19\34 670.588 4 4 4 3
24\43 669.767 5 5 5 4
5\9 666.667 1 1 1 1
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