Superpyth: Difference between revisions
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{{interwiki | {{interwiki | ||
| en = Superpyth | |||
| de = Superpyth | | de = Superpyth | ||
| es = | | es = | ||
| ja = | | ja = | ||
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{{Infobox regtemp | {{Infobox regtemp | ||
| Title = Archy; superpyth | | Title = Archy; superpyth | ||
| Subgroups = 2.3.7, 2.3.5.7 | | Subgroups = 2.3.7, 2.3.5.7, 2.3.5.7.11 | ||
| Comma basis = [[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] ( | | Comma basis = [[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] (7-limit); <br>[[64/63]], [[100/99]], [[245/243]] (11-limit) | ||
| Mapping = 1; 1 9 -2 | | Mapping = 1; 1 9 -2 16 | ||
| Edo join 1 = | | Edo join 1 = 22 | Edo join 2 = 27e | ||
| | | Generators = 3/2 | ||
| | | Generators tuning = 710.1 | ||
| Optimization method = CWE | | Optimization method = CWE | ||
| Pergen = (P8, P5) | | Pergen = (P8, P5) | ||
| Color name = Ruti | | Color name = Ruti | ||
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[5L 17s]] | | MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[5L 17s]] | ||
| Odd limit 1 = | | Odd limit 1 = 2.3.7 7 | Mistuning 1 = 9.09 | Complexity 1 = 5 | ||
| Odd limit 2 = 9 | Mistuning 2 = 15.27 | Complexity 2 = 12 | | Odd limit 2 = 9 | Mistuning 2 = 15.27 | Complexity 2 = 12 | ||
}} | }} | ||
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Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[22edo|13\22]] (~1/4 septimal comma) and [[27edo|16\27]] (~1/3 septimal comma) are the most common tunings of the generator. | Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[22edo|13\22]] (~1/4 septimal comma) and [[27edo|16\27]] (~1/3 septimal comma) are the most common tunings of the generator. | ||
If intervals of 5 are desired, the 5th harmonic is canonically mapped to +9 generators through tempering out [[245/243]], so 5/4 is an augmented second (e.g. C–D♯, a | If intervals of [[5/1|5]] are desired, the 5th harmonic is canonically mapped to +9 generators through tempering out [[245/243]], so [[5/4]] is an augmented second (e.g. C–D♯). This mapping equates the pythagorean limma, [[256/243]], to the syntonic [[81/80]], tempering out [[20480/19683]], so that 5/4 is mapped to a major third minus a limma. Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of [[7/1|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex. | ||
If intervals of 11 are desired, the canonical way is to map [[11/8]] to +16 generators, or a doubly augmented second (C–D𝄪), tempering out [[100/99]]. A simpler but less accurate way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out [[99/98]]. The latter is called '''suprapyth''', a name coined by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group | ''A few full 11-limit 896/891 temperaments'']</ref>. The [[2.3.7.11 subgroup]] restriction of suprapyth, known as [[supra]], is also notable. The two mappings unite on [[22edo]]. | |||
If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out [[91/90]]. In practice, however, this mapping only works in [[27edo]], as flatter tunings swap the sizes of [[13/12]] and [[14/13]]. An alternative mapping is –14 generators (known as [[uberpyth]]), or a doubly diminished octave (C–C𝄫), by tempering out [[144/143]]. This has a more flexible range, but the 13 tends to be tuned very sharp except in 27edo, as [[13/8]] is equated with [[18/11]]. A more practical option is to split the sharp ~3/2 into two ~[[16/13]]'s, which results in [[beatles]], and has an alternative mapping of primes 5 and 11. Alternatively, one can keep the superpyth mappings of 5 and 11 to get [[archytas clan #Thomas|thomas]]. | |||
If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out [[ | |||
[[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22. | [[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22. | ||
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|} | |} | ||
<nowiki/>* In 2.3.7-subgroup [[CWE]] tuning, <br>octave reduced | <nowiki/>* In 2.3.7-subgroup [[CWE]] tuning, <br>octave reduced | ||
</div></div> | </div></div> | ||
<div><div style="display: inline-grid;"> | <div><div style="display: inline-grid;"> | ||
| Line 145: | Line 119: | ||
<nowiki/>* In 7-limit CWE tuning, octave reduced | <nowiki/>* In 7-limit CWE tuning, octave reduced | ||
</div></div> | </div></div> | ||
== Notation == | |||
Because superpyth is generated by an octave and a fifth, [[chain-of-fifths notation]] can be used. However, if one wishes to use the 5-limit triads as bases for harmony, then much of the logic which is used in [[meantone]] cannot be used in superpyth, as superpyth does not temper out [[81/80]]. For example, the [[4:5:6|classical major triad]] on C is written as C–D♯–G rather than C–E–G as in meantone, which is awkward to notate and conceptualize. To solve this, one may want to adopt a pair of accidentals (such as ^ and v) to represent modifications by 81/80, thus notating the major triad as C–vE–G and the minor triad as C–^E♭–G. 81/80 is equated to [[28/27]], [[36/35]], and [[256/243]] in superpyth, leading to the equivalences {{nowrap| ^C {{=}} D♭ }}, {{nowrap| E {{=}} vF }}, etc. The limma (C–D♭) thus becomes the most important interval for note alterations, being around a quartertone in size and representing so many important ratios, rather than the apotome (C–C♯) as in meantone, which is a submajor second in size in superpyth. | |||
== Chords and harmony == | == Chords and harmony == | ||
{{See also| Chords of superpyth }} | {{See also| Chords of superpyth }} | ||
Superpyth contains a version of the [[5L 2s|diatonic]] scale where the major third represents [[9/7]], and the minor third represents [[7/6]]. {{W|Tertian harmony}} can thus be used, with the major and minor triads representing [[14:18:21|1–9/7–3/2]] and [[6:7:9|1–7/6–3/2]] respectively, rather than the | Superpyth contains a version of the [[5L 2s|diatonic]] scale where the major third represents [[9/7]], and the minor third represents [[7/6]]. {{W|Tertian harmony}} can thus be used, with the major and minor triads representing [[14:18:21|1–9/7–3/2]] and [[6:7:9|1–7/6–3/2]] respectively, rather than the 1–5/4–3/2 and [[10:12:15|1–6/5–3/2]] triads in meantone. However, the contrast between these triads is not as effective as the contrast between the meantone triads, as the interval between 7/6 and 9/7 is too wide, being ~140–180{{c}} in size rather than the ideal ~60–80{{c}} semitone in meantone. | ||
Perhaps a more interesting approach is for the tonic chords of superpyth to be considered the tetrad 1–7/6–4/3–3/2 ([[6:7:8:9]]) and its utonal inverse 1–9/8–9/7–3/2 (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of {{w|Steely Dan}} fame). Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add 1–9/8–4/3–3/2 (a sus2-4 chord). These three chords comprise the three ways to divide the superpyth perfect fifth into two whole tones and one septimal minor third. In the diatonic major scale, the 1–7/6–4/3–3/2 chord occurs on II, III, and VI, while its inverse occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense, though in a different way from in [[mavila]]. [[Chromatic]] or [[enharmonic]] alterations of them also exist, for example, the 1–9/8–9/7–3/2 chord may be altered to 1–9/8–11/8–3/2 (8:9:11:12), which is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 1–9/8–9/7–3/2. | |||
Another approach takes account of the fact that, in the 5-limit, the major triad can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, with the minor triad being its utonal inversion. A similar construction of septimal chords gives us 1–7/6–4/3 ([[6:7:8]]) and its inversion 1–8/7–4/3 ([[21:24:28]]). These intervals contrast by [[49/48]], similarly to how 5-limit thirds contrast by [[25/24]]. There are some issues, however. For example, the 6:7:8 chord has the root on the top rather than the bottom, and the notes may clash from being too close to each other. However, the wide voicing of these chords, those being 1–7/4–3 (4:7:12) and 1–12/7–3 (7:12:21), solve both of these issues. These triads span a twelfth. In terms of the [[chain of fifths]], these chords are simpler in superpyth than the 5-limit triads in meantone. | |||
Therefore, it may be helpful to also consider the [[9-odd-limit]] [[anomalous saturated suspension|saturated suspensions]], 1–7/6–3/2–7/4 ([[12:14:18:21]]) and 1–9/7–3/2–12/7 ([[14:18:21:24]]), which extend the chords above and are good for creating tensions and resolutions: 1–9/7–3/2–12/7 on the fifth degree creates a leading tone that wants to go to the tonic; 1–7/6–3/2–7/4 on the fourth degree creates a flat sixth that wants to go to the fifth. | |||
In meantone, the dominant seventh chord, a tempering of [[20:25:30:36|1–5/4–3/2–9/5]], is often used on the dominant to resolve to the tonic. A similar chord in superpyth is [[28:36:42:49|1–9/7–3/2–7/4]], with a leading tone at 9/7 above the perfect fifth, or [[27/14]]. This chord contains a [[49/36]] [[tritone]] between the 9/7 and 7/4, which creates tension in the chord. (However, 11-limit superpyth maps it to [[15/11]], making it a [[15-odd-limit]] [[swetismic chords|swetismic essentially tempered chord]].) | |||
== Scales == | == Scales == | ||
; 5-note mos ([[2L 3s]], proper) | ; 5-note mos ([[2L 3s]], proper) | ||
In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around [[5edo]]. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~[[28/27]] becoming a chroma. | In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around [[5edo]]. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~[[28/27]] becoming a chroma. See [[Pentatonic Functional Just System]] for a further explanation of what such a system would look like. | ||
; 7-note mos ([[5L 2s]], improper) | ; 7-note mos ([[5L 2s]], improper) | ||
In contrast to the meantone diatonic scale, the superpyth diatonic is improper. Since the fifth is sharp rather than flat in meantone, the large steps (major seconds) are wider, being around | In contrast to the meantone diatonic scale, the superpyth diatonic is improper. Since the fifth is sharp rather than flat in meantone, the large steps (major seconds) are wider, being around 220{{c}} in size. The small steps (minor seconds) are thus narrower, being around 50{{c}} (a quartertone) wide. This has the effect of large and small steps being more distinct compared to meantone diatonic, as well as stronger leading tones due to narrower small steps, though one may want to bend the leading tone down by a small step to avoid it being too close to the tonic. | ||
; 12-note mos ([[5L 7s]], improper) | ; 12-note mos ([[5L 7s]], improper) | ||
| Line 171: | Line 150: | ||
=== Scala files === | === Scala files === | ||
* [[Archy5]] – | * [[Archy5]] – in 49edo tuning | ||
* [[Archy7]] – | * [[Archy7]] – in 49edo tuning | ||
* [[Archy12]] – | * [[Archy12]] – in 49edo tuning | ||
* [[12-22a]] – in 22edo tuning | |||
* [[12-22a]] – | |||
== Tunings == | == Tunings == | ||
| Line 188: | Line 165: | ||
In general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; [[71edo]] (709.859{{c}}) and [[93edo]] with its sharp fifth of 709.677{{c}} come very close to forming closed systems of 2/7-comma. | In general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; [[71edo]] (709.859{{c}}) and [[93edo]] with its sharp fifth of 709.677{{c}} come very close to forming closed systems of 2/7-comma. | ||
27edo is also the point where superpyth tunes 5/4 to the familiar 400{{c}} major third of [[12edo]], and in sharper tunings different mappings of 5/4 arise with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to [[19edo]] (which represents [[1/3-comma meantone]] and is on the edge between septimal meantone and [[flattone]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the [[11-limit]] canonical extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. | 27edo is also the point where superpyth tunes 5/4 to the familiar 400{{c}} major third of [[12edo]], and in sharper tunings different mappings of 5/4 arise with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to [[19edo]] (which represents [[1/3-comma meantone]] and is on the edge between septimal meantone and [[flattone]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]), which map [[7/5]] wider than [[10/7]] in the superpyth mapping. Furthermore, the [[11-limit]] canonical extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11 as well as 7/5 and 10/7, and 27e-edo conflating 11/8 with 7/5. Suprapyth, on the other hand, only works in 22edo, as sharper tunings map [[11/10]] wider than [[12/11]]. | ||
Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical "modified meantones" ([[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], …), as they prioritize the tuning of 3/2 more than the accuracy of septimal harmony. The | Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical "modified meantones" ([[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], …), as they prioritize the tuning of 3/2 more than the accuracy of septimal harmony. The [[supra]] mapping of the [[2.3.7.11 subgroup]], and the quasisuper mapping of 5, work best for tunings in the range of 17c-edo to 22-edo. | ||
A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth. | A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth. | ||
Finally, it may be noted that the {{w|plastic | Finally, it may be noted that the {{w|plastic ratio}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. In fact, it is the tuning that makes ~6:7:8 become +1+1 [[delta-rational]]. | ||
=== Norm-based tunings === | === Norm-based tunings === | ||
| Line 249: | Line 226: | ||
|} | |} | ||
=== Tuning | {| class="wikitable mw-collapsible mw-collapsed" | ||
{| class="wikitable center-all left-4" | |+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings | ||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~3/2 = 709.5143{{c}} | |||
| CWE: ~3/2 = 710.0129{{c}} | |||
| POTE: ~3/2 = 710.1747{{c}} | |||
|} | |||
=== Other tunings === | |||
* [[DKW theory|DKW]] (2.3.5 superpyth): ~2 = 1200.000{{c}}, ~3/2 = 709.758{{c}} | |||
* DKW (2.3.7 archy): ~2 = 1200.000{{c}}, ~3/2 = 712.585{{c}} | |||
=== Tuning spectra === | |||
==== Archy ==== | |||
{| class="wikitable center-all left-4 left-5" | |||
|- | |- | ||
! Edo<br>generator | ! Edo<br>generator | ||
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]* | ! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]* | ||
! Generator (¢) | ! Generator (¢) | ||
! Extension | |||
! Comments | ! Comments | ||
|- | |- | ||
| '''[[7edo|4\7]]''' | |||
| | |||
| '''685.714''' | |||
| | | | ||
| '''Lower bound of 2.3.7-subgroup 9-odd-limit diamond monotone''' | |||
|- | |||
| [[12edo|7\12]] | |||
| | |||
| 700.000 | |||
| ↓ Dominant | |||
| | |||
|- | |||
| | |||
| 3/2 | | 3/2 | ||
| 701.955 | | 701.955 | ||
| | |||
| Pythagorean tuning | | Pythagorean tuning | ||
|- | |- | ||
| [[17edo|10\17]] | | [[17edo|10\17]] | ||
| | | | ||
| 705.882 | | 705.882 | ||
| | | ↑ Dominant<br>↓ Quasisuper | ||
| | |||
|- | |- | ||
| | | | ||
| 81/56 | | 81/56 | ||
| 706.499 | | 706.499 | ||
| | |||
| 1/6 comma | | 1/6 comma | ||
|- | |- | ||
| | | | ||
| 27/14 | | 27/14 | ||
| 707.408 | | 707.408 | ||
| | |||
| 1/5 comma | | 1/5 comma | ||
|- | |- | ||
| [[39edo|23\39]] | | [[39edo|23\39]] | ||
| | |||
| 707.692 | |||
| | |||
| 39d val | |||
|- | |||
| | |||
| 9/7 | |||
| 708.771 | |||
| | |||
| 1/4 comma, 2.3.7-subgroup 9-odd-limit minimax | |||
|- | |||
| [[22edo|13\22]] | |||
| | |||
| 709.091 | |||
| ↑ Quasisuper<br>↓ Superpyth | |||
| | |||
|- | |||
| | |||
| 49/27 | |||
| 709.745 | |||
| | |||
| 2/7 comma | |||
|- | |||
| [[49edo|29\49]] | |||
| | |||
| 710.204 | |||
| | |||
| | |||
|- | |||
| | |||
| 7/6 | |||
| 711.043 | |||
| | |||
| 1/3 comma, 2.3.7-subgroup 7-odd-limit minimax | |||
|- | |||
| [[27edo|16\27]] | |||
| | |||
| 711.111 | |||
| ↑ Superpyth<br>↓ Quasiultra | |||
| | |||
|- | |||
| [[59edo|35\59]] | |||
| | |||
| 711.864 | |||
| | |||
| | |||
|- | |||
| [[32edo|19\32]] | |||
| | |||
| 712.500 | |||
| ↑ Quasiultra<br>↓ Ultrapyth | |||
| | |||
|- | |||
| | |||
| 49/48 | |||
| 712.861 | |||
| | |||
| 2/5 comma, 2.3.7-subgroup CEE tuning | |||
|- | |||
| [[37edo|22\37]] | |||
| | |||
| 713.514 | |||
| | |||
| | |||
|- | |||
| [[42edo|25\42]] | |||
| | |||
| 714.286 | |||
| | |||
| | |||
|- | |||
| | |||
| 7/4 | |||
| 715.587 | |||
| | |||
| 1/2 comma | |||
|- | |||
| '''[[5edo|3\5]]''' | |||
| | |||
| '''720.000''' | |||
| ↑ Ultrapyth | |||
| '''Upper bound of 2.3.7-subgroup 7- and 9-odd-limit diamond monotone''' | |||
|- | |||
| | |||
| 21/16 | |||
| 729.219 | |||
| | |||
| Full comma | |||
|} | |||
<nowiki/>* Besides the octave | |||
==== Superpyth ==== | |||
{| class="wikitable center-all left-4" | |||
|- | |||
! Edo<br>generator | |||
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]* | |||
! Generator (¢) | |||
! Comments | |||
|- | |||
| | |||
| 3/2 | |||
| 701.955 | |||
| Pythagorean tuning | |||
|- | |||
| [[17edo|10\17]] | |||
| | | | ||
| | | 705.882 | ||
| | | 17e val | ||
|- | |- | ||
| | | | ||
| 9/7 | | 9/7 | ||
| 708.771 | | 708.771 | ||
| | | | ||
|- | |- | ||
| | | | ||
| 15/8 | | 15/8 | ||
| 708.807 | | 708.807 | ||
| | |||
|- | |||
| | |||
| 11/10 | |||
| 709.286 | |||
| | | | ||
|- | |- | ||
| Line 295: | Line 420: | ||
| | | | ||
| '''709.091''' | | '''709.091''' | ||
| '''Lower bound of 7- and | | '''Lower bound of 7-, 9-, and 11-odd-limit diamond monotone''' | ||
|- | |||
| | |||
| 11/8 | |||
| 709.457 | |||
| | |||
|- | |- | ||
| | | | ||
| Line 301: | Line 431: | ||
| 709.590 | | 709.590 | ||
| 9-odd-limit minimax | | 9-odd-limit minimax | ||
|- | |- | ||
| [[71edo|42\71]] | | [[71edo|42\71]] | ||
| Line 315: | Line 440: | ||
| 15/14 | | 15/14 | ||
| 709.954 | | 709.954 | ||
| | |||
|- | |||
| | |||
| 11/6 | |||
| 709.958 | |||
| | | | ||
|- | |- | ||
| Line 320: | Line 450: | ||
| 25/24 | | 25/24 | ||
| 710.040 | | 710.040 | ||
| | |||
|- | |||
| | |||
| 11/7 | |||
| 710.138 | |||
| | | | ||
|- | |- | ||
| Line 325: | Line 460: | ||
| | | | ||
| 710.204 | | 710.204 | ||
| | |||
|- | |||
| | |||
| 15/11 | |||
| 710.508 | |||
| | | | ||
|- | |- | ||
| Line 330: | Line 470: | ||
| | | | ||
| 710.526 | | 710.526 | ||
| | | 76bcdee val | ||
|- | |||
| | |||
| 11/9 | |||
| 710.529 | |||
| | |||
|- | |- | ||
| | | | ||
| 5/3 | | 5/3 | ||
| 710.545 | | 710.545 | ||
| | |||
|- | |||
| | |||
| 21/11 | |||
| 710.620 | |||
| | | | ||
|- | |- | ||
| Line 345: | Line 495: | ||
| 7/6 | | 7/6 | ||
| 711.043 | | 711.043 | ||
| | | | ||
|- | |- | ||
| [[27edo|16\27]] | | '''[[27edo|16\27]]''' | ||
| | | | ||
| '''711.111''' | |||
| 27e val, '''upper bound of 11-odd-limit diamond monotone''' | |||
|- | |- | ||
| | | | ||
| Line 361: | Line 511: | ||
| 711.772 | | 711.772 | ||
| | | | ||
|- | |- | ||
| [[32edo|19\32]] | | [[32edo|19\32]] | ||
| | | | ||
| 712.500 | | 712.500 | ||
| | | 32cee val | ||
|- | |- | ||
| | | | ||
| 7/4 | | 7/4 | ||
| 715.587 | | 715.587 | ||
| | | | ||
|- | |- | ||
| '''[[5edo|3\5]]''' | | '''[[5edo|3\5]]''' | ||
| | | | ||
| '''720.000''' | | '''720.000''' | ||
| ''' | | 5e val, '''upper bound of 7- and 9-odd-limit diamond monotone''' | ||
|- | |- | ||
| | | | ||
| 21/16 | | 21/16 | ||
| 729.219 | | 729.219 | ||
| | | | ||
|} | |} | ||
<nowiki/>* Besides the octave | <nowiki/>* Besides the octave | ||
== Music == | == Music == | ||