2.5.7 subgroup: Difference between revisions
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The '''2.5.7 subgroup''', or the '''no-threes 7-limit''' | The '''2.5.7 subgroup''', or the '''no-threes 7-limit''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 5, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[7/4]], [[8/7]], [[7/5]], [[28/25]], [[35/32]], and so on. | ||
The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11. | The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11. | ||
A notable subset of the 2.5.7 subgroup is the 1.5.7 [[tonality diamond]], comprised of all intervals in which 1, 5 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.5.7 tonality diamond within the octave is [[1/1]], [[8/7]], [[5/4]], [[7/5]], [[10/7]], [[8/5]], [[7/4]], and [[2/1]]. | A notable subset of the 2.5.7 subgroup is the 1.5.7 [[tonality diamond]], comprised of all intervals in which 1, 5 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.5.7 tonality diamond (which is the 7-odd-limit (1.3.5.7) with intervals of 3 removed) within the octave is [[1/1]], [[8/7]], [[5/4]], [[7/5]], [[10/7]], [[8/5]], [[7/4]], and [[2/1]]. | ||
Another such subset is the 1.5.7.25.35 tonality diamond, which adds the following intervals to the previous list: [[25/16]], [[25/14]], [[35/32]], [[64/35]], [[28/25]], and [[32/25]]. | Another such subset is the 1.5.7.25.35 tonality diamond, which adds the following intervals to the previous list: [[25/16]], [[25/14]], [[35/32]], [[64/35]], [[28/25]], and [[32/25]]. | ||
When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 5 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]]. | When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 5 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]]. | ||
In [[color notation]], this subgroup may be called '''yaza nowa''', which means that it is the intersection of 2.3.5 and 2.3.7 ("yaza"), but without 3 ("nowa"). | |||
== Properties == | == Properties == | ||
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=== Scales === | === Scales === | ||
{{todo| | {{todo|expand|inline=1}} | ||
= Regular temperaments = | |||
In the below tables, the generator of the temperament is highlighted in bold. Intervals in the tables reflect the 1.5.7.25.35 tonality diamond. It is structurally notable that they come in five clusters, each centered around one note of 6edo - [[35/32]] ~ [[28/25]] ~ [[8/7]] close to a wholetone, [[5/4]] and [[32/25]] close to a ditone, [[7/5]] and [[10/7]] close to a tritone, and so on - with intervals within each cluster separated by the commas 50/49 and 128/125. | |||
== Rank-1 temperaments (edos) == | |||
A list of edos with progressively better<sup>*</sup> tunings for the 2.5.7 subgroup: {{EDOs| 6, 15, 16, 21, 25, 27, 31, 68, 103, 134, 140, 171, 239, 379, 410, 550, 618, 789, 5902, 6691, 7480, 8269, 9058, 9847}}, and so on. | A list of edos with progressively better<sup>*</sup> tunings for the 2.5.7 subgroup: {{EDOs| 6, 15, 16, 21, 25, 27, 31, 68, 103, 134, 140, 171, 239, 379, 410, 550, 618, 789, 5902, 6691, 7480, 8269, 9058, 9847}}, and so on. | ||
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<sup>*</sup> in absolute DKW distance | <sup>*</sup> in absolute DKW distance | ||
== [[Birds]] == | |||
As [[31edo]] is very strong in the 2.5.7 subgroup so that it is a [[weakly consistent circle]] of [[5/4]]'s and [[7/4]]'s (and thus [[8/5]]'s and [[8/7]]'s) and a [[strongly consistent circle]] of [[35/32]]'s (and thus [[64/35]]'s), it makes sense for those interested in high-complexity [[fractional-octave temperaments]] to consider [[31st-octave temperaments]] (temperaments with a 1\31 [[period]]) that preserve this representation of 2.5.7, which can be seen as combining the simplificatory logics of [[didacus]], [[rainy]] and [[Quince clan|quince]]/[[mercy]], which is the 2.5.7-subgroup restriction of [[miracle]]. See [[31st-octave_temperaments#Birds]] for details on the canonical extension of it to the full [[19-limit]] that utilises 31edo's good approximation of the interval [[11/9]] and of the 13:17:19 chord. | |||
== Commas & rank-2 temperaments == | |||
{{Main|Tour of regular temperaments#Clans defined by a 2.5.7 (yaza nowa) comma}} | {{Main|Tour of regular temperaments#Clans defined by a 2.5.7 (yaza nowa) comma}} | ||
=== [[Didacus]] ([[3136/3125]]) === | |||
'''[[Didacus]]''' temperament, which is the 2.5.7 restriction of [[orion]], [[hemimean]] and [[septimal meantone]], tempers out the comma [[3136/3125]] in the 2.5.7 subgroup, which splits the major third ([[5/4]]) into two intervals of [[28/25]], and as ([[7/5]])/([[5/4]]) = 28/25 this implies 7/5 is found at (28/25)<sup>3</sup> and thus 7/4 is found at (28/25)<sup>5</sup>. It has the unique property among 2.5.7 temperaments being simultaneously extremely abundant in simple 2.5.7 harmony and surprisingly accurate. While augmented and jubilic merge the middle interval of the triplets surrounding 1\6 and 5\6 with one of the two nearby intervals, didacus makes it an exact mean between the two. | |||
The [[DKW theory|DKW]] (2.5.7) optimum tuning states ~5/4 is tuned to 388.122c, and therefore ~28/25 to 194.061c; a chart of mistunings of simple intervals is below. | |||
The [[DKW theory|DKW]] (2.5.7) optimum tuning states ~5/4 is tuned to | |||
{| class="wikitable center-1 center-2 center-3 center-4" | {| class="wikitable center-1 center-2 center-3 center-4" | ||
|+ style="font-size: 105%;" | | |+ style="font-size: 105%;" | Didacus (3136/3125) | ||
|- | |- | ||
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings* | ! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings* | ||
| Line 42: | Line 45: | ||
! Optimal tuning !! Deviation | ! Optimal tuning !! Deviation | ||
|- | |- | ||
| 35/32 || 155.140 || | | 35/32 || 155.140 || 158.427 || +3.288 | ||
|- | |- | ||
| 28/25 || 196.198 || | | '''28/25''' || 196.198 || '''194.061''' || -2.137 | ||
|- | |- | ||
| 8/7 || 231.174 || | | 8/7 || 231.174 || 229.695 || -1.479 | ||
|- | |- | ||
| | | 5/4 || 386.314 || 388.122 || +1.808 | ||
|- | |- | ||
| 32/25 || 427.373 || | | 32/25 || 427.373 || 423.756 || -3.617 | ||
|- | |- | ||
| 7/5 || 582.512 || | | 7/5 || 582.512 || 582.183 || -0.329 | ||
|- | |- | ||
| 10/7 || 617.488 || | | 10/7 || 617.488 || 617.817 || +0.329 | ||
|- | |- | ||
| 25/16 || 772.627 || | | 25/16 || 772.627 || 776.244 || +3.617 | ||
|- | |- | ||
| 8/5 || 813.686 || | | 8/5 || 813.686 || 811.878 || -1.808 | ||
|- | |- | ||
| 7/4 || 968.826 || | | 7/4 || 968.826 || 970.305 || +1.479 | ||
|- | |- | ||
| 25/14 || 1003.802 || | | 25/14 || 1003.802 || 1005.939 || +2.137 | ||
|- | |- | ||
| 64/35 || 1044.860 || | | 64/35 || 1044.860 || 1041.573 || -3.288 | ||
|} | |} | ||
<nowiki />* In 2.5.7-targeted DKW tuning | |||
=== [[Rainy]] ([[2100875/2097152]]) === | |||
[[Rainy]] is related to a number of high-limit rank 3 temperaments such as [[valentine]], [[dwynwyn]] and [[tertiaseptal]] in rank 2, and [[eros]] and [[sophia]] in rank 3, though is good as a (rank 2) pure 2.5.7 temperament also. Its generator is [[~]][[256/245]] sharpened by approximately 1{{cent}} which acts as the square root of [[~]][[35/32]], the cube root of [[~]][[8/7]] and the fifth root of [[~]][[5/4]]. Four generators reaches an ambiguous interval whose 2.5.7 interpretation is rather complex, being 2048/1715[[~]]1225/1024, which is the starting point for extensions. The most simple extension, [[valentine]], interprets this interval as a flat [[~]][[6/5]] by tempering (6/5)/(1225/1024) = [[6144/6125]], and the generator as a sharp [[~]][[25/24]], so that [[3/2]] is found at 9 generators, that is, at ([[~]][[8/7]])<sup>3</sup>, so also tempering [[1029/1024|1029/1024 = S7/S8]] = (6/5)/(2048/1715). Meanwhile, a much more accurate and complex mapping is to find [[4/3]] at 22 generators octave reduced, which is the strategy taken by [[tertiaseptal]], notable as the very high-limit [[140edo|140]] & [[311edo|311]] temperament. These two mappings of 3 merge in [[31edo]] (which serves as a trivial tuning of tertiaseptal, as another tuning of tertiaseptal is 311edo - 140edo = [[171edo]], and 171 - 140 = 31). | |||
=== 2.5.7[6 & 60] = 2.5.7-subgroup restriction of [[Waage]] ([[244140625/240945152]]) === | |||
This temperament sharpens [[~]][[28/25]] by 3.8{{cent}} to make it equal to 1\6 so that [[6edo]] is made a [[strongly consistent circle]] of 28/25's, so it is one of the [[6th-octave temperaments]]. It relates the close relation of the 2.5.7 subgroup to hexatonic structure in an intriguing way by contrasting it with ''equalized'' hexatonic structure, chosen to represent ~28/25. | |||
=== [[Cloudy]] ([[16807/16384]]) === | |||
[[Cloudy]] interprets 1\5 = 240{{cent}} as a 9{{cent}} sharp [[~]][[8/7]] so that [[5edo]] is made to be a [[strongly consistent circle]] of 8/7's, by tempering [[16807/16384|2/(8/7)<sup>5</sup>]]. | |||
=== | === [[Augmented]] ([[128/125]]) === | ||
It is debated whether [[augmented]] counts as a normal temperament or an exotemperament, though it is clear it can produce the JI-like effects of [[DR]] in good tunings of it like [[27edo]]. | |||
'''Augmentsept''' temperament tempers out the comma [[128/125]] = S4/S5. Strictly speaking, augmented is a [[5-limit|2.3.5]] temperament, with which the 2.5.7 temperament shares only the same comma, hence this version should be known under a different name. However, both temperaments make the interval [[5/4]] an exact third of the octave, which serves as the period. The generator in this case is then naturally ~8/7. | '''Augmentsept''' temperament tempers out the comma [[128/125]] = S4/S5. Strictly speaking, augmented is a [[5-limit|2.3.5]] temperament, with which the 2.5.7 temperament shares only the same comma, hence this version should be known under a different name. However, both temperaments make the interval [[5/4]] an exact third of the octave, which serves as the period. The generator in this case is then naturally ~8/7. | ||
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|- | |- | ||
| 64/35 || 1044.860 || 1018.297 || -26.563 | | 64/35 || 1044.860 || 1018.297 || -26.563 | ||
|} | |} | ||
<nowiki />* In 2.5.7-targeted DKW tuning | |||
=== | === [[Jubilic]] ([[50/49]]) === | ||
It is debated whether [[jubilic]] counts as a normal temperament or an exotemperament with respect to [[7/5]] and [[10/7]] specifically (which it equates), though it is clear it can produce the JI-like effects of [[DR]] in good tunings of it that focus on optimizing the convincingness of the [[harmonic seventh chord]] [[~]][[4:5:6:7]] and more generally on other intervals of the [[7-odd-limit]] than 7/5 and 10/7. As 7/5 needs to be sharpened (and 10/7 flattened), an optimized tuning flattens [[5/4]] and sharpens [[7/4]] so that it makes sense to temper [[225/224]] so that (5/4)<sup>2</sup> [[~]] [[14/9]], implying tempering [[64/63]], which is especially natural as it makes use of the fact that 7/4 is sharp in an optimized tuning to equate it with [[16/9]] to achieve a half-octave-period analogue of [[archy]] (2.3.7 superpyth). This leads to [[pajara]]. | |||
The [[DKW theory|DKW]] (2.5.7) optimum tuning states ~5/4 is tuned to | The [[DKW theory|DKW]] (2.5.7) optimum tuning states ~5/4 is tuned to 385.002c and therefore that ~8/7 is tuned to 214.998c, showcasing that the DKW optimum tuning can give strange results for high-damage temperaments, so check out [[jubilic]] and [[pajara]] for more reasonable tunings; a chart of mistunings of simple intervals in the DKW optimum tuning is given below to examine how DKW attempts to balance error between higher-complexity intervals of 2.5.7 not usually considered: | ||
{| class="wikitable center-1 center-2 center-3 center-4" | {| class="wikitable center-1 center-2 center-3 center-4" | ||
|+ style="font-size: 105%;" | | |+ style="font-size: 105%;" | Jubilic (50/49) | ||
|- | |- | ||
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings* | ! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings* | ||
| Line 122: | Line 131: | ||
! Optimal tuning !! Deviation | ! Optimal tuning !! Deviation | ||
|- | |- | ||
| 35/32 || 155.140 || | | 35/32 || 155.140 || 170.005 || +14.865 | ||
|- | |- | ||
| 28/25 || 196.198 || | | 28/25 || 196.198 || 214.998 || +18.799 | ||
|- | |- | ||
| | | 8/7 || 231.174 || 214.998 || -16.176 | ||
|- | |- | ||
| 5/4 || 386.314 || | | '''5/4''' || 386.314 || '''385.002''' || -1.311 | ||
|- | |- | ||
| 32/25 || 427.373 || | | 32/25 || 427.373 || 429.995 || +2.623 | ||
|- | |- | ||
| 7/5 || 582.512 || | | 7/5 || 582.512 || 600.000 || +17.488 | ||
|- | |- | ||
| 10/7 || 617.488 || | | 10/7 || 617.488 || 600.000 || -17.488 | ||
|- | |- | ||
| 25/16 || 772.627 || | | 25/16 || 772.627 || 770.005 || -2.623 | ||
|- | |- | ||
| 8/5 || 813.686 || | | 8/5 || 813.686 || 814.998 || +1.311 | ||
|- | |- | ||
| 7/4 || 968.826 || | | 7/4 || 968.826 || 985.002 || +16.176 | ||
|- | |- | ||
| 25/14 || 1003.802 || | | 25/14 || 1003.802 || 985.002 || -18.799 | ||
|- | |- | ||
| 64/35 || 1044.860 || | | 64/35 || 1044.860 || 1029.995 || -14.865 | ||
|} | |} | ||
<nowiki />* In 2.5.7-targeted DKW tuning | |||
== | == Dieses & rank-2 [[exotemperament]]s == | ||
Some important [[diesis|dieses]] in the 2.5.7 subgroup are: | |||
* [[128/125|2/(5/4)<sup>3</sup>]], the [[enharmonic diesis]] (or just "diesis"), leading to [[augmented]] if tempered. | |||
* [[16807/16384|2/(8/7)<sup>5</sup>]], the [[cloudy comma]], leading to [[cloudy]] if tempered, which can be likened in structural role to the enharmonic diesis but for prime 7 instead of prime 5. Equating it with the enharmonic diesis to form a general-purpose diesis results in [[rainy]] temperament, as their difference is the [[rainy comma]]. | |||
* [[50/49|(10/7)/(7/5)]], the [[septimal tritonic diesis]], leading to [[jubilic]] if tempered. | |||
* [[256/245|(8/7)<sup>2</sup>/(5/4)]], which can be likened in structural role to the septimal tritonic diesis, leading to [[Bapbo clan|Bapbo]] if tempered, considered below: | |||
=== [[Bapbo clan|Bapbo]] ([[256/245]]) === | |||
The '''[[Bapbo clan|bapbo]]''' [[exotemperament]] tempers out [[256/245]] = S6 * S8 = S4/S7, equating [[35/32]] with [[8/7]], and making [[5/4]] the square of 8/7, which hence serves as the generator. This comma is over 70 cents, and pathologically equates the outer edges of the clusters around 1\6 and 5\6 to each other without also equating the middle interval with them (if this is done, the tuning reduces to 6edo). Therefore, bapbo can safely be considered an exotemperament - perhaps an analogue to [[mavila]]. | |||
The [[DKW theory|DKW]] (2.5.7) optimum tuning states ~5/4 is tuned to | The [[DKW theory|DKW]] (2.5.7) optimum tuning states ~5/4 is tuned to 410.773c and therefore ~8/7 is tuned to 205.386c; a chart of mistunings of simple intervals is below. | ||
{| class="wikitable center-1 center-2 center-3 center-4" | {| class="wikitable center-1 center-2 center-3 center-4" | ||
|+ style="font-size: 105%;" | | |+ style="font-size: 105%;" | Bapbo (256/245) | ||
|- | |- | ||
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings* | ! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings* | ||
| Line 162: | Line 175: | ||
! Optimal tuning !! Deviation | ! Optimal tuning !! Deviation | ||
|- | |- | ||
| 35/32 || 155.140 || | | 35/32 || 155.140 || '''205.386''' || +50.247 | ||
|- | |- | ||
| | | 28/25 || 196.198 || 173.068 || -23.130 | ||
|- | |- | ||
| 8/7 || 231.174 || | | '''8/7''' || 231.174 || '''205.386''' || -25.788 | ||
|- | |- | ||
| 5/4 || 386.314 || | | 5/4 || 386.314 || 410.773 || +24.459 | ||
|- | |- | ||
| 32/25 || 427.373 || | | 32/25 || 427.373 || 378.454 || -48.918 | ||
|- | |- | ||
| 7/5 || 582.512 || | | 7/5 || 582.512 || 583.841 || +1.329 | ||
|- | |- | ||
| 10/7 || 617.488 || | | 10/7 || 617.488 || 616.159 || -1.329 | ||
|- | |- | ||
| 25/16 || 772.627 || | | 25/16 || 772.627 || 821.546 || +48.918 | ||
|- | |- | ||
| 8/5 || 813.686 || | | 8/5 || 813.686 || 789.227 || -24.459 | ||
|- | |- | ||
| 7/4 || 968.826 || | | 7/4 || 968.826 || 994.614 || +25.788 | ||
|- | |- | ||
| 25/14 || 1003.802 || | | 25/14 || 1003.802 || 1026.932 || +23.130 | ||
|- | |- | ||
| 64/35 || 1044.860 || | | 64/35 || 1044.860 || 994.614 || -50.247 | ||
|} | |} | ||
<nowiki />* In 2.5.7-targeted DKW tuning | |||
== Music == | == Music == | ||
{{todo| | {{todo|expand|inline=1}} | ||
[[Category:Subgroup]] | [[Category:Subgroup]] | ||