Tetrachordal otonal scale: Difference between revisions
mNo edit summary |
Things |
||
| Line 8: | Line 8: | ||
* Take two harmonic series segments (rows): 2k::4k and 3k::6k. The 3k::6k is called the '''bottom row.''' | * Take two harmonic series segments (rows): 2k::4k and 3k::6k. The 3k::6k is called the '''bottom row.''' | ||
* The rows will overlap exactly at the points of the harmonic series segment k::2k. The set of all TO scales for each k is called '''category k.''' Category 0 and 1 do not have any meaningful TO scales. | * The rows will overlap exactly at the points of the harmonic series segment k::2k. The set of all TO scales for each k is called '''category k.''' Category 0 and 1 do not have any meaningful TO scales. | ||
* Any point where the two rows overlap is called a '''transition point'''. A transition point at 4/3 or 3/2 is called '''perfect''', between 4/3 and 3/2 '''imperfect'''. Outside 4/3 ≤ c ≤ 3/2 the scale the transition is '''non-tetrachordal,''' and between that range, the TO scale can be built and thus they are optionally called '''tetrachordal''' transition points. | * Any point where the two rows overlap is called a '''transition point'''. A transition point at 4/3 or 3/2 is called '''perfect''', between 4/3 and 3/2 '''imperfect'''. Outside 4/3 ≤ c ≤ 3/2 the scale the transition is '''non-tetrachordal,''' and between that range, the TO scale can be built and thus they are optionally called '''tetrachordal''' transition points. In this article, all transition points are tetrachordal. | ||
* To build a TO scale, build a scale by starting on the bottom row and transitioning to the top row at a transition point between 4/3 and 3/2 inclusive. | * To build a TO scale, build a scale by starting on the bottom row and transitioning to the top row at a transition point between 4/3 and 3/2 inclusive. | ||
== Properties == | == Properties == | ||
* Category x's amount of transition points can be derived with the following formula:<math>J\left(x\right)=1+\operatorname{floor}\left(\frac{x}{6}\right)-I\left(\frac{\operatorname{mod}\left(x,6\right)-1}{6}\right) \text{for} \space I\left(x\right)=1-\operatorname{ceil}\left(x\right)+\operatorname{floor}\left(x\right)</math> Which for each category x = 2,3,4… the list of tetrachordial transition points starts with 1,1,1,1,2,1,2,2,2… | * Category x's amount of transition points can be derived with the following formula:<math>J\left(x\right)=1+\operatorname{floor}\left(\frac{x}{6}\right)-I\left(\frac{\operatorname{mod}\left(x,6\right)-1}{6}\right) \text{for} \space I\left(x\right)=1-\operatorname{ceil}\left(x\right)+\operatorname{floor}\left(x\right)</math> Which for each category x = 2,3,4… the list of tetrachordial transition points starts with 1,1,1,1,2,1,2,2,2… | ||
* The amount of perfect transition points of a scale can be derived with <math>K\left(x\right)=I\left(\frac{x}{2}\right)+I\left(\frac{x}{3}\right)</math>. Thus the amount of imperfect transition points is <math>J\left(x\right)-K\left(x\right)</math>. | * The amount of perfect transition points of a scale can be derived with <math>K\left(x\right)=I\left(\frac{x}{2}\right)+I\left(\frac{x}{3}\right)</math>. Thus the amount of imperfect transition points is <math>J\left(x\right)-K\left(x\right)</math>. | ||
* '''Clean scales''' are scales with that transition only once, from the bottom to the top row, and their amount is equal to the number of transition points <math>J(x)</math>. | * '''Clean scales''' are scales with that transition only once, from the bottom to the top row, and their amount is equal to the number of transition points <math>J(x)</math>. | ||
| Line 20: | Line 20: | ||
* The amount of notes a scale of a given category x has <math>S(x)</math> falls between the ranges of these functions:<math>2x+1+\operatorname{floor}\left(\frac{x-1}{3}\right) ≤ S(x) ≤ 2x+1+\operatorname{floor}\left(\frac{x-2}{2}\right) </math> And the amount of notes are closely related to the scale tree of diatonic scales. | * The amount of notes a scale of a given category x has <math>S(x)</math> falls between the ranges of these functions:<math>2x+1+\operatorname{floor}\left(\frac{x-1}{3}\right) ≤ S(x) ≤ 2x+1+\operatorname{floor}\left(\frac{x-2}{2}\right) </math> And the amount of notes are closely related to the scale tree of diatonic scales. | ||
* The prime limits of the TOs of a category k sit between the biggest primes smaller than the 3 times the smallest and biggest transition point ratios. | * The prime limits of the TOs of a category k sit between the biggest primes smaller than the 3 times the smallest and biggest transition point ratios. | ||
* All TOs are chiral, for which the right hand version is TO, the left hand is TU. | |||
The data can be summarized in this table: | The data can be summarized in this table: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! <math>x</math> !! | ! <math>x</math> !! Transition points | ||
Number of clean scales | Number of clean scales | ||
! Perfect TP !! Imperfect TP !! Total scales !! Convoluted scales !! colspan="2" | Note | ! Perfect TP !! Imperfect TP !! Total scales !! Convoluted scales !! colspan="2" | Note ranges | ||
! colspan="2" |p-limits | ! colspan="2" |p-limits | ||
|- | |- | ||
| Line 84: | Line 85: | ||
== List of categories == | == List of categories == | ||
For every provided TO, the [[EFR]] and used segment are written out explicitly. Names in italic are names never seen before on the wiki, declared here for specificity. | For every provided TO, the [[EFR]] and used segment are written out explicitly. Names in italic are names never seen before on the wiki, declared here for specificity. Only TOs are listed: to get the respective TUs, retrovert the scale. | ||
For categories with more than one scale, append letters from a up to whatever is necessary, which works for categories from 2 to 31 except 30 (using letters a-p). After that, append numbers. | For categories with more than one scale, append letters from a up to whatever is necessary, which works for categories from 2 to 31 except 30 (using letters a-p). After that, append numbers. | ||
| Line 123: | Line 124: | ||
=== Category 3 === | === Category 3 === | ||
<code>18:20:22:24:27:30:33:36</code>: | <code>18:20:22:24:27:30:33:36</code>: TO 3. | ||
Second prime TO. It is naturally a [[7edo]] detemper in the [[2.3.5.11 subgroup]], and either a [[porcupine]] ([[1L 6s]] 3|3) or [[dicot]]/[[mothra]] detemper ([[3L 4s]] 5|1 #5). | Second prime TO. It is naturally a [[7edo]] detemper in the [[2.3.5.11 subgroup]], and either a [[porcupine]] ([[1L 6s]] 3|3) or [[dicot]]/[[mothra]] detemper ([[3L 4s]] 5|1 #5). | ||
| Line 179: | Line 180: | ||
=== Category 4 === | === Category 4 === | ||
<code>24:26:28:30:32:34:36:39:42:45:48</code>. | <code>24:26:28:30:32:34:36:39:42:45:48</code>: TO 4. | ||
First composite TO, containing TO 2 as a subset. It is naturally a [[10edo]] detemper in the [[2.3.5.7.13 subgroup]], a [[negri]]<nowiki/>detemper ([[1L 8s]] 4|4) if you exclude 17/12, and a [[pajarous]] detemper, 2L 8s 6|2 #7 scale. | First composite TO, containing TO 2 as a subset. It is naturally a [[10edo]] detemper in the [[2.3.5.7.13 subgroup]], a [[negri]]<nowiki/>detemper ([[1L 8s]] 4|4) if you exclude 17/12, and a [[pajarous]] detemper, 2L 8s 6|2 #7 scale. | ||
| Line 286: | Line 287: | ||
=== Category 5 === | === Category 5 === | ||
<code>30:32:34:36:38:40:42:45:48:51:54:57:60</code>. | <code>30:32:34:36:38:40:42:45:48:51:54:57:60</code>: TO 5. | ||
'''15::20'''. | '''15::20'''. Third prime TO. Its extension is naturally a very close [[12edo]] detemper in the [[2.3.5.17.19 subgroup]], and a [[diaschismic]] detemper ([[10L 2s]] 8|2 b4) scale. | ||
{| class="wikitable" data-darkreader-inline-color="" | {| class="wikitable" data-darkreader-inline-color="" | ||
!16/15 | !16/15 | ||
| Line 437: | Line 438: | ||
=== Category 6 === | === Category 6 === | ||
It is the last category to feature no imperfect transition points. Last category to feature TO [[constant structure]]<nowiki/>s. It is a [[14edo|14c-edo]] and [[15edo]] detemper in the 2.3.5.7.11.19.23 subgroup (add-17 or add-13), and [[nautilus]] detemper ([[14L 1s]] 6|8). | It is the last category to feature no imperfect transition points and the first category to feature two transition points, and thus two scales. Last category to feature TO [[constant structure]]<nowiki/>s. It is a [[14edo|14c-edo]] and [[15edo]] detemper in the 2.3.5.7.11.19.23 subgroup (add-17 or add-13), and [[nautilus]] detemper ([[14L 1s]] 6|8). | ||
==== TO 6a ==== | ==== TO 6a ==== | ||
| Line 874: | Line 875: | ||
<code>42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84</code>''':''' TO 7. The last category to feature only one scale. | <code>42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84</code>''':''' TO 7. The last category to feature only one scale. | ||
Fourth prime TO. TU 7 is surprisingly a subharmonic interpolation of TO 3. It is naturally a [[17edo]] detemper in the no-5,17,19,add-25-29-limit subgroup, surfacing as a strange liese extension detemper ([[2L 15s]] 15|1 10b). | |||
It is also the biggest category to feature only one scale. | It is also the biggest category to feature only one scale. | ||
| Line 1,170: | Line 1,171: | ||
=== Category 8 === | === Category 8 === | ||
It contains TO 2 and TO 4. With supersets of TO 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last category to fit completely within the [[31-limit]]. TO | It contains TO 2 and TO 4. With supersets of TO 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last category to fit completely within the [[31-limit]]. TO 8a adds 23, TO 8b adds 17. | ||
The scales can be seen [[wilsec]] detemper (negri weak extension) [[19L 1s]] 10|9, but it is a very innacurate realization that only [[19edo]] through patent val can faithfully represent (well enough for TO | The scales can be seen [[wilsec]] detemper (negri weak extension) [[19L 1s]] 10|9, but it is a very innacurate realization that only [[19edo]] through patent val can faithfully represent (well enough for TO 8a albeit badly for TO 8b as it tempers together 17/12 and 35/24). Thus, it's better seen as just a linear interpolation of TO 4 in the 2.3.5.7.11.13.17.29.31 subgroup. | ||
==== TO 8a ==== | ==== TO 8a ==== | ||
| Line 1,950: | Line 1,951: | ||
==== TO 9a ==== | ==== TO 9a ==== | ||
<code>54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96:99:102:105:108</code>. The biggest | <code>54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96:99:102:105:108</code>. The biggest TO to feature a palette below the 31-limit, with [[harmonics 9-18]] as a subset, as well as TO 3. | ||
No patent val supports an equalized tuning; 19edo, the closest match, tempers together 34/27 and 35/27, as well as 17/9 and 35/128. The closest match is 22edo, which either of 22 or 22f vals provide the basic tuning for [[1L 19s]]. | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||