Parapyth: Difference between revisions
m →Theory |
|||
| (40 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
'''Parapyth''' | '''Parapyth''' is generally viewed as a no-5 2.3.7.11.13 rank-3 regular temperament {{nowrap|[[17edo|17]] & [[41edo|41]] & [[46edo|46]],}} sometimes called '''Parapythic'''. (In the strict sense, ''parapyth'' is Margo Schulter's rank-3 tuning construct inspired by medieval European and Middle Eastern music theory, where Schulter imposes commas that are to be ''observed'' as well as commas that are to be tempered out.{{citation needed}}) | ||
== | == Structure == | ||
Parapyth has two non-period generators: the fifth (which is tuned in the [[gentle region]], i.e. somewhat sharp of just but flatter than 17edo) and the "spacer" 28/27, which is equated to 33/32. This is equivalent to taking the spacer 64/63, which may be equated with 81/80 (as in [[46edo]] and [[87edo]]) or tempered smaller than 81/80 (mapping [[5120/5103]] negatively, as in [[63edo]] and [[80edo]]) assuming prime 5 is added. | |||
=== Intervals === | === Intervals === | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 24: | Line 24: | ||
|13/7 | |13/7 | ||
|1128.4 | |1128.4 | ||
| | |52/27 | ||
|- | |- | ||
!-6 | !-6 | ||
| Line 47: | Line 47: | ||
|784.1 | |784.1 | ||
|11/7 | |11/7 | ||
|'''840.3''' | |class="thl"|'''840.3''' | ||
|'''13/8''' | |class="thl"|'''13/8''' | ||
|- | |- | ||
!-3 | !-3 | ||
| Line 71: | Line 71: | ||
|496.0 | |496.0 | ||
|4/3 | |4/3 | ||
|'''552.2''' | |class="thl"|'''552.2''' | ||
|'''11/8''' | |class="thl"|'''11/8''' | ||
|- | |- | ||
!0 | !0 | ||
|1143.8 | |1143.8 | ||
|27/ | |27/14, 64/33 | ||
|'''0''' | |class="thl"|'''0''' | ||
|'''1/1''' | |class="thl"|'''1/1''' | ||
|56.2 | |56.2 | ||
|28/27, 33/32 | |28/27, 33/32 | ||
| Line 85: | Line 85: | ||
|647.8 | |647.8 | ||
|16/11 | |16/11 | ||
|'''704.0''' | |class="thl"|'''704.0''' | ||
|'''3/2''' | |class="thl"|'''3/2''' | ||
|760.2 | |760.2 | ||
|14/9, 99/64 | |14/9, 99/64 | ||
| Line 103: | Line 103: | ||
|911.9 | |911.9 | ||
|22/13 | |22/13 | ||
|'''968.1''' | |class="thl"|'''968.1''' | ||
|'''7/4''' | |class="thl"|'''7/4''' | ||
|- | |- | ||
!4 | !4 | ||
| Line 118: | Line 118: | ||
|24/13 | |24/13 | ||
|1119.9 | |1119.9 | ||
|21/ | |21/11 | ||
|1176.1 | |1176.1 | ||
|63/ | |63/32 | ||
|- | |- | ||
!6 | !6 | ||
| Line 138: | Line 138: | ||
| | | | ||
|} | |} | ||
(* Cent values in 2.3.7.11.13 CE tuning. The 104edo tuning (~3/2 = 703. | (* Cent values in 2.3.7.11.13 CE tuning. The [[104edo]] tuning (~3/2 = 703.846c, ~28/27 = 57.692c) is close.) | ||
== Scales == | |||
=== Diachrome, Penthesilia[12] === | |||
The ternary scale ''[[diachrome]]'' is defined as one of the following patterns: | |||
* 5sC: LsLsLmsLsLsm (subst 2m(5L5s)) | |||
* 5sR: LmsLsLsLmsLs | |||
* 5sL: LsLsLsmLsLsm | |||
Diachrome can be given a Parapyth tempering: | |||
* The L step becomes 12/11; | |||
* The m step becomes 22/21~104/99~256/243; | |||
* The s step becomes 28/27~33/32~1053/1024. | |||
The tempered tuning thus has the mappings | |||
* 3/2 = 3L + m + 3s, | |||
* 7/4 = 4L + m + 5s, | |||
* 11/8 = 2L + m + 3s, | |||
* 13/8 = 3L + 2m + 4s. | |||
If 144/143 is not tempered out, Parapyth distinguishes m + s from L by tuning the former to 13/12 and the latter to 12/11. | |||
The 5sL version of diachrome tempered to parapyth is known by [[Margo Schulter]] under the name "Penthesilia[12]". | |||
Scale Workshop links: | |||
* [https://sw3.lumipakkanen.com/scale/xTx-LlSdY 5sRA Aeolian (46edo 6:3:2 Parapyth chromedye)] | |||
* [https://sw3.lumipakkanen.com/scale/xTxNfkoaW 5sRA Aeolian (87edo 11:6:4 Parapyth chromedye)] | |||
== Supporting temperaments == | |||
[[Rodan]] (Parapyth + [[Slendric]], 41 & 46) is arguably the most important rank-2 temperament that supports Parapyth (besides the gentle-fifth-generated 2.3.7.11.13 temperament Leapfrog, which see [[Gentle tuning]]). It splits the spacer 28/27 into two 64/63's. As an aside, having Rodan also enables [[penslen]] in a Parapyth tuning. | |||
== Extensions == | == Extensions == | ||
If prime 5 is added to 2.3.7.11.13 Parapyth via [[Aberschismic]] tempering equating 64/63 to 81/80, the resulting 2.3.5.7.11.13 temperament is called Pele (41 & 46 & 58). | If prime 5 is added to 2.3.7.11.13 Parapyth via [[Aberschismic]] tempering equating 64/63 to 81/80, the resulting 2.3.5.7.11.13 temperament is called Pele (41 & 46 & 58). However, note that there is no canonical 5 extension applying broadly to Parapyth, as 17edo lacks a good prime 5. | ||
If the diatonic augmented fourth is equated to 23/16, the resulting 2.3.7.11.13.23 temperament is called Skidoo (17 & | If the diatonic augmented fourth is equated to 23/16, the resulting 2.3.7.11.13.23 temperament is called Skidoo (17 & 41 & 46). [[63edo]] is a particularly good Skidoo tuning and also adds prime 29. | ||
Prime 31 | Prime 31 is canonically added by equating 28/27~33/32 with 32/31. | ||
== Patent vals == | |||
:''Main article: [[Parapyth/Patent vals]]'' | |||
{{Navbox regtemp}} | {{Navbox regtemp}} | ||
{{Cat|Temperaments}} | {{Cat|Temperaments}} | ||
Latest revision as of 03:54, 13 April 2026
Parapyth is generally viewed as a no-5 2.3.7.11.13 rank-3 regular temperament 17 & 41 & 46, sometimes called Parapythic. (In the strict sense, parapyth is Margo Schulter's rank-3 tuning construct inspired by medieval European and Middle Eastern music theory, where Schulter imposes commas that are to be observed as well as commas that are to be tempered out.[citation needed])
Structure
Parapyth has two non-period generators: the fifth (which is tuned in the gentle region, i.e. somewhat sharp of just but flatter than 17edo) and the "spacer" 28/27, which is equated to 33/32. This is equivalent to taking the spacer 64/63, which may be equated with 81/80 (as in 46edo and 87edo) or tempered smaller than 81/80 (mapping 5120/5103 negatively, as in 63edo and 80edo) assuming prime 5 is added.
Intervals
| # spacers | -1 | 0 | +1 | |||
|---|---|---|---|---|---|---|
| # fifths | Cents* | JI | Cents* | JI | Cents* | JI |
| -7 | 1015.9 | 1072.1 | 13/7 | 1128.4 | 52/27 | |
| -6 | 519.9 | 576.1 | 39/28 | 632.3 | 13/9 | |
| -5 | 23.9 | 64/63 | 80.1 | 22/21 | 136.3 | 13/12 |
| -4 | 727.9 | 32/21 | 784.1 | 11/7 | 840.3 | 13/8 |
| -3 | 231.9 | 8/7 | 288.1 | 13/11 | 344.3 | 11/9 |
| -2 | 935.8 | 12/7 | 992.0 | 16/9, 39/22 | 1048.3 | 11/6 |
| -1 | 439.8 | 9/7 | 496.0 | 4/3 | 552.2 | 11/8 |
| 0 | 1143.8 | 27/14, 64/33 | 0 | 1/1 | 56.2 | 28/27, 33/32 |
| 1 | 647.8 | 16/11 | 704.0 | 3/2 | 760.2 | 14/9, 99/64 |
| 2 | 151.7 | 12/11 | 208.0 | 9/8, 44/39 | 264.2 | 7/6 |
| 3 | 855.7 | 18/11 | 911.9 | 22/13 | 968.1 | 7/4 |
| 4 | 359.7 | 16/13 | 415.9 | 14/11 | 472.1 | 21/16 |
| 5 | 1063.7 | 24/13 | 1119.9 | 21/11 | 1176.1 | 63/32 |
| 6 | 567.7 | 18/13 | 623.9 | 56/39 | 680.1 | |
| 7 | 71.6 | 27/26 | 127.9 | 14/13 | 184.1 | |
(* Cent values in 2.3.7.11.13 CE tuning. The 104edo tuning (~3/2 = 703.846c, ~28/27 = 57.692c) is close.)
Scales
Diachrome, Penthesilia[12]
The ternary scale diachrome is defined as one of the following patterns:
- 5sC: LsLsLmsLsLsm (subst 2m(5L5s))
- 5sR: LmsLsLsLmsLs
- 5sL: LsLsLsmLsLsm
Diachrome can be given a Parapyth tempering:
- The L step becomes 12/11;
- The m step becomes 22/21~104/99~256/243;
- The s step becomes 28/27~33/32~1053/1024.
The tempered tuning thus has the mappings
- 3/2 = 3L + m + 3s,
- 7/4 = 4L + m + 5s,
- 11/8 = 2L + m + 3s,
- 13/8 = 3L + 2m + 4s.
If 144/143 is not tempered out, Parapyth distinguishes m + s from L by tuning the former to 13/12 and the latter to 12/11.
The 5sL version of diachrome tempered to parapyth is known by Margo Schulter under the name "Penthesilia[12]".
Scale Workshop links:
Supporting temperaments
Rodan (Parapyth + Slendric, 41 & 46) is arguably the most important rank-2 temperament that supports Parapyth (besides the gentle-fifth-generated 2.3.7.11.13 temperament Leapfrog, which see Gentle tuning). It splits the spacer 28/27 into two 64/63's. As an aside, having Rodan also enables penslen in a Parapyth tuning.
Extensions
If prime 5 is added to 2.3.7.11.13 Parapyth via Aberschismic tempering equating 64/63 to 81/80, the resulting 2.3.5.7.11.13 temperament is called Pele (41 & 46 & 58). However, note that there is no canonical 5 extension applying broadly to Parapyth, as 17edo lacks a good prime 5.
If the diatonic augmented fourth is equated to 23/16, the resulting 2.3.7.11.13.23 temperament is called Skidoo (17 & 41 & 46). 63edo is a particularly good Skidoo tuning and also adds prime 29.
Prime 31 is canonically added by equating 28/27~33/32 with 32/31.
Patent vals
- Main article: Parapyth/Patent vals
