Diachrome: Difference between revisions
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'''Diachrome''' ( | '''Diachrome''' or '''chromedye''' (denoted dia5s or 5s in [[groundfault]]'s [[aberrismic theory]] systematic naming) is a set of three 5L 2m 5s [[scale pattern]]s: | ||
* 5sL: LsLsLsmLsLsm | * 5sL: LsLsLsmLsLsm | ||
* 5sR: LmsLsLsLmsLs | * 5sR: LmsLsLsLmsLs | ||
* 5sC: LsLsLmsLsLsm | * 5sC: LsLsLmsLsLsm (''interchroid'' structure) | ||
5sL and 5sR are [[chiral]] pairs, and 5sC is achiral. The three chiralities are also determined by the number of ms and sm substrings they have. | 5sL and 5sR are [[chiral]] pairs, and 5sC is achiral. The three chiralities are also determined by the number of ms and sm substrings they have. | ||
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=== Balance === | === Balance === | ||
The 5sC pattern, LsLsLmsLsLsm, is | The 5sC pattern, LsLsLmsLsLsm, is an even-regular scale according to the [[Ternary scale theorems#|classification of ternary balanced scales]]. In particular, it is (as an abstract scale word) MV3 but not SV3. | ||
== Diachrome in edos == | == Diachrome in edos == | ||
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Diachrome is interesting for having at least four notable JI interpretations. In all the interpretations below, L + s = 9/8, and m = 256/243. | Diachrome is interesting for having at least four notable JI interpretations. In all the interpretations below, L + s = 9/8, and m = 256/243. | ||
=== 7-limit[5120/5103] === | === 7-limit [5120/5103] === | ||
In the 7-limit, diachrome has two JI tunings which are very similar and can be identified by tempering out [[5120/5103]], the 5. | In the 7-limit, diachrome has two JI tunings which are very similar and can be identified by tempering out [[5120/5103]], the 5.8{{c}} gap between 81/80 and 64/63. These commas are notable for being the two most common interpretations for aberrisma scale steps in [[aberrismic theory]]. | ||
* The 2.3.5 tuning has L = 10/9, m = 256/243, s = 81/80. | * The 2.3.5 tuning has L = 10/9, m = 256/243, s = 81/80. | ||
* The 2.3.7 tuning has L = 567/512, m = 256/243, s = 64/63. | * The 2.3.7 tuning has L = 567/512, m = 256/243, s = 64/63. | ||
The tempered tuning thus has the mappings | The tempered tuning thus has the mappings | ||
* 3/2 = 3L + m + 3s, | * 3/2 = 3L + m + 3s, | ||
* 5/4 = 2L + s, | * 5/4 = 2L + s, | ||
* 7/4 = 4L + 2m + 3s. | * 7/4 = 4L + 2m + 3s. | ||
The property of tempering out 5120/5103 thus lends 41edo, 46edo, 53edo, and 58edo some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason. | |||
The property of tempering out 5120/5103 thus lends 41edo (6:3:1), 46edo (7:3:1), 53edo (8:4:1), and 58edo (9:4:1) some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason. | |||
=== 2.3.7.11.13 parapyth === | === 2.3.7.11.13 parapyth === | ||
Diachrome can be given a [[parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering: | Diachrome can be given a [[parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering: | ||
* The L step becomes 12/11 | * The L step becomes 12/11; | ||
* The m step becomes | * The m step becomes 22/21~104/99~256/243; | ||
* The s step becomes 28/27~33/32~1053/1024 | * The s step becomes 28/27~33/32~1053/1024. | ||
The tempered tuning thus has the mappings | The tempered tuning thus has the mappings | ||
* 3/2 = 3L + m + 3s, | * 3/2 = 3L + m + 3s, | ||
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* 11/8 = 2L + m + 3s, | * 11/8 = 2L + m + 3s, | ||
* 13/8 = 3L + 2m + 4s. | * 13/8 = 3L + 2m + 4s. | ||
By not tempering out 144/143, parapyth distinguishes m + s from L by tuning the former to 13/12 and the latter to 12/11. | By not tempering out 144/143, 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]". | The 5sL version of diachrome tempered to parapyth is known by [[Margo Schulter]] under the name "Penthesilia[12]". | ||
=== 2.3.11.19.23/5.31[17 & 24] === | === 2.3.11.19.23/5.31 [17 & 24] === | ||
There's also a diachrome tempered by an extension of [[Rastmic_clan#Neutral | neutral]] (2.3.11.19.23/5.31[17 & 24]): | There's also a diachrome tempered by an extension of [[Rastmic_clan#Neutral | neutral]] (2.3.11.19.23/5.31[17 & 24]): | ||
* The L step is 12/11 | * The L step is 12/11; | ||
* The m step is 256/243~128/121~93/88~19/18 | * The m step is 256/243~128/121~93/88~19/18; | ||
* The s step is 33/32~32/31~95/92 | * The s step is 33/32~32/31~95/92. | ||
The tempered tuning thus has the mappings | The tempered tuning thus has the mappings | ||
* 3/2 = 3L + m + 3s, | * 3/2 = 3L + m + 3s, | ||
| Line 60: | Line 66: | ||
* 23/20 = L + m, | * 23/20 = L + m, | ||
* 31/16 = 5L + 2m + 4s. | * 31/16 = 5L + 2m + 4s. | ||
Note that L = m + s in this tuning, and it makes the scales nonstrictly [[Rothenberg propriety | proper]]. | Note that L = m + s in this tuning, and it makes the scales nonstrictly [[Rothenberg propriety | proper]]. | ||
=== 2.3.7.13.19.23[17 & 19 & 41] === | === 2.3.7.13.19.23 [17 & 19 & 41] === | ||
A diachrome is also available in the 2.3.7.13.19.23[17 & 19 & 41] tempering: | A diachrome is also available in the 2.3.7.13.19.23[17 & 19 & 41] tempering: | ||
* The L step is 13/12 | * The L step is 13/12; | ||
* The m step is 256/243~96/91~19/18 | * The m step is 256/243~96/91~19/18; | ||
* The s step is 28/27~27/26 | * The s step is 28/27~27/26. | ||
The tempered tuning has the mappings | The tempered tuning has the mappings | ||
* 3/2 = 3L + m + 3s, | * 3/2 = 3L + m + 3s, | ||
| Line 73: | Line 81: | ||
* 19/16 = L + m + s, | * 19/16 = L + m + s, | ||
* 23/14 = 3L + 2m + 4s. | * 23/14 = 3L + 2m + 4s. | ||
Unlike the above parapyth tempering, this one sharpens s~28/27 slightly towards 27/26 instead of flattening it towards 33/32, sharpens 32/27 slightly towards 19/16 instead of flattening it to 13/11. | Unlike the above parapyth tempering, this one sharpens s~28/27 slightly towards 27/26 instead of flattening it towards 33/32, sharpens 32/27 slightly towards 19/16 instead of flattening it to 13/11. | ||
It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper. | It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper. | ||
=== Pele temperament === | |||
In the [[pele]] temperament ([41 & 46 & 58]), following interpretations are available: | |||
* L = 10/9, m = 21/20~22/21~256/243, s = 56/55~64/63~66/65~81/80; | |||
* L = 11/10, m = 21/20~22/21~256/243, s = 40/39~45/44~50/49; | |||
* L = 12/11, m = 21/20~22/21~256/243, s = 28/27~33/32~65/63; | |||
* L = 13/12, m = 21/20~22/21~104/99~256/243, s = 27/26. | |||
== External links == | |||
* [https://sw3.lumipakkanen.com/scale/XvmXDTFp0 5sRA Aeolian (46edo 7:3:1 Aberschismic chromedye)] | |||
* [https://sw3.lumipakkanen.com/scale/Xvmsw1KBx 5sRA Aeolian (46edo 6:3:2 Parapyth chromedye)] | |||
[[Category:Aberrismic theory]] | [[Category:Aberrismic theory]] | ||
[[Category:Rank-3 scales]] | [[Category:Rank-3 scales]] | ||