diff --git a/documentation/commas.md b/documentation/commas.md index 05cd61e..a51ded1 100644 --- a/documentation/commas.md +++ b/documentation/commas.md @@ -9,6 +9,9 @@ These inflections always go in the direction of the arrow so `6/5` ~ `m3^5h` ins 4. [Semifourth bridges](#semifourth-bridges) 5. [Tone-splitter bridges](#tone-splitter-bridges) 6. [Lumi's irrational bridges](#lumis-irrational-bridges) +7. [Neutral bridges revisited](#neutral-bridges-revisited) +8. [Bridging highlights](#bridging-highlights) +9. [FJS revisited](#fjs-revisited) ## Helmholtz-Ellis 2020 + Richie's extension @@ -101,21 +104,26 @@ The neutral FJS master algorithm can be generalized to semiquartals. Semifourths | `17q` | `sqrt(70227/6553)` | `[-8 5/2 0 0 0 0 1>` | `+59.843` | | `19q` | `sqrt(1083/1024)` | `[-5 1/2 0 0 0 0 0 1>` | `+48.491` | | `23q` | `sqrt(14283/16384)` | `[-7 3/2 0 0 0 0 0 0 1>` | `-118.793` | -| `29n` | `sqrt(841/768)` | `[-4 -1/2 0 0 0 0 0 0 0 1>` | `+78.6` | -| `31n` | `sqrt(233523/262144)` | `[-9 5/2 0 0 0 0 0 0 0 0 1>` | `-100.077` | -| `37n` | `sqrt(4107/4096)` | `[-6 1/2 0 0 0 0 0 0 0 0 0 1>` | `+2.322` | +| `29q` | `sqrt(841/768)` | `[-4 -1/2 1>@2.3.29` | `+78.6` | +| `31q` | `sqrt(233523/262144)` | `[-9 5/2 1>@2.3.31` | `-100.077` | +| `37q` | `sqrt(4107/4096)` | `[-6 1/2 1>@2.3.37` | `+2.322` | ## Tone-splitter bridges Tone-splitter master algorithm is basically the plain FJS master shifted by a semioctave (with the semiapotome as the radius of tolerance). -| Prime | Comma | Monzo | Size in cents | -| ------ | ----------------- | ----------------------| ------------- | -| `5t` | `sqrt(2025/2048)` | `[-11/2 2 1>` | `-9.776` | -| `7t` | `sqrt(441/512)` | `[-9/2 1 0 1>` | `-129.219` | -| `11t` | `sqrt(121/128)` | `[-7/2 0 0 0 1>` | `-48.682` | -| `13t` | `sqrt(169/162)` | `[-1/2 -2 0 0 0 1>` | `+36.618` | -| `17t` | `sqrt(289/288)` | `[-5/2 -1 0 0 0 0 1>` | `+3.000` | +| Prime | Comma | Monzo | Size in cents | +| ------ | ------------------- | -------------------------| ------------- | +| `5t` | `sqrt(2025/2048)` | `[-11/2 2 1>` | `-9.776` | +| `7t` | `sqrt(441/512)` | `[-9/2 1 0 1>` | `-129.219` | +| `11t` | `sqrt(121/128)` | `[-7/2 0 0 0 1>` | `-48.682` | +| `13t` | `sqrt(169/162)` | `[-1/2 -2 0 0 0 1>` | `+36.618` | +| `17t` | `sqrt(289/288)` | `[-5/2 -1 0 0 0 0 1>` | `+3.000` | +| `19t` | `sqrt(29241/32768)` | `[-15/2 2 0 0 0 0 0 1>` | `-98.577` | +| `23t` | `sqrt(529/512)` | `[-9/2 0 0 0 0 0 0 0 1>` | `+84.823` | +| `29t` | `sqrt(7569/8192)` | `[-13/2 1 1>@2.3.29` | `-68.468` | +| `31t` | `sqrt(8649/8192)` | `[-13/2 1 1>@2.3.31` | `+46.991` | +| `37t` | `sqrt(1369/1458)` | `[-1/2 -3 1>@2.3.37` | `-54.521` | ## Lumi's irrational bridges @@ -161,3 +169,59 @@ The semiapotome `2l` is a handy companion of the neutral inflections. It's equiv | `n7_5n` | `9/5` | `n7^5n` | `15/8` | If for some reason you find this concept appealing the vocalized names of intervals should follow [Color notation](https://en.xen.wiki/w/Color_notation) but in place of *white* we have *microwave* for the smaller intervals and *x-ray* for the larger ones. So `4/3` is a *'microwave-fourth'*, *'mu-fourth'* or *'µ4'* while `3/2` is an *'x-ray-fifth'*, *'ex-fifth'* or *'x5'*. + +## Neutral bridges revisited + +Neutral commas are repeat here for your convenience. + +| Prime | Comma | Monzo | Size in cents | +| ------ | ----------------------- | ----------------------------| ------------- | +| `5n` | `sqrt(25/24)` | `[-3/2 -1/2 1>` | `+35.336` | +| `7n` | `sqrt(54/49)` | `[-1/2 -3/2 0 1>` | `-84.107` | +| `11n` | `sqrt(242/243)` | `[1/2 -5/2 0 0 1>` | `-3.570` | +| `13n` | `sqrt(507/512)` | `[-9/2 1/2 0 0 0 1>` | `-8.495` | +| `17n` | `sqrt(8192/7803)` | `[-13/2 3/2 0 0 0 0 1>` | `-42.112` | +| `19n` | `sqrt(384/361)` | `[-7/2 -1/2 0 0 0 0 0 1>` | `-53.464` | +| `23n` | `sqrt(529/486)` | `[-1/2 -5/2 0 0 0 0 0 0 1>` | `73.387` | +| `29n` | `sqrt(864/841)` | `[-5/2 -3/2 1>@2.3.29` | `-23.355` | +| `31n` | `sqrt(2101707/2097152)` | `[-21/2 7/2 1>@2.3.31` | `+1.878` | +| `37n` | `sqrt(175232/177147)` | `[7/2 -11/2 1>@2.3.37` | `-9.408` | + +## Bridging highlights + +[Cole](https://en.xen.wiki/w/User:2%5E67-1) recommends the following set of bridging commas: + +| Prime | Comma | Cents | Reduced harmonic | Spelling | Against C4 | +| ----- | ------------ | --------- | ---------------- | ---------- | ---------- | +| `5n` | `√25/24` | `+35.336` | `5/4` | `n3^5n` | `Ed4^5n` | +| `7q` | `√49/48` | `+17.848` | `7/4` | `n6.5^7q` | `εd4^7q` | +| `11t` | `√121/128` | `-48.682` | `11/8` | `P4.5^11t` | `γ♮4^11t` | +| `13n` | `√507/512` | `-8.495` | `13/8` | `n6^13n` | `Ad4^13n` | +| `17t` | `√289/288` | `+3.000` | `17/16` | `P1.5^17t` | `η♮4^17t` | +| `19n` | `√361/384` | `-53.464` | `19/16` | `n3^19n` | `Ed4^19n` | +| `23t` | `√529/512` | `+28.274` | `23/16` | `P4.5^23t` | `γ♮4^23t` | +| `29q` | `√841/768` | `+78.600` | `29/16` | `n6.5^29q` | `εd4^29q` | +| `31f` | `31/32`* | `-54.964` | `31/16` | `P8^31f` | `C5^31f` | +| `37q` | `√4107/4096` | `+2.322` | `37/32` | `n2.5^37q` | `αd4^37q` | + +The choice is based on [hemipyth](https://en.xen.wiki/w/Hemipyth)[10] 2|2(2). A comma is constructed from nearest note to the harmonic. + +*) This means that 31 is not associated with an irrational bridge. The simplest irrational choice would be `√8649/8192` for `31/16` = `P7.5^31t` (`ζ♮4^31t`). + +## FJS revisited + +Rational commas are repeated here for your convenience. + +| Prime | Comma | Monzo | Size in cents | +| ----------- | ----------- | ---------------------- | -------------- | +| `5` | `80/81` | `[4 -4 1>` | `-21.506` | +| `7` | `63/64` | `[-6 2 0 1>` | `-27.264` | +| `11` | `33/32` | `[-5 1 0 0 1>` | `+53.273` | +| `13` | `1053/1024` | `[-10 4 0 0 0 1>` | `+48.348` | +| `17` | `4131/4096` | `[-12 5 0 0 0 0 1>` | `+14.73` | +| `19` | `513/512` | `[-9 3 0 0 0 0 0 1>` | `+3.378` | +| `23` | `736/729` | `[5 -6 0 0 0 0 0 0 1>` | `+16.544` | +| `29` | `261/256` | `[-8 2 1>@2.3.29` | `+33.487` | +| `31`, `31f` | `31/32` | `[-5 0 1>@2.3.31` | `-54.964` | +| `31c` | `248/243` | `[3 -5 1>@2.3.31` | `+35.261` | +| `37` | `37/36` | `[-2 -2 1>@2.3.37` | `+47.434` |