EDOs with multiple prime mappings
Posted: Wed Sep 28, 2016 4:30 pm
Cryptic Ruse has raised in this facebook thread, the question of how best to notate those EDOs below 72 that have no single best prime mapping but instead have multiple mediocre mappings. To see a list of 13-prime-limit mappings for 7-edo, click on the following link. Change the number after "ets=" to see the corresponding edo.
http://x31eq.com/cgi-bin/rt.cgi?ets=7&limit=13 Thanks to Graham Breed for this.
Cryptic makes a good point that we should have notations for these that do not favour one mapping over another. He proposes to use a 3-limit notation having Sagittal accidentals for semi-apotome, quarter-apotome, semi-limma and quarter-limma. There is some discussion of what these accidentals might look like in this Sagittal forum thread.
I have worked through all the standard EDO notations up to 72 in http://sagittal.org/sagittal.pdf and I note the following cases where we have used a prime mapping that does not agree with _all_ the mappings given for that ET by Graham's app.
I note that the following EDOs have multiple mappings which can't even agree on their mapping of the prime number 3.
23, 25, 28, 30, 33, 35, 42, 45, 47, 52, 54, 59, 64, 66, 71.
That means that even a 3-limit fractional-comma notation will not be able to avoid favouring one mapping over another if it attempts a native-fifth notation for them.
The solution chosen for these, in the standard Sagittal EDO notations, is to notate them as a subset of the EDO that is 2 times (or sometimes 3 times) larger. Although in a few cases (23, 33, 45, 47, 64) we provide native fifth notations as well. In the case of 23, 33 and 47 these do not use any prime 3 accidental. i.e. they do not use sharps or flats.
http://x31eq.com/cgi-bin/rt.cgi?ets=7&limit=13 Thanks to Graham Breed for this.
Cryptic makes a good point that we should have notations for these that do not favour one mapping over another. He proposes to use a 3-limit notation having Sagittal accidentals for semi-apotome, quarter-apotome, semi-limma and quarter-limma. There is some discussion of what these accidentals might look like in this Sagittal forum thread.
I have worked through all the standard EDO notations up to 72 in http://sagittal.org/sagittal.pdf and I note the following cases where we have used a prime mapping that does not agree with _all_ the mappings given for that ET by Graham's app.
EDO Sag uses Maps agree only primes 21 7 2 3 27 5 13 2 3 5 7 33 11 2 34 11 2 3 5 13 38 11 2 3 5 39 5 11 2 3 45 13 2 47 various 2 48 11 23 2 3 51 5 7 2 3 55 11 2 3 60 7:11 2 3 5 64 13 2 65 5 7 11 2 3 5 67 7:11 13 2 3 7 68 55 11 2 3 5 7 13 69 11 13 2 3 5 70 5 11 2 3I note that several of these have a problem only with the use of as the 11-M-diesis 32:33. But already has a secondary role in Sagittal as a semi-apotome symbol, and so, if it is read as such, the problem goes away. However, up 'til now, this symbol would only have been used if it is both the semi-apotome _and_ the 11-M-dieisis in the EDO's obvious mapping.
I note that the following EDOs have multiple mappings which can't even agree on their mapping of the prime number 3.
23, 25, 28, 30, 33, 35, 42, 45, 47, 52, 54, 59, 64, 66, 71.
That means that even a 3-limit fractional-comma notation will not be able to avoid favouring one mapping over another if it attempts a native-fifth notation for them.
The solution chosen for these, in the standard Sagittal EDO notations, is to notate them as a subset of the EDO that is 2 times (or sometimes 3 times) larger. Although in a few cases (23, 33, 45, 47, 64) we provide native fifth notations as well. In the case of 23, 33 and 47 these do not use any prime 3 accidental. i.e. they do not use sharps or flats.