### Quadritikleismic (68&72 13-limit)

Posted:

**Mon Mar 08, 2021 7:35 am**Hello everyone, i am pretty new to Sagittal, less new to xenharmonic alliance or the regular tuning paradigm. Some musical experience, mostly from church choir and amateur playing, nothing professional. Would say i have a good musical ear and a keen mathematical mind though.

After having noticed that standard Sagittal notation for 13-limit JI uses the symbols , , , and for 81/80, 64/63, 33/32 and 27/26 (down, down, up, down) as standards, and noticing that 33/32 is almost exactly a fourth root of 9/8 and 27/26 is almost exactly a cube root of 9/8, thus being likely to be perceived as a quarter tone and a third tone respectively, i wondered about what temperament we would get by setting 27/26 = 26/25 = 25/24 (thus (27/26)^3 = 9/8) and setting 36/35 = 33/32 and 36/35 * 33/32 = 35/33 (thus (33/32)^4 = 9/8).

The answer is Quadritikleismic temperament, a rank 2 (linear) 13-limit temperament, supported by 72edo, 68edo and 140edo.

Having commas 676/675, 625/624, 385/384 and 9801/9800 tempered to unison, among many others.

I now wonder which accidentals would be best for using in this temperament, besides and ? It should preferrably be a good compromise between 72edo and 68edo, and possibly 140edo.

Are and still good alternatives for 5 and 7 commas, or are there better ones? Should the 11 and 13 commas also be changed perhaps?

The idea is for this temperament to possibly be a good microtemperament for 13-limit music performance, especially real time performance, if there is any interest for such in 13-limit almost JI based on diatonicism plus pretty simple extensions.

Just an idea, i haven't really had the opportunity to train myself very much in any instrument except voice yet, although planning to learn recorder, piano/synth and guitar better, and hopefully be able to retune such instruments to e.g. Quadritikleismic, and other tunings/temperaments.

After having noticed that standard Sagittal notation for 13-limit JI uses the symbols , , , and for 81/80, 64/63, 33/32 and 27/26 (down, down, up, down) as standards, and noticing that 33/32 is almost exactly a fourth root of 9/8 and 27/26 is almost exactly a cube root of 9/8, thus being likely to be perceived as a quarter tone and a third tone respectively, i wondered about what temperament we would get by setting 27/26 = 26/25 = 25/24 (thus (27/26)^3 = 9/8) and setting 36/35 = 33/32 and 36/35 * 33/32 = 35/33 (thus (33/32)^4 = 9/8).

The answer is Quadritikleismic temperament, a rank 2 (linear) 13-limit temperament, supported by 72edo, 68edo and 140edo.

Having commas 676/675, 625/624, 385/384 and 9801/9800 tempered to unison, among many others.

I now wonder which accidentals would be best for using in this temperament, besides and ? It should preferrably be a good compromise between 72edo and 68edo, and possibly 140edo.

Are and still good alternatives for 5 and 7 commas, or are there better ones? Should the 11 and 13 commas also be changed perhaps?

The idea is for this temperament to possibly be a good microtemperament for 13-limit music performance, especially real time performance, if there is any interest for such in 13-limit almost JI based on diatonicism plus pretty simple extensions.

Just an idea, i haven't really had the opportunity to train myself very much in any instrument except voice yet, although planning to learn recorder, piano/synth and guitar better, and hopefully be able to retune such instruments to e.g. Quadritikleismic, and other tunings/temperaments.