"Find POTE = T/T"
It doesn't seem to handle the case where the period is a fraction of an octave. For example, try this diaschismic mapping: [⟨2, 2, 7, 15], ⟨0, 1, -2, -8]]. We know the period is half an octave. Indeed,
T = [599.44659699 703.03185125]
Now POTE = T/T gives
POTE = [1200. 1407.36176623]
Is this correct?
Besides, since I don't see a definition of TE generator, I don't know if it should be octave reduced, period reduced or left as is. Changing the basis definitely changes the result. Does TE generators specify a basis or whatever maps are accepted?
From my perspective, this seems more intuitive:
- Find the TE octave: (TV)1, that is, the first entry of TV.
- POTE = T/(TV)1
POTE = [600. 703.68088312]
- OK everything's solved. It should be made more clear that TE generators and TE map are different. While TE gens change on basis, never does TE map. FloraC (talk) 05:55, 25 June 2020 (UTC)
Is someone able and willing to make a video? Or is there an article with illustrations that enables even non-mathematicians like me to follow the process? Thanks in advance for your help. --Xenwolf (talk) 13:49, 6 December 2020 (UTC)
This article fails to justify why POTE is a good idea. The Kees Height is a complexity measure on JI intervals, not an error metric on tempered intervals, so I can't see how the resulting tuning is optimal in this sense.
As I understand it, the POTE tuning simply finds the least-squares tuning under the TE norm, and then does a completely ad-hoc adjustment to get the octaves just. In constrast, the constrained TE tuning is actually optimal under the TE norm, in the subspace where the octave is just. It seems like a much better candidate for a 'standard' tuning.
- Make sure to ask Paul Erlich. It's a shame he isn't here to help. I added some of his words but that was a tip of the iceberg. FloraC (talk) 01:39, 19 December 2021 (UTC)
OK, after reading up on exterior algebra, POTE just seems wrong. On the page Tenney-Euclidean tuning a justification is given (which should be moved here anyway):
- "The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object."
This only works for rank-1 (eg edos) though. In general, for a rank-r temperament, you have some subspace/hyperplane of dimension r. By only looking at the line from 0 to T, you are missing all the other degrees of freedom that you can optimize. (Again, something CTE does correctly.)