Rasputin wrote: ↑
Mon Feb 19, 2018 9:33 pm
What range does a guitar temperament really need to cover?
Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A#
Just the bit with the asterisks? You get the relative natural minors for free but have to think about those raised sixths and sevenths.
Edit: the asterisks moved when I pressed submit - they are supposed to start under Bb and finish under E
Well, I thought I'd have a look at designing a guitar temperament that mathematically optimised the scales Bb thro' E to Just temperament (JT). i.e. how close you could get the intervals of each of those 7 tone scales to match JT (at the expense, of course, of the other keys which would be way out). It's interesting what happens when you let a computer optimise things without any preconceived bias. For the base key, for which I chose A, to pin it to A440, all the middle intervals (major third to major sixth) came out pretty close (within 2 cents) of JT as did the major 7th, except for the 5th and minor 6th which were 6 cents flat. The octaves were forced to pure. The big errors were on minor seconds (36 cents sharp, which sounds a lot, except that equal temperament minor seconds are nearly 30 cents sharp of JT and are not exactly a well worn interval anyway) the major 2nd and minor 3rd were about 12 cents flat of JT and the minor 7th around 20 cents flat. The keys will all sound different to each other, of course, because of the differently sized intervals, so there will be "key colours".
Just for laughs, I thought I'd see how close I could get to this temperament using straight frets and only nut and saddle compensation. The answer to that was nowhere near close enough! So this would definitely need some sort of wiggly frets implementation. The TT guys may well have done this already.
I think I'll find someone with a programmable temperament keyboard and listen to what the various keys sounds like before I go any further!
Another thing that doing exercises like this demonstrates is that accurate ET is a pretty good all round compromise. (It's a still shame about those 3rds, though).