[ofalkaed]

>Interestingly also, the notes used by modern keyboards and all modern instruments

Vast majority of fretted instruments since the death of the lute are untempered.

Edit: Which is not to suggest that lutes were tempered. Lutes and other tied fret instruments allow for unequal fret spacing so you can temper one string at the cost of more notes being more off from the temperament on other strings, or the frets being at an angle so you could find a bit of a compromise. But often they were EDO or in the ancient tradition of fretted instruments, close enough for rock and roll.

[moefh]

Do you mean equal-tempered?

I never heard someone describe a tuning system as "untempered", but I guess it would mean something like just intonation -- which sounds really great for playing anything in a specific key but falls horribly apart if you try to change the key (which is why it has seen very little use since the renaissance).

[ofalkaed]

Equally divided octave (EDO) with no tempering which is distinct from Equal temperament. Tempered scales are generally EDO with tempering. Other methods like just intonation don't really need to be tempered and generally are not in my experience, but it has been years since I was into just intonation and may just not remember. Historically speaking, the advantage of justly tuned scales is there is no need to temper it because it is already just and perfect, things may have changed in 20th century as far as just intonation is concerned and tempering, don't recall.

Edit: ET and EDO are essentially the same in the case of most fretted instruments, I am dredging long forgotten stuff from memory here and somewhat off above.

Edit2: Refreshing my memory some and seeing how much things have become muddled in my head over the years. Clearly I did not even consider what came out of my memory and just regurgitated it verbatim. ET scales are not tempered but do not mean EDO. Guitar and the like are both ET and EDO. ET and EDO are untempered in the sense that notes are not shifted slightly away from the EDO/ET as they are on the piano and many instruments.

[moefh]

I don't see how you can divide the octave equally and not end up with equal temperament: that's exactly what equal temperament is!

> Tempered scales are generally EDO with tempering.

That's not historically accurate. EDO wasn't used until very recently (about the middle of the 19th century I think), tempering was used way before that.

For example, the first widely used temperament (which became popular in the Renaissance) was the quarter-comma meantone, which shrinks each fifth (from the natural 3/2) so that the major thirds are perfectly 5/4. The name "quarter-comma" means that the amount of shrinkage is 1/4 of the "syntonic comma", which is the difference you get beteween going up 4 fifths (e.g. C->G->D->A->E) and a major third plus 2 octaves (C->E->E->E). Those final Es can only be the same if you shrink the fifth or stretch the third (or both). What this tempering does is shrink each fifth by 1/4 of the difference (so that going up 4 fifths closes it) and doesn't touch the major third. That means the major thirds are beautiful, and the fifths are a little off. For a chosen key, that is -- everything sounds horrible as soon as you try to change the key too far away from the chosen key.

In the Baroque period a lot of other temperaments were invented, the Werckmeister temperaments were very widely used in (what is today) Germany for example (a lot of people believe Bach had one of these in mind when writing the Well-Tempered Clavier). Those temperaments were also defined by how much each fifth is changed from the "normal" 3/2, but each fifth was to be changed by some different amount in some complicated way.

It was only much later that EDO (12-TET, or "equal temperament") started to be widely used. You can think of it (and people do!) as a "temperament" because it just means you shrink the fifth from the "normal" 3/2 = 1.5 to be instead 2^(7/12) =~ 1.4983, so that going up 12 fifths lands you exactly 7 octaves above (since 2^(7/12)^12 = 2^7). That also means that the octave is divided exactly equally, because going up 12 fifths goes through every one of the 12 notes before going back to the original note.

[ofalkaed]

EDO on fretted instruments goes back to at least the 16th century and was essentially the standard well before the mid 19th. Equal temperament is an EDO scale whose divisions approximate justly tuned scales. The western 12TET scale is not actually 12TET or 12EDO, we temper the scale itself and tweak some notes to make it work better unless you play fretted instruments and then it is up to the guitarist to make small adjustments in their playing technique so their untempered 12TET is in tune with the pianos tempered 12TET.

I admitted to making a mess in that post.

[moefh]

> The western 12TET scale is not actually 12TET or 12EDO, we temper the scale itself and tweak some notes [...]

I think you and I must be using words differently. To me (and to Wikipedia, and everything else I've ever read, including[1] which I just consulted to make sure I'm not crazy), 12TET is a way to specify by how much you have to multiply the frequency of the first note of the scale to get the other notes' frequencies. Wikipedia[2] has a table with the numbers for 12TET (the column "Decimal value in 12-ET"), but it's very simple: you just multiply the value of the preceding note by 2^(1/12). If you take 12TET and adjust/change the notes a bit, then it's not 12TET anymore.

> EDO on fretted instruments goes back to at least the 16th century

I'd love to see a reference for that. I just consulted [1], it has a chapter called "Non-Keyboard Tuning" and it doesn't mention that (although admittedly it spends most of its time talking about violin, with a ton of references to stuff that Mozart said). The book does say that equal temperament was known for centuries before it was used, but the people who first discovered it simply didn't think it sounded good.

[1] "How Equal Temperament Ruined Harmony (and Why You Should Care)" by Ross W. Duffin

[2] https://en.wikipedia.org/wiki/12_equal_temperament

[ofalkaed]

Try the wikipedia pages for Equal Temperament and Musical Temperament, they explain all of this.

Here is a 1688 Stradivari[1] guitar with fixed frets and a EDO octave, they were reasonably common by that point. Much of the information regarding this is looking at the fixed frets that many lutes and guitars had applied to their soundboard and comparing that to how composers used those fixed frets, either the tied frets adhere to the scale of the fixed frets or they are out of tune. The history of EDO/ET in fretted instruments goes back to at least Vincenzo Galilei[2] (father of Galileo) who developed the rule of 18 for fret spacing. If memory serves we have a few early steel string instrument (cittern, bandora, orpharion) from around ~1600 with equal spaced frets and this orpharion[3] looks it but it is difficult to tell from that photo. Going back earlier things get more difficult since we have so few intact and unaltered instruments but we do have a fair amount of ingravings and art plus writing on the topic such as Galilei's.

There is a paper going into great depth on all this that is just out of reach in my memory and I can't seem to trick the search engines to give it to me, I will post it if I remember/find it. No time to dig more right now.

[1]https://lsaguitarshop.substack.com/p/gear-27-the-stradivariu...

[2]https://en.wikipedia.org/wiki/Vincenzo_Galilei#Acoustics_and...

[3]https://i0.wp.com/earlymusicmuse.com/wp-content/uploads/2017...

[ofalkaed]

To offer something better than this mess and correct. Fretted instruments are tehnically unequally tempered do to the physics the string, the fret board is EDO but in fretting the string we stretch it and raise its pitch. Fretted instruments use various compensation tricks to lessen this effect but most notes are off from both 12EDO and 12TET, we get the open string and the 12th fret but the rest are off by varying amounts.

[brudgers]

An untempered instrument would be one that is tuned to maintain the perfect intervals of a specific root tone.

Temperament is adjusting tuning for musical practicality. 12 TET is simply one set of compromises/benefits in a constellation of alternatives.

https://en.m.wikipedia.org/wiki/Musical_temperament

[ofalkaed]

Thank you for offering something clearer than the mess I made. Been 25 years since I studied this stuff and finally learned to just accept (and love) the modern standard.