I've just tried something similar, using a computer to generate harmonic series based on different fundamentals covering an octave and comparing these to the notes on a keyboard tuned to a Werckmeister temperament.
Rather than comparing all the harmonic series, I've concentrated on the ones that we use the most frequently - the 2nd, 3rd, 4th, 5th, 6th, 8th, 9th and 10th. I've put up the graph on my flickr page: http://www.flickr.com/photos/flowtik/4018720867/ There are clear differences between how in-tune the different crooks are - the worst have a mean discrepancy of around 7 cents from the notes in the Werckmeister scale, and the best has a mean discrepancy of about 2 cents. So which is this marvellous crook with the great intonation? C, D, F or G?! To my great surprise it turned out to be B-flat - which is strange because this is the rarest of the crooks we use... Or is it? It has puzzled me since I was very small that trumpets and clarinets are generally pitched in B-flat rather than C like the strings and piano. I wonder whether the answer could be that this key worked best before the days when orchestras were all tuned to ET? Kit > On Sat, Oct 17, 2009 at 00:54, Kit Wolf <[email protected]> > wrote: >> >> I'm interested to hear you talk about the trumpet here, as the sources >> I've read on temperaments tend to focus on string and keyboard >> instruments. It frustrated me that they didn't mention brass and >> woodwind >> more often, but I have never been very clear in my own mind what I >> expected them to say about them. In other words, I couldn't really >> formulate a question, but I feel sure there must be an interesting >> answer! >> >> Take the situation where a valveless brass instrument that 'naturally' >> produces notes in the harmonic series plays with an orchestra or >> keyboard >> that is tuned to a different (but historically plausible) temperament: >> if >> I understand you correctly, 'D' is one of the keys where the discrepancy >> between the brass instruments and the other instruments is likely to be >> minimised? Are there any other keys that work particularly well or >> particularly badly, or are we able to lip everything enough that the >> natural discrepancy doesn't really matter? Perhaps our ability to >> hand-stop is the reason horns seem to be crooked in so many more keys >> than >> trumpets? >> >> These are just examples of the ill-formed questions running through my >> mind - I'd be interested to hear any comments about playing horns in the >> days before equal temperament. >> > > > After writing my comment and reading your reply and questions I > started calculate and think more on the subject. I know, wrong order > of doing things ;) > > I've looked up the intervals in Werckmeister and Valotti, popular > temperaments for 18th century baroque, and the distances in cents come > out like this: > (I hope the table comes out alright ;) > > F: W V D: W V > just equal > G 198 196 E 198 196 > 204 200 > A 390 392 F# 396 396 > 386 400 > Bb 498 498 G 504 506 > 498 500 > C 702 698 A 696 698 > 702 700 > D 894 894 B 900 894 > 884 900 > E 1092 1090 C# 1098 1098 > 1088 1100 > > > So actually F comes out as slightly closer to the natural harmonic > series, which maybe explains the "natural" or "peasant" affect > associated with it. > > In retrospect and after calculating I have to rethink my statement and > would have to conclude that other factors have to do with the (at > least for me real) relative discomfort of playing in F. It is probably > a culmination of the differences in the temperament and the tuning > tendencies/problems of other instruments that make F so much harder to > play in. The F on a baroque flute for instance is a notoriously bad > note, and our sounding D (written A, normally played wide open and > lipped up) will tend to be too flat relative to the D of the strings > which is based on pure fifths. > In these temepraments F lies a bit higher on the keyboard than in > equal temperament and D slightly lower, so that could be a > contributing factor as well. > -- Sometimes my Email program gives the wrong return address. If you have any trouble replying to me, use '[email protected]' and not 'n802...' Sorry for any confusion -- Sometimes my Email program gives the wrong return address. If you have any trouble replying to me, use '[email protected]' and not 'n802...' Sorry for any confusion _______________________________________________ post: [email protected] unsubscribe or set options at https://pegasus.memphis.edu/cgi-bin/mailman/options/horn/archive%40jab.org
