One resource I had intended to include with my earlier posts on intonation here and here was a great one page handout showing exactly where each note of a chord needs to be in order to be perfectly in tune according to just intonation. I picked up this handout several years ago while attending the Round Top Festival Institute. I plan to post more in the future about Round Top, but in short it is a great summer music festival set up in a similar way to Aspen, Music Academy of the West, and various other programs. When I attended, all participants received free room and board for the entire six week festival.
Getting back to the handout, it is constructed to show all possible triads and seventh chords, with indications beside each chord tone showing how many cents that tone needs to be adjusted. All of the chords are constructed with “C” as the root, but the same principles would apply regardless of transposition. For example, the first chord is a major triad.
Notice that the third must be lowered 14 cents, which is further than many students are willing to go, and the fifth needs to be just a shade higher (2 cents). Looking at a minor triad, you can see that the third must be quite high (16 cents), with the fifth in the same place as in the major triad.
Looking at seventh chords, a problematic sonority for many groups is the dominant seventh.
In this case, the third needs to be where it was in the major triad, and the seventh needs to be quite low (31 cents). Although in practice you wouldn’t necessarily need to think, “ok, I have the third of a major chord, so now I am lowering my pitch by exactly 14 cents,” I think this chart is very useful because it helps you visualize just how much some of these chord tones need to be maneuvered around in order to ring true. Try playing each pitch of these chords against a drone and see just how much you need to bend things around on your particular horn, mouthpiece, etc. for things to really be in tune – you might be surprised!
If you are interested in the complete handout you can download it here. In Tune Chords
One word about copyright issues – I would love to be able to give credit to whomever produced this wonderful handout, but I have not been able to find any information on the creators. The handout lists Mr. Ward Widener and his AccuTone tuner as the responsible party, but the only information I can find on him is this old HTML webpage which lists him as the manufacturer of the tuner mentioned and lists his company as Widener Engineering in Austin, TX. The bottom line of the handout also names “Jack Holland Productions” in St. Louis, MO, but it is unclear what connection they have to either Ward Widener or the handout itself. If anyone out there has any additional information I’d love to hear about it.
For now, feel free to download and use this handy reference; I hope you find it as useful as I have.
Although it’s probably not directly practical for horn playing, this post reminds me of the music of Ben Johnston. He was a student of Harry Partch and, as one might expect, a partisan of a particular system of just intonation. At first blush his music sounds slightly out-of-tune to our equal-tempered ears, but after a bit of listening, the harmonies begin to sound amazingly rich and deep (listening to, say, Chopin after Johnston makes equal-temperament sound quaint and contrived). His Suite for Microtonal Piano is a good example.
Thanks for the comment Josh. It is very interesting to consider the extent to which our notions of consonance and dissonance are influenced by various systems of intonation and the musical system (s) to which we are accustomed. I remember Gunther Schuller once remarking that after so many years of composing “fully chromatic” music, those harmonies which others found dissonant actually sounded consonant to his ears. In addition, I think your comment is actually very applicable to horn playing, given the pieces in our literature (Ligeti, Schuller, Hill, etc.) which call for 1/4 tone tunings.