My original attempts consisted of just the graphical part. I was unaware of Mark McDunn's chart (and of Anders Larson's) until many years later. I developed the ETSP Chart back in the 1960s, while a graduate student in aerospace engineering at Cornell University, and playing in the Cornell Symphony Orchestra under Karel Husa. This is all useful pedagogy, especially for beginning students exploring upper harmonics and alternate positions.ĮTSP Charts are also useful for comparing various different single-valve and (dependent or in-line) dual-valve tunings of attachments-where all positions of slide-alone and attachment(s) are immediately obvious.Ī more comprehensive hard-copy description of the construction of the ETSP Chart, including equations for harmonics and the underlying “curves” for the semitone, whole-tone and other interval between-adjacent-harmonic small-increment patterns, and single-valve continuous chromatic extension to pedals, is available by contacting me (Benny Leonard) via email-enclosing a convenient postal address-at Also note that the non-shaded regions indicate where alternate positions are available none are available where the coloured shadings touch each other. Note that between-adjacent-harmonic whole-tone scales change direction at the ninth harmonic at the seventeenth harmonic for the chromatic scales. Experiment with other scales and arpeggios while “reading from” the corresponding tones (black dots) in the Chart. Also some lip-slurred chromatic scales “against the grain” in the upper treble clef-good for emboucher building! Also some lip-slurred diminished-seventh arpeggios, tracing minor-third increments between adjacent harmonics. While looking at the ETSP Chart, try playing some lip-slurred whole-tone scales: major-second increments between adjacent harmonics “against the grain” (up and down) from around the top of the bass clef up into the treble clef. Although different real instruments played by the same player (and the same instrument played by different players) will differ from this “theoretical” natural-harmonic model, the correct practical positions on well-designed instruments will not vary by more than a few millimetres. See the ETSP Chart for the Bb trombone (without attachment), below.

A more precise graphical portrayal of Equitempered Tones (vertical axis) versus Slide Position extension (horizontal axis) results in a good representation of where to find equitempered tones-the ETSP Chart. So all these tones require slide-position adjustment for equitemperament, as shown precisely in Mark McDunn’s diagram.Īnders Larson’s Digitaltrombone chart (from the second to the twelfth harmonic) is qualitatively correct, but quantitatively fairly “approximate”. essentially half-way between reference positions) the thirteenth is 41¢ sharp the fifteenth is 12¢ flat (2¢ sharper than the fifth harmonic or 14¢ flatter than the third). twice as sharp as the third) the eleventh and twenty-second are 49¢ flat (i.e. In these fixed reference positions, compared to their nearest (in frequency) equitempered tones, the third, sixth, twelfth, and twenty-fourth natural harmonics are all 2¢ sharp the fifth, tenth, and twentieth are all 14¢ flat the seventh and fourteenth are 31¢ flat the ninth and eighteenth are 4¢ sharp (i.e. The reference positions correspond to fixed positions for natural harmonics one, two, four, eight, and sixteen. We must add seven new (different) positions for each odd harmonic above that. Mark McDunn’s “The 51 Positions” chart goes from the first natural harmonic (pedal) up to the sixteenth harmonic. When they get note-wise close there is just not enough room to display everything on paper, but the mind's eye can zoom in on the detail to follow the paths. And most charts do not well handle the compression of partials as they go up. The ear can then recognize in aural context there are position paths available for play, not just notes.Īnyway, this chart does not support that kind of thing.

So melody becomes a path of connected positions in the chart. (Lower in his visual model is higher in pitch-don't get that part.)Īnyway, the internal visual model allows one to think visually about lines, and Greg does that in his chart.

The best representation of such chart I have seen is in Greg Waits' book, but for some reason his internal model is upside down relative to mine. What I mean by internal model is an internal visualization of the position chart. This chart does not align with my internal model, though it seems pretty accurate for someone with a horn that behaves in that way on his own face.

Kids trombone position chart