Spectrum of brass tones of same fingering

Question

On valve brass instruments, a single fingering may be used to play different notes. For example on the trumpet, leaving all valves open (up) allows the player to play the following notes:

C - G - C - E - G - Bb - C - ...

(The first playable interval is a 5th instead of an octave, but that is another question.)

The wave created when playing a low C has a specific spectrum that determines the timbre.

What is the spectrum of the G just above this low C?

1. the timbre is very similar, so the spectrum should also be very similar. If low C is a weighed sum of the harmonics of the fundamental (C - G - C - E - G - Bb - C - ...), the spectrum of G should have the same components transposed (G - D - G - B - D - F - G - ...).

2. G is played on the same length of tubing as the low C, therefore the harmonics present should be the ones allowed by just this length of tubing. Then the spectrum of G should be G - C - E - G - Bb - C - ....

These two points of view are incompatible. I have a feeling that 2 is wrong, but I can't see why the tube length of C would allow such a strong D (a ninth above the low C) in the spectrum of G.

Answer 1

Your first assumption is (mostly) right. Trumpet physics are actually fairly complicated: The basic tube has an open and closed end which by itself would only produce odd numbered harmonics with a quarter wavelength fundamental. However the tapered mouthpiece tapered bell change the harmonics spacing so it gets much closer to the natural harmonic progression.

At higher frequencies the trumpet harmonics spectrum gets fairly dense so there is always a useful harmonic near by when you need it. Also the tapering makes the resonance not super sharp so you can actually move the frequencies around a bit.

So the horn by itself is capable of supporting a fair amount of frequencies. Which ones are actually contained in a specific note depends by the excitation. If the excitation itself doesn't contain a certain harmonic, the horn won't amplify it either. In your example of C the excitation from your lips already contains the harmonic series (C, G, C, E ...) and the horn just amplifies them. Same for the G (G, D, G, B ...). While playing the G the horn is perfectly capable of reproducing an E, your lips just don't excite it.

Another nit-picky remark: It's actually not entirely correct to call the harmonics by note. This is exactly true only for the octaves the other notes are slightly off (harmonic tuning vs. temperate tuning). For example, the 6th harmonic (minor 7) is off by about 2% or 31 cents.

Answer 2

In enthusiastic agreement with Hilmar's explanation of excitation and resonance for an open-valved trumpet, here are additional details and a chart about relationships between resonant frequencies, timbral components and pitch. This follows the question's example of an open-valved middle G played on a trumpet. Corresponding notes for other brasses are listed at the bottom.

Summary of the notation used here and a disclaimer about the scope of “timbre”:

C D E F G A B = below-clef C through mid-clef B       "Timbre" is considered
,X X 'X "X ^X = consecutive octaves of note X         here only in terms of
     X-       = frequency near note X but flatter     which frequencies are the
     X--      = frequency near X- but flatter yet     most significant, not even
    X/Y       = frequency between notes X and Y       their relative amplitudes.

Resonant frequencies of an open trumpet:      C  G 'C 'E- 'G 'Bb-- "C "D "E- "F/F# "G ...
Harmonics of ,C  (octave below low C):    ,C  C  G 'C 'E- 'G 'Bb-- "C "D "E- "F/F# "G ...
Harmonics of  C  (low C):                     C    'C     'G       "C    "E-       "G ...
Harmonics of  G  (middle G):                     G        'G          "D           "G ...

For the timbre of a trumpet's open-valved G, the question suggests two possibilities.

Possibility 1. The timbre is very similar [to that of low C], so the spectrum should also be very similar. If low C is a weighed sum of the harmonics of the fundamental (C-G-C-E-G-B♭-C-...), the spectrum of G should have the same components transposed (G-D-G-B-D-F-G-...).

Possibility 1 is indeed true for the sustained timbre of G, after the attack has developed into a stable tone, with the complete list of timbral components being the harmonics of G: G 'G "D "G "B- ^D ^F-- ^G .... (The lists of harmonics in Possibility 1 should be interpreted as if they include one more C or G at their corresponding heads.)

Possibility 1 applies to a sustained timbre because the player's lips are steadily vibrating at the frequency of G, which, mathematically being a periodic signal, mainly produces harmonics of G but not other frequencies. Those harmonics are selectively amplified because each one matches some resonant frequency of the open trumpet. A trumpet cannot amplify a frequency that is not supplied by the lips, such as C or E- in this case, even though it may be a resonant frequency.

Possibility 2. G is played on the same length of tubing as the low C, therefore the harmonics present should be the ones allowed by just this length of tubing. Then the spectrum of G should be G-C-E-G-B♭-C-....

Possibility 2 is true for the timbre of the attack of G, which in fact includes low C in the complete list of components: C G 'C 'E- 'G 'Bb-- "C "D "E- .... This is because the attack of any note is driven by virtually every frequency (best online reference found so far, any better leads?), not just harmonics of the intended note. Thus, energy at every resonant frequency of the open trumpet is present for amplification during the first split second, so much that played C and G are fairly indistinguishable before they begin to settle into their individual sustained timbres (as in Possibility 1).

Possibility 2 is also a primary cause for the difference in sound between the attacks of an open-valved G and a 1st-and-3rd-valved G. The timbral attack components of a 1st-and-3rd-valved G contain the overtones of ,,G (not of ,C, like open-valved G): ,G D G B- 'D 'F-- 'G ....

Chart of resonances   (captioned below)

    Lipped pitch    Resonant frequencies of an open trumpet  (scroll sideways for more)                                                    (Lipped
    ------------    ---------------------------------------------------------------------------------------------------------------------   pitch)
                    C     G    'C    'E-   'G   'Bb--  "C    "D    "E-  "F/F#  "G   Ab/A  "Bb--  "B-   ^C   ^Db    ^D   ^Eb   ^E-   ^F--

            'G                             'G                                  "G                                  ^D                         ('G)
                                            3 . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . 9 . . . . . . . . . .

            'E-                      'E-                           "E-                           "B-                          ^E-             ('E-)
                                     5/2 . . . . . .  . . . . . . 10/2 . . . . . .  . . . . . . 15/2 . . . . . .  . . . . . . 20/2 . . . .

            'C                 'C                      "C                      "G                      ^C                     ^E-             ('C)
                                2 . . . . . . . . . . . 4 . . . . . . . . . . . 6 . . . . . . . . . . . 8 . . . . .  . . . . . 10 . . . . .

             G            G                'G                "D                "G                "B-               ^D               ^F--       (G)
                         3/2 . . . . . . . 6/2 . . . . . . . 9/2 . . .  . . . 12/2 . . .  . . . 15/2 . . .  . . . 18/2 . . .  . . . 21/2 . .

             C      C          'C          'G          "C          "E-         "G         "Bb--        ^C          ^D         ^E-              (C)
                    1 . . . . . 2 . . . . . 3 . . . . . 4 . . . . . 5 . . . . . 6 . . . . . 7 . . . . . 8 . . . . . 9 . .  . . 10 . . . . . .

            ,C      C     G    'C    'E-   'G   'Bb--  "C    "D    "E-  "F/F#  "G   Ab/A  "Bb--  "B-   ^C   ^Db    ^D   ^Eb   ^E-   ^F--      (,C)
Frequency
    ratio   1/2     1    3/2   4/2   5/2   6/2   7/2   8/2   9/2  10/2  11/2  12/2  13/2  14/2  15/2  16/2  17/2  18/2  19/2  20/2  21/2

Semitones        12    7     5    3.9   3.2   2.7   2.3    2+   1.8   1.7   1.5   1.4   1.3   1.2   1.1    1+    1    1-    .9    .8

Columns show which frequencies, if supplied by a player's lips, are amplified by resonance in an open trumpet. Rows show which frequencies are actually supplied by the lips for the first six pitches that may be played with an open trumpet, including pedal ,C for completeness. Note that the fundamental frequency of pedal ,C is absent from the resonant frequencies, allowing at best a rough gutless sound, especially on cylindrical-bore brasses such as trumpet and trombone.

Corresponding notes for other brass instruments. The notes described here as C and G for trumpet correspond, in terms of harmonics, not actual pitch, to:
•  French horn – bass clef C and G.
•  Other transposing brasses in treble clef – same C and G as for trumpet throughout this post.
•  E♭ tuba, bass clef – below-clef E♭ and in-clef B♭.
•  F tuba – below-clef F and in-clef C.
•  C tuba – below-clef C and clef-bottom G .
•  B♭ tuba – below-clef B♭ and F.
•  Other B♭ brasses, bass clef – in-clef B♭ and F.

_{This post stems from an understanding of trombone acoustics.}