# Playing 440 Hz, what are the harmonics for a trumpet? For a flute?

Playing 440 Hz, what are the approximate harmonics for a trumpet? For a flute? This to help students understand the differences when those instruments play the same note.

I've been to many website, including University of New South Wales. I would like spectrum in percentages. For example: --- Flute => 440 Hz - 100% | 880 Hz - 33% | 1320 Hz - 8% | 1760 Hz - 13% | 2200 Hz - 12% | 2640 Hz - 2% | 080 Hz - 12% --- Trumpet => 440 Hz - 100% | 880 Hz - 100% | 1320 Hz - 53% | 1760 Hz - 75% | 2200 Hz - 85% | 2640 Hz - 40% | 3080 Hz - 32% --- Same harmonics - different harmonic amplitudes.

I prefer percentages. Many students have no idea what dB means. Percentages they do understand.

``````I posted this question a few days ago.  I'm curious if no one has pursued this type of
question.  Trumpets, etc. have been around for decades.
``````
• You'll find plenty of waveform pictures of the sustain portion of trumpet and flute notes. Flute tends more to a sine wave, with predominately even-order harmonics. Trumpet is a rather more complex sound with more odd-order harmonics. But you might be surprised at how SIMILAR the sustained part of the two instruments sound, not how DIFFERENT. A lot of the characteristic of an instrument is in the initial attack. This displays as a very complex - even chaotic - waveform, difficult to analyse. I'm afraid your 'measure the harmonics of the sustain portion' approach may not be terribly useful. Commented Nov 9, 2020 at 23:50
• (continued) This fact was exploited by the 'hybrid synthesis' generation of keyboards, notably the Roland D50 and the Yamaha SY range. Computer memory for samples was still limited and expensive, so they sampled the attack, synthesised the sustain and release. It was remarkably successful! This approach continues in the much-respected Kurzweil keyboards that manage to produce remarkably realistic instrumental sounds from remarkably low amounts of sample memory. Commented Nov 9, 2020 at 23:50
• @LaurencePayne you may be thinking of clarinet, which (thanks to the cylindrical bore and half-open characteristic) has only odd harmonics. But that doesn't really apply to flute, which though also cylindrical is sufficiently open on both ends to support both even and odd harmonics – like trumpet, but with much lower amplitude. I don't think there's any instrument that has predominantly even harmonics, at least there's no simple physics reason why this would happen. Commented Nov 10, 2020 at 0:11
• What kind of students? High school? University? What prevents them from understanding dB? What class? Physics? Music? Would they understand difference between power and amplitude percentages? What exactly are you trying to teach or demonstrate? Commented Nov 10, 2020 at 5:35
• "I prefer percentages. Many students have no idea what dB means". That's understandable, but on the other hand, using a linear scale (%) instead of a logarithmic scale (dB) is misleading, as the difference between 0.1% and 10% is a lot more from an acoustical standpoint than the difference between 10% and 100%. This is the reason for using dB or phon.
– mins
Commented Apr 18, 2021 at 16:54

You can find detailed acoustic information on both trumpet and flute (and many others) on the Music Acoustics website for the University of New South Wales. Here are the pages for

In particular, the trumpet spectra

and the flute spectra

Similar information is available in The Physics of Music and Musical Instruments, by David Lapp, Chapter 3, "Modes, Overtones, and Harmonics" (pp. 27-39). (This site appears to be directly related to the UNSW site above insofar as it links to it directly for "further information".)

Again the trumpet spectrum

The first graph is a histogram of the power spectrum for that musical instrument. The first bar on the left is the power of the fundamental frequency, followed by the overtones. Be careful when comparing the relative strength of the overtones to the fundamental and to each other. The scale of the vertical axis is logarithmic. The top of the graph represents 100% of the acoustic power of the instrument and the bottom of the graph represents 80 dB lower than full power (†10-8 less power). The second graph is a superposition of the fundamental wave together with all the overtone waves. Three cycles are shown in each case.

The corresponding flute graphs are not provided, but there are similar graphs for a variety of other instruments.

I did not find graphs corresponding to the two instruments at 440Hz

You may also find another post from SE Music Practice and Theory helpful.

• I don't see how this answers the question. Thanks! Commented Jan 10, 2021 at 14:03
• @Clyde if a detailed graph of acoustic spectrum of a flute with db on the y axis and all the harmonic peaks clearly marked doesn't answer your question, what would? What am I missing here? Commented Apr 18, 2021 at 17:37

I'm adding this answer to show how using harmonics amplitude to characterize a sound can lead to problems.

The spectrum of a periodic signal can be computed using a Fourier transform, which purpose is to transform a sum of sinusoidal components into a sum of complex exponents of e.

The coefficients of the spectrum are therefore complex numbers. The real part of a coefficient is the frequency relative amplitude and the imaginary part is its phase offset which determines the amplitude of the component at time 0 (more). Most of the time spectrums are shown with only the amplitude information visible and offsets are not mentioned.

If all harmonics started at time 0 at their maximum like a cosine function, offsets would be equal and everything would be simple. That's usually not the case, causing alterations in the waveform. For example, let's create a wave at 440 Hz with overtones this way:

The harmonics parameters are:

• Frequencies: 440, 880, 1320, 1760, 2200 (Hz)
• Amplitudes: 1.0, 0.8, 0.22, 0.4, 0.55
• Phase offsets: 0.0, 0.4, 0.8, 0.7, 0.3 (unit = pi radians)

As visible these waves don't start at their maximum, except the fundamental (the sum doesn't start at its maximum either).

On the right hand side this is the spectrum as usually represented. I actually obtained this spectrum with a Fourier transform (DFT) of the blue wave. The graph shows only the amplitudes of the components, not the phase offsets.

Now, this spectrum corresponds to an infinity of different waves. It's easy to demonstrate this by reconstructing a wave from the amplitudes only and another from both the amplitudes and the phase offsets. To do this I performed an actual inverse DFT of the spectrum, once using only the real part of the coefficients, once using the whole coefficients:

Reconstructing waves from real and complex coefficients of the DFT

The first graph shows both reconstructions, and the resulting waves are indeed different (the blue wave is identical to the original wave)

The other rows show respectively the complex and real spectrums. They are actual spectrums obtained by performing DFTs on each reconstructed wave. They are identical, except the phase offsets.

Note the wave reconstructed without the phase offsets assumes all harmonics have a 0 offset, and therefore the sum has also a 0 offset, and starts at its maximum.

An infinity of different waves can be constructed from the spectrum amplitudes, just by varying the values of the (usually unknown) phase offsets.

Conclusion: If you want to characterize sounds from their spectrums, you need to know the harmonics amplitudes, as you asked for, but also the phase offsets. Getting the offsets might be a bit more difficult than the amplitudes.

• In many situations, though, the relative phase of the harmonics doesn't affect the auditory perception of a sound. See here for a review and discussion of the literature. Commented Apr 18, 2021 at 18:12
• @MichaelSeifert: Honestly I wasn't aware of such aspect, thanks for pointing out.
– mins
Commented Apr 19, 2021 at 1:24