musical thoughts

This is an article I posted on the forum at the SuperSite. Everyone talks about tone quality, but no one does anything about it (apologies Mark Twain). I am beginning the process of doing something. And bringing science to an area that lacks it.

I read a lot of posts where people discuss the tone of a Rhodes. So far I haven’t seen much in the way of attempts to quantify it or study it empirically. So this is the beginning of my attempts to study the Rhodes tone.

All this analysis was done using Audacity, which is free and excellent. I recorded my 1976 suitcase 73 direct in mono using my Edirol UA-25 at 44.1kHz, 24bit sample size, and normalized to -3dB. Tone controls were set flat, tremolo off. While I’m giving you an MP3 to listen to, I did all the analysis before any compression.

To begin, I started with a single note, F3, played with dynamics forte.

The first step on the journey is to visualize the waveform.

This is Audacity’s default waveform display. The amplitude scale is linear. You can visually determine that the decay of the sound follows a curve–this is approximately an exponential decay curve (found in nature in various places).

The sustain carries on a long time–you can hear the note for almost 20 seconds. But we need some better data here–you can’t really quantify “long sustain.” Let me pause for a minute and introduce the dB scale. By themselves, decibels (dB) have no dimension. They express the ratio of two quantities as a logarithm, which makes certain kinds of math easier. To avoid confusion with various ways of measuring sound, I’ll stick to using dB to indicate a ratio between two amplitudes (also known as a dynamic range). The dynamic range of human hearing is about 120 dB. A 10dB difference is perceived as about a doubling of volume.

Audacity will let you display a waveform against an amplitude scale in decibels. Here 0dB indicates the loudest sample possible in whatever system you’re using. There’s also a handy feature called “normalize” which scales your sampled waveform to -3dB, evenly increasing everything by a consistent amount. So a 30dB decay time would be the same before and after normalization, it’s just easier to see it after normalization since everything is scaled up. The sample I used was normalized before analysis.

Here’s the note in log (that’s dB) scaled amplitude display:

Here you can easily read that a 30dB decay takes about 10 seconds (for this note). This is quantifiable, and a repeatable measurement you can use for comparisons, adjusting your voicing, etc.

A straight line decay in this scale would be precisely exponential. You can see that the sustain portion of the wave is very closely exponential in amplitude while the earlier attack portion decays faster.

Why would that be? We can look at a few more measurements to investigate.

Another way to visualize the same waveform is as a frequency plot.

We still have time on the horizontal axis here, but the vertical scale is now frequency, with amplitude being depicted by color. White is the highest amplitude, and it “cools off” through yellow into red and blue as amplitude drops.

This view explains quite nicely what’s happening. The higher-frequency components of the tone decay much more quickly than the lower-frequency components. By 7 seconds, most of the overtones (higher-frequency components of the tone) are decayed enough that they don’t appear on the plot.

What are these components, how do they arise, and what role do they play in the Rhodes tone? Let’s look at a few more graphs.

This is an amplitude versus frequency plot done on the first 100 milliseconds of the note.

One thing I don’t like about this display is that the amplitude extends down to -80dB or so. You should make a mental note to ignore everything below about -40dB on this plot. I was going to edit it out, but decided against it. Just keep in mind we only want the top half of this plot. One thing I do like is that each division on the horizontal scale is exactly an octave (2x the frequency).

Notice a few things that are easy to see on this plot:
1. There are a bunch of somewhat regularly-spaced “humps” in the response. These correspond to the fundamental pitch (which in this case is F3, in the neighborhood of 175Hz) and the overtone series that results from integer multiples of the fundamental frequency (350, 525, 700, 875, 1050 and so on).

2. The “first harmonic” (which is at 350Hz) is actually louder than the fundamental. (Recall all that stuff in chapter 4 of the service manual about tine position with respect to the pickup–I voiced mine this way, yours can vary.) So are the second, third, and fourth harmonics, at least for the 100 milliseconds or so covered by this plot.

3. There’s a hump at around half the fundamental frequency. I’m not going to attempt to explain that, I will just note it and move on for now.

This is an amplitude versus frequency plot done on the 100 milliseconds following 7 seconds into the note.

Notice a couple things that are easy to see on this plot:
1. The harmonics are mostly below the -40dB threshold, but you can still spot them. So you could characterize the tone at this time as “pure fundamental” at least by comparison to the attack tone.
2. That “undertone” hump that showed up in the attack at half the fundamental frequency, it’s shifting lower (which is odd).

Here’s an overlay of both to help you see the change in the frequency response of a Rhodes tone from the attack to the sustain segments of a tone:

Hopefully, this gets you thinking about what the words like “warm” or “bell-like” mean in terms of frequencies, time, and overtone content. Visualizing the tone in terms of frequency like this should help you get a grasp of what equalizers (including the tone controls on your piano) can do to affect a Rhodes tone.

One more chart for you to help get the sense of what pitches are associated with overtones. this is the exact same attack analysis, the first 100 milliseconds of the note you’re hearing in the mp3.
click for a lovely big chart

This one is big for readability, so I linked it to let you open it in a new window. The horizontal pitch scale is stretched a bit in this one, and the note names for the first 10 harmonics are included.

Cool things you can learn from this diagram:
1. The 3rd, 4th and 5th harmonics from a major triad that belongs to the fundamental. This is a big part of why the major triad is a stable sound in western harmony.
2. Higher harmonics are less prominent than lower ones in general, but they all contribute to a tone quality.
3. You could EQ all the harmonics out of a tone, but it’s tough to put them in with EQ if they aren’t there. So voicing is really important.
4. The 7th harmonic (Eb) is in there at levels damn close to the fundamental. Again, that was my voicing choices that made this happen–your piano may not sound at all like mine.
5. The harmonic series theoretically extends to infinity. Physical limitations prevent you from perceiving all of it, but the overtone obtained by 35 times the fundamental frequency is theoretically present.

I hope this article has helped you to visualize a little better what goes into the unique Rhodes tone. Let me know if you like this kind of thing and I’ll work on some more topics, bringing the science.