Representing E12 electronic component values musically

Kragen Javier Sitaker, 02020-11-17 (updated 02020-12-26) (16 minutes)

Resistors, capacitors, and sometimes even inductors are conventionally manufactured to have a set of “preferred values” in each order of magnitude: the E3 series, 10 22 47; the E6 series, which intercalates 15 33 68; the E12 series, which additionally intercalates 12 18 29 39 55 82; and longer series. So it’s common to see a 220-ohm resistor or a 220-μF capacitor, but you’ll very rarely see a 200-ohm resistor or a 200-μF capacitor.

Despite appearances, these values are fairly evenly spaced — just in exponential space. 82÷68 ≈ 1.206; 29÷22 ≈ 1.318; 18÷15 = 1.2; and so on. The intervals all approximate 101/12 ≈ 1.2115.

Component identification — knowing whether this resistor you have is a 2k2, a 22k, or just broken — is a big challenge for salvaging electronics. Especially for us old colorblind humans. Usually it’s good enough to know which E12 value something is. Microcontrollers that can easily distinguish components are easy to come by, but getting their output into a useful form requires some kind of screen, and those are comparatively rarer and more finicky to interface with.

The humans can easily distinguish notes of the 12-tone equal temperament scale traditionally used for Chinese music (though this is far easier when presented as intervals rather than as bare notes), and there’s a pleasing perceptual correspondence between the 12 tones in an octave and the 12 E12 values in a decade. And speakers are cheap and easy; sometimes, as in surface-mount MLCCs using piezoelectric X7R dielectrics and similar, they’re even included by accident. Auditory output also has real advantages for high temporal resolution that the humans can perceive.

So I propose that a component-identifying smart-tweezer program should map measured component values to the musical scale in this way and alternately play a couple of notes through a speaker, separated by the appropriate interval. One of the notes can be something like a flute playing a standard frequency, like 261.62557 Hz for A440 middle C (C₄ except in MIDI), to give a point of reference, while the other plays after it as a string instrument or something, with each decimal order of magnitude of component values mapped to one musical octave. That is, represent them with a fundamental frequency of k 2log₁₀ v, where v is the value to represent and k is some proportionality constant.

Psychoacoustics

The humans can easily hear the eight octaves from 32.70 Hz up to 8372 Hz, while the next octave and a half up is mostly audible only to larval humans; the next octave down from 32.70 Hz to 16.35 Hz is audible in its pure form only if it is very loud, but if provided with rich harmonic content, it is clearly comprehensible even when the fundamental is too low to hear or even not reproducible by the speakers. Their hearing sensitivity is about 1 Hz for complex tones up to 500 Hz and about 0.6% above 1000 Hz; between 1kHz and 2kHz their perception of frequency is least distorted by amplitude, while between 2kHz and 5kHz they are most able to detect sounds. This suggests that it’s probably best to stick to the lower octaves as much as possible, even down below 20 Hz, since their harmonics will populate the most sensitive regions of hearing more densely.

However, we start to lose relative precision once we go below 500 Hz; at 500 Hz the frequency precision is about 0.2%, at 250 Hz only about 0.4%, at 100 Hz only about 1%, at 65.41 Hz about 1.5%, and at 20 Hz only about 5%. After being transformed through the inverse of the representation function above, these perceptual imprecisions are respectively 0.7%, 1.3%, 3.4%, 5.2%, and 14%. At higher frequencies errors climb again but not so high.

Resistors, with high resistances at high frequencies

Resistors generally used by the humans range from 1Ω to 1MΩ, six orders of magnitude apart, with most being in the 100Ω–100kΩ range, only three orders of magnitude apart. Generally low-value resistors are used for precise measurement (linear conversion between voltage and current), and current limiting, while high-value resistors are used for less-precise applications like pullups, pulldowns, capacitor bleeders, and protection. This suggests maybe positioning 1Ω around 110 Hz (A₂, A below small C or A above deep C), giving the following correspondences:

This of course gives k = 110 Hz.

This assignment is a compromise between not “wasting” the lowest octaves on little-used low resistances that require Kelvin probing to measure accurately, assigning the best precision to values in the 1–100 Ω range where it often matters the most, and not assigning megohm values to totally inaudible frequencies.

Resistances above a few megohms might be best represented by some additional gimmick, like using a different musical instrument.

Capacitances, and a better resistance scale

Capacitances are trickier because they span a wider range; common capacitors are in the 47 pF to 470 μF range, though up to 22000 μF is not unheard of — though anything above 1 μF probably isn’t very precise, because the piezoelectric and electrolytic technologies used at those higher capacitances aren’t very precise. (And then there are supercaps, up to 100 F — that is, 100 000 000 000 pF.) In the 47 pF–4.7 μF range, though, we have only five orders of magnitude, which seems quite manageable.

It’s unclear whether to use high frequencies for high capacitances (like, microfarads) or low capacitances (picofarads). Arguments in favor of using them for low capacitances include:

  1. In real life smaller capacitances do things at higher frequencies. 0.01 μF is 1 kΩ at about 16 kHz; 100 pF doesn’t drop to 1 kΩ until 1.6 MHz. Whether in an LC tank, an RC oscillator, or an RC filter, using a smaller capacitor means your frequencies go up.
  2. Also, high capacitances tend to be physically larger, and, psychologically, larger things should make deeper sounds, while tinier things make tinnier sounds.

Arguments in favor of using high frequencies for high capacitances include:

  1. Precision is more important for low capacitances, which are largely used for tuning things, than for high capacitances, which are more used for bypassing things. If we put 47 pF at 880 Hz so that perceptual precision is best around 100 pF, and go down from there, then at 0.047 μF we’re already at 110 Hz, and by 4.7 μF we’re at 27.5 Hz. What on Earth would we do with 470 μF? The noise of an idling motorcycle without a muffler, at 6.875 Hz? And we’d be wasting a couple of audible octaves that correspond to capacitances too small to measure reliably under ordinary conditions, because they’re swamped by parasitics.
  2. You’ll be able to transfer your numerical ear learning from resistance to capacitance more easily. (“Oh, that C# is 2.2kΩ. That means it’s 0.022 μF.”)
  3. Higher resistances move RC oscillators and filters to higher frequencies too, so what gives? (Higher resistances move RL circuits to lower frequencies, but nobody uses RL circuits.)
  4. The difference in response frequencies of capacitors is vastly understated by the frequencies you hear.

The rebuttal arguments in favor of using high frequencies for low capacitances include:

  1. Maybe high resistances should be represented as low frequencies too, both because they slow down RC circuits and because we think of higher resistances as being “larger”, so maybe they should speak in deeper voices too? (Also, really high resistances, like 10 MΩ and up, tend to be used with high voltages, so they are physically large, both for power dissipation and creepage allowance. Still, lots of physically large resistors I have here are 10-Ω current-sensing shunts or 25-Ω heating elements, not multi-megohm high-voltage protection resistors.)

So I propose this scale for capacitances:

So, for example, 120 pF would be about G#₄, 415.30 Hz, one half-step deeper than A₄, because it’s one step higher on the E12 scale; 150 pF would be about G₄ (392.00 Hz), and 220 pF would be about F₄ (349.23 Hz). These are approximate because the E12 scale is approximate: 120 pF would be more accurately 416.50 Hz, 150 pF more accurately 389.44 Hz, and 220 pF more accurately 347.03 Hz. So 150 pF, for example, is flat of G₄ by 11.3 cents, a difference audible to trained musicians and possibly somewhat painful, but hard for the untrained ear to detect, though probably possible in this region of best sensitivity. A perfect G₄ would be closer to 146.8 pF, an error of -2.2% from the nominal value.

Correspondingly, I propose this one for resistances, equating a microfarad to a megohm, a nanofarad to a kilohm, and a picofarad to an ohm, as if we were interested in an electrical signal of 160 kHz:

Note that this also solves the problem of what to do with arbitrarily high resistances, although you have to be careful you aren’t swamping the whole audio spectrum with harmonics from a spurious detection of a 100-GΩ resistor that’s really leakage through your probe insulation or something.

It might be reasonable to do, say, a square wave for capacitance and a repeatedly plucked Karplus–Strong string (or two, slightly offset in frequency, like a piano) for resistance, so that you can play them at the same time if you’re doing an ESR measurement. Using different envelopes will help the humans hear them as two separate sounds rather than one sound with a discordant timbre.

Inductors

Now we’re faced with another consistency dilemma: do we represent large inductances with low pitches or with high pitches?

In favor of low pitches:

  1. Large inductances, like large capacitances, are used at low frequencies. A 60-Hz fluorescent light ballast might be an entire henry. A millihenry has a kilohm of reactance at 160 kHz; at 16 MHz you only need 10 μH to get to 1kΩ.
  2. Inductors with large inductances are physically large, so should ring deeply, like a church bell, not like a jingle bell. Also they tend to have heavy magnetic cores.
  3. It would be numerically consistent with the systems suggested above for capacitances and resistances.
  4. Of course in LC circuits increasing the inductance lowers the frequency.

In favor of high pitches:

  1. Steal underwear.
  2. ???
  3. Profit!
  4. In some metaphorical or Steinmetzian sense a large inductance is the opposite of a large capacitance.

I think the underwear gnomes lose this one.

If we try to use the same 160-kHz signal frequency to figure out where to set the equivalence, we get this:

This seems reasonable.

It’s probably worthwhile to use a different instrument again for inductor detection, perhaps an FM-synthesized bell sound or something.

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