Speed of playing notes in different octaves

Question

Is it common for pieces to have lower notes played slower? Do instruments with lower registers typically play slower?

I ask because notes in lower octaves are closer together in frequency. For example, A0 to A#0 is about 1.6 Hz different while A1 to A#1 is about 3.3 Hz different - this is just a fundamental consequence of the doubling of frequencies between successive octaves.

At least with Fourier analysis, to distinguish two frequencies only different by 1.6 Hz you would need 1.25 seconds of the note being played. (that is 2*1/1.6 where the 2x is to handle the Nyquist frequency issue).

For A1 (delta of 3.3 Hz) you would need about 612 ms, for A2 (delta of 6.5 Hz) about 305 ms, for A3 (delta of 13 Hz) about 153 ms, ...

Is human perception able to distinguish notes better than this and thus not as limited by the speed of low notes as a Fourier Transform would be? Do lower notes usually get played slower to accommodate for this? With the logic above, A3 can only be played 6.5 bps or 392 bpm (eighth notes at 180 bpm).

Answer 1

You are right. When it comes to measuring frequencies (which is at the core of listening), there is the uncertainty principle: The shorter you measure your signal (or the shorter the signal is) the less accurate it can be measured. This is a fundamental principle and it is not depending on whether one uses a fast fourier transformation or a mechanical device (ears) for measuring.

It is stated that the uncertainty of the frequency times the uncertainty of the time is roughly 1. That means that a sine wave with 100 Hz can be estimated to within 1% accuracy if one hears it for 1 second. A sine wave with 1000 Hz can be estimated to within 1% accuracy in a tenth of a second.

There is a really great resource available online, unfortunately it is in German. It loosely translates in the introduction "The uncertainty principle": "Notes are written... as if pitch and duration could be created completely independent from each other. However, experienced composers know for a long time that the low notes of an organ or a tuba have to persist for a certain time to be perceived as well-sounding. Sequences of such low notes are therefore only playable at limited speed." (from: Karrenberg U. (2017) Das Unschärfe–Prinzip. In: Signale – Prozesse – Systeme. Springer Vieweg, Berlin, Heidelberg).

The German wikipedia on pitch also points out that from the uncertainty principle "it follows that in music practice intonation accuracy is much more important (because audible) for slow passages (long notes) than for fast passages (short notes). String players often claim - to the surprise of the layman- that it is by no means easier to play slow pieces."

Todd Wilcox' statement on the overtones is relevant here: Because of the presence of overtones, we can estimate pitch faster than if one would listen to pure sine waves, but still one needs more time to estimate the pitch of a low note.

Regarding Topo Morto's statement about the cilia in the ear detecting soundwaves directly without doing a fourier analysis: It takes time for the cilia to tune into resonance. If they are exposed to a pure sound wave for a long time, only the cilia with the right resonance frequency will vibrate. In the onset of the soundwave several cilia "around" the right frequency will start to vibrate, making it impossible to get the exact pitch of a short note.

Answer 2

The first thing I suggest you think about is the fact that a musical note is not a single frequency. Depending on the timbre of the instrument, there can be hundreds of frequencies present in a sound wave even when only one note is played on one instrument. We perceive the pitch of a note through our ears analyzing all of the frequencies simultaneously, not just the lowest frequency.

Also, human pitch perception is not accomplished by our ears or brains doing fast Fourier transforms. It’s an analog electromechanical analysis engine, and we have two of them. Our ears do have a sort of “integration time” if you will, and like an FFT, the integration time is shorter for higher frequencies, but again we are using a lot of midrange and high frequencies (often called “overtones” when perceiving the pitches of low notes.

Answer 3

It is true that low instruments don't 'speak' as easily or as clearly and high instruments, and therefore have to work harder to achieve that clarity. Double basses can sound behind because of this, and they have to almost anticipate in order to sound on the beat (especially when playing arco). This is less of a problem on the piano because of the way in which the notes are produced, but still clarity is harder work in the lower octaves.

Answer 4

Is it common for pieces to have lower notes played slower? Do instruments with lower registers typically play slower?

Yes, for a number of reasons. It often takes longer for a low-frequency resonator to settle into stable oscillation; It's often physically harder to change frequency quickly on a larger instrument (which tend to be the lower sounding ones); and the harmony of most styles of music tends towards slower motion in the bass voices.

I ask because notes in lower octaves are closer together in frequency. For example, A0 to A#0 is about 1.6 Hz different while A1 to A#1 is about 3.3 Hz different - this is just a fundamental consequence of the doubling of frequencies between successive octaves. At least with Fourier analysis, to distinguish two frequencies only different by 1.6 Hz you would need 1.25 seconds of the note being played. (that is 2*1/1.6 where the 2x is to handle the Nyquist frequency issue).

I'm not sure human pitch perception is quite as dependent on length of sound as you suggest there, as the cilia in the human ear can detect pitch components directly without needing to do a 'fourier analysis' as such. However it is true that we are less precise in our ability to distinguish the pitches of very low sounds in general. If you look up information on experiments to discern the pitch difference limen of the human ear you will find more information. If I recall correctly, the difference limen - the minimum difference we can hear - is around 1Hz through much of the audible range, becoming much bigger in the higher frequencies (where we are also less precise).

Todd's point about musical notes not only consisting of the fundamental frequency is also very relevant.

Answer 5

In a musical instrument which produces sound by plucking or striking, the note will start sounding essentially immediately at the proper pitch. Instruments which are bowed or blown, however, act as systems which produce a certain amount of "random" sound and then selectively amplify parts that are at the correct pitch. Each time a reed or string vibrates back and forth, or each time a puff of air makes a round trip through the column, it will be amplified by a certain amount before the next cycle. If it takes 10ms for a string to make an oscillation or for a buff of air to make a round trip, the signal would only get boosted about 10% as often as if it only took 1ms.

Because the amount of "random" sound energy at various pitches is somewhat unpredictable, the time required for sustained-sound instruments to speak can be likewise unpredictable. On some instruments this effect can be minimized by having a player play notes with a sharp attack that is stronger than the sustained sound will be, but on others it can't. It's not uncommon, for example, to have an organ where low notes sometimes start speaking instantly and sometimes take a noticeable fraction of a second. Such an effect would not be an issue with an instrument played percussively, however. A bass drum or timpani will produce very low frequency, but still have an attack that's almost as sharp as a triangle.

Answer 6

Most of the answers seem to interpret "play notes slower" differently than I would.

All notes in the score - assuming orchestra or chamber ensemble -- must be played at the speed indicated. Imagine the piccolo taking off at lightspeed while the contrabassoon plods along! What does happen is that the composer takes the response time of each instrument into account and tries to avoid giving the lowest-pitched instruments runs of extremely short notes.

OTOH, the flutes, trumpets, violins can play very long, slow passages as desired.

Answer 7

Right observation, wrong reason. Yes, low notes 'speak' slower than high ones on many instruments. Basically, there's more air to get moving. But a low note can be perceived as quickly as a high one. It's the same fallacy that suggests that a low note can't be heard in a small room, because there isn't room for a complete waveform. Obviously rubbish, else listening on headphones would have a severe bass cut!