# Speed of playing notes in different octaves

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).

• only be played at 6.5bps? Oct 23, 2018 at 19:27
• @TimH saying "one Mississippi" to count seconds has 5 syllables per second, that is only a bit faster. Albeit, it is on the fast end, but also not that far from middle C4. One octave lower is down to 3.25 bps. Oct 23, 2018 at 19:50
• Keep in mind that the human hearing range is starting 20 Hz for an average person.The perception of notes below 20 Hz comes down to feeling rather than hearing.
– Ian
Oct 24, 2018 at 6:45
• It seems to me that the reason that this seems so amazing is that we intuitively understand exactly what you're asking: that it's harder to play low notes fast. Oct 24, 2018 at 12:24
• "notes in lower octaves are closer together in frequency" - it's a little senseless to compare the absolute difference in counting Hz when the notes are not "closer together" from the proportional point of view. The ratio is the same. Oct 24, 2018 at 20:05

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.

• I like that PDF you linked to. What book is that? Oct 24, 2018 at 7:02
• It is written on the last page: Signale - Prozesse -Systeme, Eine multimediale und interaktive Einführuung in die Signalverarbeitung. Karrenberg, U. ISBN: 978-3-642-01863-3 Oct 24, 2018 at 7:08
• This is exactly what I was looking for. It is interesting to hear (no pun intended) that the cilia also have this problem as well - I suspected as much but am not completely familiar with them. I would be interesting to see if there was a study that actually measured the ability for individuals to properly identify notes (that were pure waves) based on duration of the note (you would need to find several people with "perfect pitch"). I do think that Todd's comment that the overtones are used to help identify the lower notes is also important though as no instrument is really a pure note. Oct 29, 2018 at 13:15
• There are such studies. Check out phys.org/news/2013-02-human-fourier-uncertainty-principle.html Oct 29, 2018 at 16:47

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.

• I had forgotten about the "filling in" our brains do, for example with the missing fundamental illusion. Do you know what the approximate integration times are for some example notes? Oct 23, 2018 at 19:26
• This is true, and interestiing, but I don't think it addresses the fundamental question, which really is looking at the fundamental (sorry!) pitch issued from each instrument. Oct 24, 2018 at 13:41
• @CarlWitthoft the fundamental pitch from the instrument isn't actually what we hear when we listen to such an instrument, though, even if the instrument plays a pure sine wave we're actually perceiving a range of overtones. Especially the first overtone. Oct 24, 2018 at 14:28

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.

• Interesting comment about the clarity which is essentially what I getting at - low notes being accurately detected as a particular note when played quick vs slow. Oct 25, 2018 at 21:25

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.

• I have never noticed instruments that play lower notes generally being played slower. I can think of several pieces of music across several genres with very fast bass lines. “Confutatis” from Mozart’s Requiem, so many symphonies, almost all of Bootsy’s bass lines from his time with James Brown, same with almost all the Rage Against The Machine bass lines. Beethoven’s “Moonlight” sonata (third movement). The list of very fast, very low notes seems endless. Oh yeah, “Hysteria” by Muse. Oct 23, 2018 at 19:35
• There are two good points being made here. Setting up a wave of lower frequencies may take more time/effort (as suggested by @topomorto), but maybe this is a very minor effect - possibly only noticeable in the extreme low end if at all - compared to the speed of notes played on average. It is also possible that the bass lines that move quickly (@ToddWilcox) focus on particular instruments that are capable of faster switching but on average they are still slower. Oct 23, 2018 at 19:46
• @ToddWilcox I'm not totally sure that instruments that play lower notes are generally played slower, but I thought in the context of the OP's question, the more relevant thing is the maximum speed at which a sequence of distinct notes can be played. And I'm pretty sure that in most families of instruments, the higher, smaller ones have the advantage there. Bass guitar has strings that are further apart and that take more energy to excite with a full tone, and the frets are further apart; everything about it slows you down. Oct 23, 2018 at 20:22
• There are some basslines that are quite dense in terms of 'events' - pull-offs, muted strokes, etc - but in terms of being able to play sequences of defined pitched notes, even a mediocre lead guitarist can shred faster than a very fast bassist. Brass and wind likewise have physical limitations to attack speed at lower pitches. Keyboard instruments aren't so limited, but stringed keyboard instruments can still 'suffer' from inharmonicity on the lower strings. I'd probably look to electronic instruments if I wanted a well-defined bassline with high 'note density'. Oct 23, 2018 at 20:25
• @ToddWilcox there sure are many fast bass parts around, but a) they usually aren't “pure bass” sounds – in symphonic bass parts, the double basses would have a hard time getting it melodically clear if they weren't doubled by the cellos an octave higher; fast piano bass and electric bass benefits from overtone-rich attack close to the bridge; and even so e-bass often requires distortion to be properly distinguishable b) those bass parts generally aren't that fast, only as fast as still makes musically sense. A good other example is in Dance Of Eternity by Dream Theater: ... Oct 24, 2018 at 10:34

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.

• Very interesting about the nature of the sound production and how it relates to the sometimes unpredictable nature of timing. I am curious though: some of the low instruments you suggest like a bass drum don't seem to me to be "tuned" instruments and thus establish the note being played is unimportant. Is this true? Oct 25, 2018 at 21:21
• @thaimin: Timpani are generally tuned to defined musical pitches. Bass drums produce multiple frequencies without a single distinct pitch, but I mention them because most of their frequencies are lower than those of timpany. Oct 25, 2018 at 23:46

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.

• I do understand that high notes can be played slowly. I was more asking about the maximum speed. Sorry for the confusion You do address my concern though that frequently the composer / conductor / etc attempt to have the lower instruments play at a slower maximum speed. Oct 25, 2018 at 21:15

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!