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Let say I sing or play song in C major. I feel that C note as home, or in other words, it has the quality of the tonic note for me. I stop playing the song. After 24 hours or any longer period, without listening or playing any other songs I will have 3 possibilities

  1. I still will have in mind C as tonic.
  2. I will have in mind random note as a tonic.
  3. I will not have any feeling of tonal center. And any random note for me will not have any context. This random note will not have any scale degree at all for me.

Which of these three possibilities more likely to happen?

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Remember that music isn't the only thing that has pitch. Most sounds in the natural world have a pitch as well, and a random sound outside could temporarily get stuck in our ears as a possible tonic.

For me—and keep in mind I don't have absolute pitch—I often find myself singing in B♭. To make a long story short, I think this is because a number of devices around my house are "in" B♭: my microwave, coffee grinder, and lawnmower all hum a B♭.

I say all of this because, in order to really answer this question, we'd need a complete vacuum of sound to really test it. As it stands, there is an immeasurable number of outside pitch influences that could, however temporarily, stand as tonic.

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    Haha, next time you’re buying an electrical appliance you should ask if it’s in Bb to complement your orchestra. – Creynders Sep 20 '19 at 14:57
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    Is it possible that because the standard electrical hum in the United States is 60Hz (~B♭1), most of your electrical devices hum a B♭, resulting in you singing in B♭? (Europe and Asia, if I'm not mistaken, run at 50Hz instead) – user45266 Sep 20 '19 at 16:07
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    Annoyingly sharp of G in the EU, yup. – Tetsujin Sep 20 '19 at 18:15
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    Umm, a transposing lawnmower! Sounds interesting. Mine's at concert pitch, although it's occasionally at football pitch... And I had it tuned this season - goes better still. – Tim Sep 20 '19 at 19:23
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    @Tetsujin 50 Hz is actually closer to G than 60 Hz is to B♭. Both are better described as quarter-tones between G and G♯, and B♭ and B, respectively. – leftaroundabout Sep 21 '19 at 11:46
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If you have absolute pitch, then no.1 will be the answer.

No.2 for most people, when they will still be able to whistle or hum the tune in question, but having a random note as tonal centre/root/tonic. It won't really matter to them.

Any 'random note' will not have the feel of any 'tonic' until you decide on one - the same as original or not, doesn't matter. So no. 3 is virtually no.2, I think.

Whatever key it gets sung in later is of little consequence to most people - as long as it's sung in tune! It only really matters when other players (not singers, particularly) are involved, as a common key makes everything sound so much better...

Or have I missed the point of this question?

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  • Isn‘t the point that we don‘t have a tonal centre without minding a certain tune or interval like a fourth up or a minor third down or any other musical motif? – Albrecht Hügli Sep 20 '19 at 21:24
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"Random" tonal center

This will depend on several factors. As @Tim mentioned, if you have absolute pitch, then your conjecture (1) is correct, and any tune in C will remain in C for you, and likewise for the other 11 keys.

As for your conjecture (2), this is most likely, but it isn't nearly as random as you may think! Even if we avoid listening to music for an extended period, there are sounds all around us:

  • the buzz of AC circuits, also known as mains hum, power line hum (60 Hz in USA (~B half flat), 50 Hz in most of the rest of the world (~G half sharp)).

  • If you are in a more rural setting, you may hear birdsong (varies by species) and the chirping of insects (varies by ambient temperature!).

  • In a more urban or industrial setting, the rumbling of vehicles and machinery will have characteristic frequencies as well.

The more ear training you have, the more your brain will take these sounds and contextualize them into pitches and intervals, and these will form the basis of the "random" tonal center in your head. For example: if you are in USA, you will hear a mains hum of 60 Hz in your office. Outside, a semi truck is idling (call it ~39 Hz for this scenario). These two notes are B and D# (roughly), producing a B major chord in your head. If you start humming while listening to those sounds, the tune will likely be in B major. Note that if you heard the exact same truck outside your UK office, your tonal center would be Eb major instead.

Conjecture (3) is mostly for non-musicians, who as far as I can tell by talking to them, somehow don't have songs in their heads at all times (??), which is really hard for me to even consider.

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    "If you start humming while listening to those sounds, the tune will likely be in B major" Do you know this to be true? Have there been any studies to that effect? I'm not sure I buy that the tonal center is extremely biased towards those certain frequencies, especially because a friend of mine with absolute pitch goes walking with me sometimes, and whenever we find someone singing or whistling or humming, they tell me what key they're in, and so far it's been a random assortment of tonal centers (and more closely related to the original key of the song than to any particular note). – user45266 Sep 20 '19 at 16:12
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    @user45266 My example was just that, an example. I also provided several other potential sources of pitch above which would lead to any number of different tonal centers. My point was not that everyone always hums in B major, but that upon hearing certain frequencies, one can be "steered" in the direction of keys related to those frequencies. Right now, I can hear a computer fan humming at F#3, and so the melodies in my head are using that pitch so as not cause too much dissonance. – dissemin8or Sep 20 '19 at 16:32
  • My answer is very close to yours. – Albrecht Hügli Sep 20 '19 at 21:17
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As Richard says, any noice in our environment, e.g. the „brumm“ of a car, a plane or even a bee or a fly can become a new tonic. I assume it is also depending of the range of our voice and the range of a song we have in mind whether a tone like the pitch fork A will become the tonic or the dominant of an eventual tune. I am speaking from my experience trying to acquire perfect pitch and always getting disturbed by environmental music or noise. The tonal center can also get lost simply by listening a modulating music piece. Sometimes I wake up in the morning and have still the sound of the fork in the right pitch in my ear. But this doesn‘t mean that when reading sheetmusic I find automatically the right pitch even I am used to derive the initial interval of a song from the fork pitch. This is always an activ act of concentration.

So to answer the question:

People with perfect pitch will find or keep the same tonal centre.(1)

All others - like me - are helplessly lost even if they are willing to find the same centre as you are asking. Randomly they may keep it. (2)

If we don‘t have a music piece in mind there will be no tonal centre. (3)

It sometimes happens that I try to find the A=440 (concert pitch) without thinking of a song. But then this A is rather the dominant of D than the tonic of A as I have probably the circle of fifths in mind or the upbeat of a fourth to a possible song. (4)

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I quite like Richard's answer, but to offer a more narrow one: the concept of a tonal center is meaningless in the absence of music. There's nothing physically real about a tonal center; from a pure physics perspective it doesn't describe any property of vibrations in air. When we talk about a tonal center, we are talking about a very specific and socially constructed understanding of sounds that have been organised and/or presented in a particular way, as music.

Your point three is probably equivalent to this view.

I would personally describe the phenomenon Richard observes as a "potential tonal center for music yet to come, set by ambient sounds", but I can imagine that there are many contexts where it is convenient to treat this unwieldy phrase and concept as being essentially identical to a tonal center in my narrower sense of the word.

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It's your voice. It is in my case. The note which i can hum with the least effort in expiration is G2. Unconsciously you do that.

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