Final answer:
When moving up an octave on a keyboard, the next occurrence of the same letter name will vibrate at a frequency twice that of the original note. This is rooted in the science of acoustics and resonance. Beats are heard when two adjacent notes with slightly different frequencies produce a warbling sound due to interference.
Step-by-step explanation:
If you take any pitch on the keyboard, the next occurrence of the same letter name going towards the right (up) will vibrate at a frequency that is twice as high. This is due to the octave relationship where each octave increase results in a doubling of frequency. For example, if a note at middle C (approximately 261.63 Hz) is played, the next C higher on the keyboard will vibrate at about 523.25 Hz. This concept is part of the physical study of sound waves, known as acoustics, and the phenomenon is rooted in the harmonic series which is fundamental to the physics of musical instruments.
When discussing piano strings and pitch, resonance is also an important concept. Resonance occurs when a system, such as a string on a piano, oscillates with greater amplitude at a particular frequency, which is the string's natural frequency. You can observe this when you produce sound waves from your voice and a piano string resonates, as it has the same natural frequency as the sound waves of your voice.
Another important concept in sound wave physics is beats. Beats occur when two sounds with slightly different frequencies are played together, resulting in a fluctuating sound wave due to constructive and destructive interference. This can often be heard when two adjacent piano keys are struck simultaneously, resulting in the characteristic warbling sound.