Pitch, Intervals, Tonality

Sounds are vibrations (waves) usually transmitted in air or water. Pitch is one of the fundamental properties of sound that can be defined objectively as the frequency of vibration.  The musical term "tone" sometimes refers to pitch and sometimes to timbre, sometimes to both properties together. Tone is a versatile word in music.

The descriptions higher and lower refer to pitch frequency.  If you model sound production with a vibrating string or an organ pipe, then you would say longer and shorter instead of lower and higher. The names of notes could be anything. When you work with electronic keyboards, the keys are numbered; midi recordings show notes as numbers; low pitches are smaller numbers and high pitches are larger. If you model sounds with an electronic oscillator, then pitches are described by frequency as cycles per second (Hertz = Hz). We could refer to frequency as faster and slower. Slower frequency waves are longer; faster waves are shorter. Sound waves of less than 20 Hz are perceived by skin sensors. Pitch differences greater than 2 Hz are perceived as different notes.

The audio frequency range is 0 to 20,000 Hz. Voice perception depends on  a smaller range of about 30 to 5000 Hz.

A pitch of frequency 440 Hz is described as concert A, the A above middle C on the piano. Multiples of that frequency are also A. Higher As are 880 Hz, 1760 Hz and lower As, descending, are 220 Hz, 110 Hz, and 55 Hz. An A octave  spans the intervals between between two adjacent As.

Octaves are divided into 12 notes, each described as a semitone. Much music occurs within two octaves 220 to 880 Hz. Most singers are comfortable in this range. Several instruments are easiest to play in this range - violin,  trumpet, clarinet, alto saxophone and oboe are examples.

Scales are patterns of single semitone and 2 semitone intervals. The piano keyboard divides the octave into 12 equal intervals known as equal temperament. The grand piano has 88 keys and middle C is C5. All my electronic keyboards have 61 keys and middle C is C3. The midi note number for middle C is 60,  no matter what keyboard you are using.

If you play all the white keys on a piano C, D, E, F, G, A, B, and higher C you have a C major scale. The sequence of semitones is 2,2,1,2,2,2,1. The C minor scale is C, D, Eb, F, G, Ab, Bb and higher C. Music that only uses the notes of a scale is described as diatonic.

The term tone is used to describe both a note, a defined pitch, and also the intervals between the notes. Tonality refers to the sound of a scale. The tonic note is the first or root note of the scale. All major scales have the same sequence of intervals but they start and end on different tonic notes.

A composer declares the tonality of a piece by starting and/or ending on the tonic, often approaching the tonic note from the leading note which is a semitone below the tonic. Harmonies that utilize mostly the notes of a scale also declare tonality.

The pitch of a vibrating string depends on three factors:

The length of the string: longer string = lower pitch;  shorter =higher pitch. Frequency is inversely proportional to the length. A string twice as long will produce a tone of half the frequency (one octave lower).

Tension: less tension (looser) = lower pitch; greater tension (tighter) = higher pitch. The frequency is proportional to the square root of the tension.

Density: Thicker, heavier strings vibrate more slowly that finer lighter strings. Frequency is inversely proportional to the square root of the density:

In practice, string instruments are tuned by adjusting tension. The strings are supplied with the right density and length to achieve the desirable range of frequencies. Pitch is determined by finger pressure on the string which changes the length.

Tunings for string instruments:

violin G, D, A, E

viola, cello, tenor banjo, mandolin C, G, D, A

double bass, bass guitar E, A, D, G

guitar E, A, D, G, B, E

The sine wave is a pure tone with no harmonics. The square wave is packed with different frequencies. Natural sounds are mixes of different frequencies. A casual description of the sensation of pitch is best since when you examine the experience of sound more closely, you realize that humans vary in their experience of sound waves and it is difficult to tell if these experiences are comparable. Perfect pitch describes people who can identify the pitch of a sound and can differentiate different pitches accurately. This is a valuable ability for musicians, although not all musicians enjoy perfect pitch.

It is the mix of frequencies that make natural and musical sounds so complex. In the first analysis, a sound made by an instrument has a fundamental frequency that allow us to say it is, for example, C on piano. In addition, there is a mix of other frequencies or harmonics that give the sound its timbre or tone. A vibrating string in the piano changes its harmonic structure from the moment it is struck until the string stops vibrating. Its fundamental frequency remains more or less the same, but its timbre varies.

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