Missing Harmonics

See holytones.m

spectrum 1: 24 successive harmonics, with a rolloff of 3dB/oct. spectrum 4: the first 6 harmonics are present: 7, and 9 are left out. Why omit more higher harmonics than lower?

• Critical band is 100 at low frequencies
• changes gradually to around 20% of the frequency (minor 3rd) at higher frequencies

If 2 sinusoids of the spectrum are

• separated by several CBs, they don't interact and can be heard separately.
• closer they are in frequency, they sound more fused.
• within a CB, they sound harsh.

Successive lower harmonics of low pitches are separated by more than a CB.

The higher the harmonic number, the closer the harmonic spacing in CBs.

spectrum 7: higher partials are spaced more than a CB apart (holy) spectrum 8: same number of partials as 7, but higher partials are deliberaltely moved closer together

Violin tends to approximate such an effect:

• coupling between string and soundboard rises and falls rapidly with frequency
• valleys tend to eliminate higher harmonics that are very close in frequency

Why holy tones 5, 6, and 7?

• leaving out partials produces a coloration--may have similar ``holiness'' but still sound different because different partials are omitted.

Spectrum 9 has a distinctive ``octavely'' quality.

Organs use this effect:

• different holy tones are produced using a mixture of an 8-foot stop, a 4- foot stop, and a 2-foot stop.
• the 4-foot (an octave up) and the 2-foot (two octaves up) stops add higher harmonics, at two and four times the harmonic spacings of the 8-foot stop.
• Organ demonstration: Click to listen

Orchestation:

• doubling an instrument “doubles” a harmonic at one or two octaves above.
• Bolero (Ravel) with trumpet and flute in octaves. Click to listen
• La Mer (Debussy) with trumpet and english horn in unison Click to listen