Music Scales dictionary - SCALAMUSICA Version 6.10 - A Complete Dictionary & tool for  every music scale in the equal tempered system. Visit FORUM SCALORUM!

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Intervals and scales

over-tones, timbre, intervals between natural harmonics, outer and inner intervals, the tempered tone system, beats

Where do scales come from?

Over-tones

Every physical body that forces the air to swing in such a regular manner that we interpret it as a tone will most likely, simultaneously, produce several more tones of varying strength. If the tone (fundamental tone, pedal tone, ground-tone) has the frequency f (We measure frequencies in p/s or Hz, periods per second), then these tones have the exact frequencies 2f, 3f, 4f and so on. F1, f2, f3 are called partial tones and f2, f3, f4 are called over-tones or harmonics. Note that these tones are produced without any outer modification of the body like shortening a string or stoppening a hole on a wood-wind instrument.

These tones may, however, also be produced by manipulating various musical instruments, a fact that is known from time immemorial. A string that has the pedal tone f Hz that is shortened to a half, a third etc will also produce the tone frequencies 2f, 3f, 4f compared to the ground-tone of the entire string. The same situation occurs in different wind instruments. Whichever the fundamental is we recognize the intervals between these harmonics, a fact that makes them natural constants.

Timbre

We don’t hear these harmonics as seperate tones (they are much weaker than the pedal tone) but we percieve their gathered impression, the timbre
, which is why different instruments sounds so different, let alone human voices. A tuning forc for instance have almost no over-tones at all, a flute has a strong 2f, an oboe has relatively strong 5f and 4f and a clarinet has 7f, 8f and 9f stronger than the other harmonics.

Intervals between natural harmonics

In the picture you can see an attempt to write the first partial tones as notes. Note that all tones separated by an octave (1f-2f, 3f-6f, 5f-10f etc) has the interval ratio 2 i.e. exactly 12 HS (= 12 tempered half tone steps). (For this and the following calculations use the Sds intervals calculator
). The interval between 2f and 3f is a fifth comprising 7,02 HS, the interval 3f to 4f is a fourth of 4,98 HS, the interval 4f to 5f is major third of 3,86 HS, the intervals 5f-6f and 6f-7f are both minor thirds of 3,16 HS and 2,67 HS respectively. The mere fact that nature has two different minor thirds suggest the impossibility of notating natural over-tones. It becomes yet more evident when we look upon the intervals 10f -11f and 11f -12f. In the table of natural harmonics is shown the intervals measured in HS (tempered half ton steps), no less than four intervals we do percieve as minor seconds but they are all dissimilar.

Outer and inner intervals

When we write the first seven intervals of the harmonics as Sds systematic letter codes we obtain the interval row LGEDCCAAAA (see the table of systematic interval names) but only the octave L has an even number in HS, namely 12. Next interval, G, the fifth, has a size of 7,02 HS (or 702 cents) as we saw in the table of natural harmonics .The letter code is said to represent the outer interval size and the exact interval represent the inner interval size. We hear for instance both the intervals between the fifth and sixth or between the sixth and seventh harmonics as minor thirds, which we denote C but their real inner sizes, that we hear by ear, are different. See also Exotic scales.

The tempered tone system

Without bothering too much of the underlying history we now can say that the need arouse of dividing the octave in twelve exactly similar intervals. The main reason for this was a need to perform music in different keys. In the older days people had to use the natural intervals but now all the tones that we used in western music were adjusted so that the interval ratio between every two adjacent notes became exactly similar. Use the Sds intervals calculator! All the tones in this chromatic scale have their exact equivalent in the octave above having the frequency 2f. The modern pianoforte and all modern synthetical instruments are examples of instruments tuned this way.

Beats

When two notes f1 and f2 have a comparatively similar frequency a new tone arise having the frequency f2 - f1 (the upper frequency minus the lower frequency of the interval). If the difference is quite small, only a few p/s (or Hz), we hear it as a softly rising and falling sound, a so called beat.
The tone a (below middle c) is conventionally determined to have the frequency 440 Hz. From this we calculate the tone d above middle c, 5 HS upwards, to have the frequency 587,33 Hz. The natural f4 over 440 Hz is 4*440=1760Hz and the third harmonic over the tone d has the frequency 3*587,33 = 1761,99 Hz. Thus the difference between these naturals is 1,99 Hz (almost twice a second) and we hear it as a beat having the frequency of about 2 p/s (2 Hz). This beat appears only between tones in the tempered tone system and is dependent of the frequencies. The natural “pure“ fourth however between 3f and 4f allways has an interval of 4,98 HS but a tempered fourth has (or should have if tuned properly) 5 HS. In the same way we can easily calculate the natural fifth having an interval of 7,02 HS (inner interval) while a tempered fifth has exactly 7 HS (outer interval). The beat property is used to produce a synthetical vibrato in some instruments like in accordeons and organs and is the most important aural aid when tuning different string instruments.