In the playing of music, the ability to associate the correct pitch (or frequency) with a given note is actually of little importance. There are very few musicians who can name the corresponding note when they hear a sound isolated from its context. This fact often surprises non-musicians: it would be like saying that a painter cannot recognise the individual colours that they use to create their paintings. Actually, in the playing of music, it is much more important to be able to recognise the interval formed between two notes and to recognise and use its quality of being "consonant" or "dissonant". The terms consonance and dissonance seem to introduce elements of subjectivity (which we will see are definitely present) in the judgement of various intervals. However, there also are (objective) physical aspects and (intersubjective) psychoacoustic aspects that have allowed all civilisations in different times and places to determine privileged intervals to use in their music. The absolute first is the interval of the octave.

After Galileo's inestimable contribution, it is now known that the pitch of a note is intimately connected to the vibration frequency of the sound wave that strikes the tympanum. Therefore, we can ask what relationship must there be between frequencies to give rise to an interval of an octave. The following table presents frequency differences and ratios:

${\displaystyle f_{1}}$	${\displaystyle f_{2}}$	${\displaystyle f_{2}-f_{1}}$	${\displaystyle {f_{2}}/{f_{1}}}$
55	110	55	2
110	220	110	2
220	440	220	2

The table leaves no doubts: the perceived interval is an octave if the ratio between the two frequencies is exactly double. In general, to our perceptual system:

two intervals are considered equal if the ratio (and not the difference) of the frequencies of the sounds of the interval is identical.

Two of the most consonant intervals that have been determined are the perfect fifth, for which the frequency ratio is 3/2 (instead of 2/1 like the octave), and the perfect fourth for which the frequency ratio is 4/3. If we want to construct a musical scale based on a purely mathematical criterion with all its "steps" (intervals) being equal, which begins from the lowest note of the octave and ends with the highest, we must use sounds with frequencies in geometric progression and not in arithmetic progression). We observe that in the normal steps of a scale, the difference (and not the ratio) between absolute pitches (with respect to the base of the scale) of two consecutive steps remains constant! Obviously, this mathematical criterion does not always produce such "consonant" intervals as the octave. Historically, the choosing of frequencies (and, therefore, of notes) to insert in an octave that will compose a musical scale has been based on aesthetic criteria related to the consonance of the intervals within the scale itself. The history of how various civilisations produced different divisions of the octave is absolutely fascinating and interwoven with considerations regarding "numerology", aesthetics and the technicality and physicality of constructing musical instruments. This is discussed in more detail in the following paragraphs and on the pages regarding the Pythagorean scale, Natural scale and Temperament scale.

How is the amplitude of an interval measured?

If we really want to construct a musical scale, we need to measure the amplitude of an interval between two consecutive steps of the scale. Conventionally, the frequency ratios of the steps of a scale are in reference to the first step of the scale (called the tonic of the scale). For example, let's suppose we have a scale in which the ratios are those indicated in this table (which is a Pythagorean scale based on the C tonic).

Frequency ratio with respect to C	1:1	9:8	81:64	4:3	3:2	27:16	243:128
Note	C	D	E	F	G	A	B

For example, if we need to measure the interval between G and A, i.e. the ratio of their frequencies, we must calculate:

${\displaystyle {\frac {f_{\rm {La}}}{f_{\rm {Sol}}}}={\frac {{\frac {27}{16}}\cdot f_{\rm {Do}}}{{\frac {3}{2}}\cdot f_{\rm {Do}}}}={\frac {27}{16}}\cdot {\frac {2}{3}}={\frac {9}{8}}}$.

We must agree that the calculations are not always easy (especially when there are ratios containing large numbers) however, above all, it is not trivial to recognise the identity of the two intervals. For example, the interval between A and B is exactly equal to the one between G and A just calculated above (try it to believe it!).

Because the ratio and not the difference of frequencies plays a fundamental role in determining an interval, it is possible to use a logarithmic scale, which is based on the division of the octave into 1,200 equal intervals called cents.

${\displaystyle ({\textrm {1\ cent}})^{1200}={\textrm {1\ ottava}}=2}$,

therefore,

${\displaystyle {\textrm {1\ cent}}={\sqrt[{1200}]{2}}\approx 1.00057779}$.

Why exactly 1,200 equal parts?

The cent scale is based on our modern musical scale. Therefore, the answer can be found on the page Equal temperament.

The advantage of the logarithmic scale can be appreciated when we need to measure the amplitude of an interval made up of two consecutive subintervals.

The measurement of the interval between two frequencies in cents ${\displaystyle f_{1}}$ e ${\displaystyle f_{2}}$ is given by

${\displaystyle 1200\log _{2}\left({\frac {f_{1}}{f_{2}}}\right)\approx 3986\log _{10}\left({\frac {f_{1}}{f_{2}}}\right)}$ cents.

Therefore, if the first interval measures

${\displaystyle \Delta _{1}=1200\log _{2}\left({\frac {f_{1}}{f_{2}}}\right)}$ cents

and the second interval measures

${\displaystyle \Delta _{2}=1200\log _{2}\left({\frac {f_{2}}{f_{3}}}\right)}$ cents,

then the interval obtained "joining" the two intervals measures

${\displaystyle \Delta _{3}=1200\log _{2}\left({\frac {f_{1}}{f_{3}}}\right)}$ cents.

Thanks to the properties of logarithms, we can write

${\displaystyle 1200\log _{2}\left({\frac {f_{1}}{f_{3}}}\right)=1200\log _{2}\left({\frac {f_{1}}{f_{2}}}\cdot {\frac {f_{2}}{f_{3}}}\right)=1200\log _{2}\left({\frac {f_{1}}{f_{2}}}\right)+1200\log _{2}\left({\frac {f_{2}}{f_{3}}}\right)}$;

i.e.

${\displaystyle \Delta _{3}=\Delta _{1}+\Delta _{2}\;}$.

Adopting the logarithmic scale, the amplitude of the entire interval is equal to the sum of the amplitudes of the subintervals that it is made up of.

For example, the scale we choose as a model becomes very simple if the intervals are expressed in cents:

Frequency ratio with respect to C (in cents)	0	204	408	498	702	906	1110	1200
Note	C	D	E	F	G	A	B	C

If we want to recalculate the amplitude of the G-A interval, we only have to take the difference between the intervals (in cents) from A to G with respect to C. The equality of the intervals of various consecutive steps is now completely evident: subtracting is much easier than dividing! In particular, it is easy to see that in this scale (which is actually a Pythagorean scale), there are only two types of intervals with amplitudes of 204 and 90 cents.

In-depth study and links

• Several frequently asked questions about scales in the section Questions and answers about musical scales.
• If you want more information on what is covered on this page, visit the pages regarding the various musical scales.
• A helpful online convertor between frequency ratios and the cent scale can be found at http://www.sengpielaudio.com/calculator-centsratio.htm.

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