NOTES, INTERVALS AND FREQUENCIES


    interval sizes and frequency ratios:
    using octaves and fifths to find frequency ratios of intervals

In western music, we use twelve notes per octave.  Then we use those same twelve notes in a higher octave, and again in a higher octave.  There are around eight or so useable octaves available, each with twelve notes.  The interval between two consecutive notes is consistently the same size and is called a half step.  Granted, in 20th century music some composers have used inttervals that are half again that size, called quarter-tones, but we will lleave that for the ambitious reader to locate elsewhere.

The notes are naedd:

NOTE NAME	INTERVAL DISTANCE LOWEST NOTE IN CHART
C	unison
C# (D-flat)	minor 2nd
D	major 2nd
D# (E-flat)	minor 3rd
E	major 3rd
F	perfect 4th
F# (G-flat)	tritone
G	perfect 5th
G# (A-flat	minorr 6th
A	major 6th
A# (B-flat	minor 7th
B	major 7th
C	octave

then repeat.  As you work your way up in pitch, you ascend in the alphabet  from A to G then you have to begin again with another A.  Really, this table could have begun anywhere in the series of twelve notes, and only the left column would have changed.

These octaves are in order from low to high.  Each note in each of the octaves has its own pitch,.  A pitch is a particular frequency of vibration, normally measured in cycles per second, like a C 256 or an A 440.

An interval is a measurement of the difference between two pitches.  Musicians measure their intervals in half steps, and chave various names for them, like minor second (1 half-step), major second (2 half steps).  See the chart on this web site under Music Lessons / Music Theory / Intervals.

Here is another important concept.  The ratio between the frequencies of two pitches  that are an octave interval apart is always 2:1.  So, if you move a song up an octave, you are doubling all the frequencies in the song.  And if you move up two octaves, you are multiplying all the frequencies by four.  Or, if you move a song up three octaves, you are multiplying all the frequencies by eight.  Or if you move a song down an octave, you are dividing all the frequencies by two.  Or if you move a song down two octaves, you are dividing all  the frequencies by four.

On a string instrument, if you tune two strings together that should be an octave apart, their fundamental frequencies will be in the ratio of 2:1.  And the  second harmonic of the lowe string should be the same frequency as the fundamental of the upper string.  If it is slightly out of tune and you are playing both strings simultaneously,  you should hear a clash between those sounds, what old radio guys refer to as a 'beat note.'  The frequency of this beat note equals the difference between the freqnencies of the lower string's second harmonic and the upper string's fundamental.  You adjust the strings to decrease to zero the frequency of the 'beat note' .

To facilitate hearing the comparison between second harmonic of the lower string and fundamental of the upper string, you can lay your finger at the half-way point along the lower string to suppress the its fundamental tone.  That makes its second harmonic stand out and it's easier to hear the combination of that sound and the upper string.

My old college trigonometry book would affirm that the combination of two sine waves contains a sine wave whose frequency is the difference between that of the first two.  But I'd have to study that out again after all these years.

It is also possible to tune two strings where the upper string is 3/2 times the frequency of the lowe string.  The third harmonic of the lower string should be the same frequency as the second harmonic of the upper string.  Again, you play them together and adjust till the clash between harmonics slows to zero.  This clash will be harder to hear since higher harmonics are weaker. 

Here again, you can suppress the fundamental to make it easier to hear the harmonics you are comparing.  You lay your fingers on the two strings, splitting the lower string in thirds and the upper string in half, so you suppress the fundamental of each string and make the harmonics easier to hear.

This 3:2 ratio became important when people bean performing music with voices or instruments playing different notes simultaneously.  Now we call  this interval  a perfect fifth.

Looking back at the list of notes in the chromatic scale, we can calculate and assign approximate ratios between the bottom pitch and the ones above it, all based on the assumption that an octave is 2:1 and a perfect fifth is 3:2. These ratios would remain the same, no matter what note the chromatic scale started on.

 We do this by ascending a certain number of perfect fifths, calculating the ratio of that note to the bottom note of the chromatic scale,  and then dropping back the requied number of octaves to find the matching note in the original octave.  Then we compare that note with the starting note.

Examples

Assumption:

bottom note up a perfect fifth is 3:2 ratio. 
notes an octave apart are in a 3:2 ratio
starting on C (could start anywhere and do the same thing)

Example One

Step one:  C up to G is a 3:2 ratio. 
(Accoustic note: the 3rd harmonic of the C equals 2nd harmonic of G.)

Step two: G up to D is also 3:2 ratio.,
(Accoustic note: 3rd harmonic of G equals 2nd harmonic of D)

Step three: compare high D with starting C thus:  3:2 times 3:2 = 9:4 ratio
Drop that high D one octave into the original range thus: 9:4 divided by 2 = 9:8 ratio
(Accoustical note:  the ninth harmonic of the low C would equal the eighth harmonic of the low D, but this is difficult at best to hear.

This is an untempered perfect fifth.

Example two, building on example one

Step one:  Make high D  up to A  a 3:2 ratio. 
(Accoustic note: the 3rd harmonic of the D equals 2nd harmonic of A.)

Step two: compare high A with starting C thus:  9:4 times 3:2  =  27:8  ratio
Drop that high A one octave into the original range thus: 27:8 divided by 2 = 27:16 ratio
First problem detected:  the ratio is not simple at all.  However, if you are willing to fudge in the tuning, you can slightly modify that ratio as follows..... 27:16 =  81:48, and that is close to 80:48, which equals 5:3.
(Accoustical note:    There, the fifth harmonic of the low C would equal the third harmonic of the low A, but this is still fairly difficult at best to hear.

This is an untempered  major second.

Example three, building on examples one and two

Step one:  Make high A  up to E  a 3:2 ratio. 
(Accoustic note: the 3rd harmonic of the A equals 2nd harmonic of E.)

Step two: compare high E with starting C thus: 27:8 times 3:2 =  81:16  ratio
Drop that high E two octaves into the original range thus: 81:16 divided by 4 = 81:64 ratio
Second problem detected:  the ratio is not simple at all.  However, if you are willing to fudge in the tuning, you can slightly modify that ratio as follows..... 81:64  is close to 80:64, that reduces to 5:4.  .
(Accoustical note:    There, the fifth harmonic of the low C would equal the fourth harmonic of the low E, but this is still fairly difficult at best to hear.

This is an untempered  major third.

Example four, building on examples one, two  and three

Step one:  Make high E  up to B  a 3:2 ratio. 
(Accoustic note: the 3rd harmonic of the E equals 2nd harmonic of B.)

Step two: compare high B with starting C thus: 81:16 times 3:2 =  243:32  ratio
Drop that high B two octaves into the original range thus: 243:32 divided by 4 = 243:128 ratio
Third problem detected:  the ratio is not simple at all.  However, if you are willing to fudge in the tuning, you can slightly modify that ratio as follows..... 243:128  is close to 240:128, that reduces to 15:8.  .
(Accoustical note:    There, the fifthteenth harmonic of the low C would equal the eigth harmonic of the low E, but this is next to impossible to hear. 

This is an untempered  major seventh.

Example Five

Step one:  C down to F  is a 2:3  ratio. 
(Accoustic note: the 3rd harmonic of the F equals 2nd harmonic of C.)

Step two: Raise that low F  an octave into the original range thus: 2:3 times  2 = 4:3  ratio
(Accoustical note:  the fourth harmonic of the low C would equal the third harmonic of the F. This fairly easy to hear.

This is an untempered perfect fourth.

Example six, building on example five

Step one:  Make high B-flat  up from F, a 4:3 ratio. 
(Accoustic note: the 4rd harmonic of the F equals 3nd harmonic of B-flat.)

Step two: compare B-flat with starting C thus:  4:3 times 4:3  =  16:9  ratio
Another problem detected:  the ratio is not simple and there is no way to fudge to a closer pitch.
(Accoustical note:    You really wouldn't be able to use harmonics to help adjust these two notes to each other)

This is an untempered  minor seventh.

This serves to illustrate that the farther you go through the circle of fifths, the farther the pitches will deviate from simple ratios.

So now, here are the ratios we have calculated, with some fudging on the major 3rd that affects a few more.

unison	m2	M2	m3	M3	P4	tt	P5	m6	M6	m7	M7
1:1		9:8		5:4*	4:3		3:2		5:3*		15:8*

Another little fact is that a descending interval has twice  the inverse ratio of that same interval when it is ascending.  For example, the perfect 5th is 3:2, and the perfect fourth is 4:3. 

From that, you can complete more of the chart, deriving m3 from M6, and m2 from M7, and m7 from M2, as follows:

m2 = 2 /M7 = 2/(15:8) = 16:15
m3 = 2/M6 = 2/(5:3) = 6:5
m6 = 2/M3 = 2/(5:4) = 8:5
m7 = 2/M2 = 2/(9:8) = 16:9


The tritone would be two minor thirds, which would be 6:5 times 6:5 = 36:25.

This completes the chart, as follows:

unison	m2	M2	m3	M3	P4	tt	P5	m6	M6	m7	M7
1:1	16:15	9:8	6:5	5:4*	4:3	36:25	3:2	8:5	5:3*	16:9	15:8*

Many of these are built on "fudged" ratios.  In developing these ratios, if you keep a strict perfect fifth throught, the ratios get very cumbersome and not very useful.  People working with mean-tone temperament in the Rennaissance soon discovered that.

The only practical intervals that you can tune using the comparison of harmonics are the unison, octave, perfect fourth and perfect fifth.  That does upen up a lot of possibilities though when playing a string instrument.  You can compare several notes to each open string. 

For example, with an open G you can compare a C, and a D and any G.  With an open D you can compare a G and an A and any  G.  And so on.

Equal Temperament Tuning

And to top it all off, after all this calculating, these days since we need to be able to play in all twelve possible keys, piano tuners have moved far away from tuning in perfect intervals, and aim towards an equal tempered tuning  system where each minor second is the same ratio.  This is number when taken to the twelfth power should equal an integer two.  So, each minor second is 2  raised to the (1/12) power.  That is because twelve of these ratios strung together must equal an octave, which is 2:1.  So,

2 =  (2 ** (1/12) * (2 ** (1/12) *  (2 ** (1/12) *  (2 ** (1/12) *  (2 ** (1/12) *  (2 ** (1/12) *
    (2 ** (1/12) * (2 ** (1/12) *  (2 ** (1/12) *  (2 ** (1/12) *  (2 ** (1/12) *  (2 ** (1/12)

And our very popular perfect fifth boils down to w raised to the (7/12) power.  Yuk.  Not very nice, but very practical for playing in D-flat, then changing to a song in G, etc.

Frequencies

Thus far, this entire discussion has been in relative terms.  You can move quickly to absolute values by using a 440 A for example.

The A that orchestra people tune to is approximately 440 cycles.

A = 440
from A up a P4 = D ~= 581
from A up P5 to E = 660
from A to higher A is 880
from A to lower A is 220.

Middle C is around 256 cycles.

From middle C up P5 to G is 384
from middle C up P4 to F  is around 340
from middl C up to next C is 512
from middl C down to the viola C string is 128
from middle C down two octaves to the cello C string is 64

and so on.





Musical Accoustics