Notes That Sound Good Together

Where Math meets Music

Eastward ver wonder why some note combinations sound pleasing to our ears, while others brand us cringe?� To empathize the answer to this question, you�ll kickoff need to empathize the wave patterns created by a musical instrument.� When you lot pluck a string on a guitar, it vibrates dorsum and forth.� This causes mechanical energy to travel through the air, in waves.� The number of times per second these waves hitting our ear is chosen the �frequency�.� This is measured in Hertz (abbreviated Hz).� The more waves per 2d the higher the pitch.� For instance, the A annotation below middle C is at 220 Hz.� Eye C is at about 262 Hz.

At present, to understand why some notation combinations sound improve, let�due south first look at the moving ridge patterns of ii notes that sound skilful together.� Let�south use centre C and the G just to a higher place it as an example:

At present permit�southward expect at two notes that audio terrible together, C and F#:

Exercise you discover the departure between these two?� Why is the first �consonant� and the second �dissonant�?� Notice how in the kickoff graphic there is a repeating blueprint: every three^rd wave of the G matches up with every two^nd wave of the C (and in the second graphic how there is no pattern).� This is the clandestine for creating pleasing sounding note combinations: Frequencies that match upwardly at regular intervals (* - Please see footnote about complications to this rule).

Now permit�s look at a chord, to detect out why it�due south notes sound good together.� Hither are the frequencies of the notes in the C Major chord (starting at middle C):

�� C � 261.half-dozen Hz
�� Eastward � 329.6 Hz
�� Thousand � 392.0 Hz

The ratio of E to C is about 5/4ths.� This means that every 5^thursday wave of the E matches up with every 4^th wave of the C.� The ratio of G to E is about 5/4ths as well.� The ratio of M to C is virtually 3/ii.� Since every note�s frequency matches upwardly well with every other note�s frequencies (at regular intervals) they all audio good together!

Now permit�south look at the ratios of the notes in the C Major key in relation to C:

C � 1
D � 9/8
E � 5/4
F � 4/three
G � three/2
A � 5/3
B � 17/nine

To tell y'all the truth, these are judge ratios.� Remember when I said the ratio of E to C is about 5/4ths?� The actual ratio is not 1.25 (five/4ths) but i.2599.� Why isn�t this ratio perfect?� That�s a good question.� When the 12-annotation �western-style� scale was created, they wanted not only the ratios to be in tune, but they likewise wanted the notes to go up in equal sized jumps.� Since they couldn�t have both at the same time, they settled on a compromise.� Here are the bodily frequencies for the notes in the C Major Fundamental:

Note	Perfect Ratio to C	Bodily Ratio to C	Ratio off by	Frequency in Hz
Middle C				261.6
D	9/eight or 1.125	1.1224	0.0026	293.7
E	five/iv or one.25	1.2599	0.0099	329.6
F	4/3 or i.333�	one.3348	0.0015	349.2
K	3/ii or 1.5	1.4983	0.0017	392.0
A	5/3 or 1.666�	1.6818	0.0152	440.0
B	17/9 or i.888�	1.8877	0.0003	493.9

You can see that the ratios are not perfect, but pretty close. The biggest difference is in the C to A ratio. If the ratio was perfect, the frequency of the A in a higher place middle C would be 436.04 Hz, which is off from 'equal temperament' by about 3.96 Hz.

The previous list shows only the 7 notes in the C Major key, not all 12 notes in the octave.� Each note in the 12 note scale goes upwards an equal amount, that is, an equal corporeality exponentially speaking.

Hither is the equation to figure out the Hz of a note:

�� Hertz (number of vibrations a second) = half-dozen.875 x two ^ ( ( 3 + MIDI_Pitch ) / 12 )

The ^ symbol means �to the ability of�. The MIDI_Pitch value is according to the MIDI standard, where middle C equals sixty, and the C an octave below it equals 48. As an instance, allow�s figure the hertz for middle C:

�� Hertz = 6.875 x two ^ ( ( 3 + sixty ) / 12 ) = 6.875 x 2 ^ 5.25 = 261.6255

The next notation up, C#, is:

Hertz = six.875 ten 2 ^ ( ( 3 + 61 ) / 12 ) = 277.1826

And the side by side note, D, is:

�� Hertz = 6.875 x 2 ^ ( ( 3 + 62 ) / 12 ) = 293.6648

The spring between C and C# is 15.56 Hertz, the jump between C# and D is xvi.48 Hertz. Although the Hertz jump is not equal between the notes, it is an equal leap in the exponent number and it sounds similar an equal jump to our ears going upwardly the scale. This gives a nice smooth transition going upwards the scale.

Another important characteristic of the scale is that it jumps past 2 times each octave. The A below middle C is at 220 Hertz, the A above middle C is at 440 Hertz, and the A above that is at 880 Hertz. This ways that you can motility notes into different octaves and nonetheless have them audio consonant. For instance, allow�s take the case of centre C and G once again, except move G into the adjacent octave. We all the same take center C at 261.6Hz, simply G is now at 784 Hz. That gives a ratio from Thousand to C of virtually three/1 (twice the original ratio of 3/2). The waves however encounter upwardly at regular intervals and they still sound consonant! Another nice characteristic of having an equal exponential bound is that y'all can showtime a scale on whatever notation you wish, including the black keys. For instance, instead of C,D,E,F,Thousand,A,B, you can start on, say, D# and take D#,F,1000,G#,A#,C,D as your scale with the same cracking sounding combinations of frequencies.

At a certain point frequency ratios are besides keen to sound consonant. It takes also many waves for them to match up, and our ears simply can�t seem to notice a regular pattern. At what point is this? The simple respond is when the ratio�south numerator or denominator gets to virtually 13. For instance, C# has a frequency ratio to C of well-nigh 18/17ths. That�s simply too many waves before they meet upwardly, and you can tell that immediately when you play them together.

And so now you lot�re thinking that we have a scale that goes upwards in even steps and has reasonably authentic ratios, nosotros�re all set, right? Actually, at that place are a lot of dissenting opinions on the subject. Think those not-quite-authentic ratios? Ane reason for this was for instruments to be able to be tuned once, and sound reasonably good in all keys. Some of the grumpier musicians still mutter, though, saying that equal temperament makes all keys sound as bad. If you tune to but 1 item key, y'all can get those ratios perfect (since the human ear can notice a deviation of 1Hz, being off by several Hz tin exist a problem!).

Maybe more importantly, though, is that there are a lot of undiscovered frequency combinations that can�t be played in the confining 12-note system. Many alternative scales used in India accept up to 22 notes per octave. If yous�re not satisfied with the standard western scale, there are lot of alternative tuning methods bachelor, such as 'Just Intonation' and 'Lucy Tuning'. With modernistic digital equipment, these alternate tunings have become much easier to implement. We should hear some new and incredibly interesting music come out of these tuning methods equally they are gradually accepted into the mainstream.

* - The frequency ratio theory of consonance does not e'er agree true. See this other article for an explanation.

Here are some links if you lot�d like to explore this topic further:

�� Just Intonation Network

�� Just Intonation explained

�� Lucy Tuning

�� But vs Equal Temperment � �harmonic tuning� described

�� American Festival of Microtonal Music