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1)Determine if the interval C4 to G4 is dissonant or consonant

step 1: Find the frequency of each individual note

C4) fn= fo X (a)^n

f9= 440Hz(1.059463)^-9

*note that n is negative because on a piano, C4 is lower than A4*

f9= 261.6Hz....C4=261.6Hz

G4) fn=fo X (a)^n

f2= 440Hz(1.059463)^-2

f2= 392Hz......G4=392Hz

step 2: Find the beat frequency of the two notes:

fbeat = |f2– f1|

= |392-261.6|

=130.4Hz

130.4Hz is greater than 60Hz (consonant range) so the interval from C4 to G4 (perfect 5th) is consonant (pleasing)

Dissonant and Consonant Intervals relating to Math

Graphing: P5, M2,P8,M7

Sound Energy

Major 2nd: C-D

Perfect 5th: C-G

notice the pattern?

Octave: C-C

Major 7th: C-B

  • Vibrations of matter produce sound energy
  • As humans, we can only hear the vibrations that fall between 20vibrations-2000vibrations/second (20Hz-2000Hz)
  • each individual note has a distinct frequency that it creates

The pitch A4 is used in western music as a tuning and reference pitch. It is known as the central pitch with a frequency of 440Hz.

the formula for calculating the frequencies of the 12 pitches is:

fn = f0 x (a)^n

1) fo refers to the frequency of one set pitch (A4= 440Hz)

2) n refers to the number of half steps you are from the set pitch A4

* if you are at a higher note than A4, n is positive; if you are at a lower note than A4, n is negative*

3) a = 2^(1/12) this is about 1.059463

4) this formula is derived from the "equal tempered scale" which was created so that no matter what key something was played in, it could sound equally good or bad

using the frequency of A4, we can determine the frequency of all of the other 12 chromatic pitches used in western music

Interval Frequency Ratios

Important Terms:

Graphs of Sine

What Next?

  • The graph of sine serves as a model for our frequencies of intervals
  • using the frequencies, we can graph each interval of any two pitches using a graph of sine and determine if the four intervals are dissonant or consonant
  • the formula used for graphing the notes:

y=sin(f(2piX))

*f is the frequency of the note*

example:

  • Another way we can determine if intervals are consonant or dissonant is by understanding the ratios between two seperate pitches.
  • The ratio system of the notes allows us to analyze the notes more easily
  • 4 intervals: Perfect 5th, Major 2nd, Perfect Octave, Major 7th
  • ratios: P5(3:2), M2(8:9), P8(2), M7(15:8)

Example of Consonance Vs Dissonance

fn = f0 X (a)^n

1)find the freqency of C5

C5 is greater than A4 so n is positive

from A to C there are 3 half steps so n=3

Now set up equation

f3= 440hz(1.059463)^3

f3= 523.3Hz

the frequency of C5 is 523.3Hz

Now that we understand how to find the individual frequencies of notes, we can find the difference or beat frequency between two seperate notes

  • Interval: In music, an interval refers to the distance in pitch between two notes
  • dissonance: lack of harmony among musical notes
  • consonance: Harmonically pleasing to the ear
  • Frequency: The measurement of the number of times that a repeated event occurs per unit of time (seconds)
  • measured in Hertz(Hz)

Example:

Beat Frequency

rules:

Conclusion:

To find the beat frequency between two notes, take the absolute value of the second frequency(f2) minus the first frequency(f1)

fbeat = |f2 – f1|

As a general rule of musical consonance and dissonance:

1) any beat frequency between 10Hz and 60Hz is considered dissonant

2)any beat frequency greater that 60Hz is considered consonant

In conclusion, musical tonality of dissonance/consonance can be determine using various methods as explained. The Beat Frequency, ratios, and sine graphs all serve as models of the relationship between intervals. All of the standards for consonance/dissonance are related to Western Music are not always defined the same way in Eastern or other cultural music studies.

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