Normalizing the Musical Scale

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Frequency is measured on an exponential function scale (E.g., 2^x).

Each octave is exactly twice the size of the previous octave.

A given octave may span from 110 Hz to 220 Hz. (span=110 Hz)

The next octave will span from 220 Hz to 440 Hz. (span=220Hz)

The third octave spans from 440 Hz to 880 Hz. (span=440 Hz), ad infinitum.

Each successive octave spans twice the frequency range of the previous octave.

Therefore, frequency (Hz) can be understood as inhabiting an exponential (2^x) frame of reference.

This diagram underscores the exponential nature of octaves, when measured on a linear frequency scale.

1 Perception
2 Normalizing an exponential scale into a linear one
3 See also
4 Source
5 External links

[edit] Perception

However, human ears interpret all octaves as equally-sized, in linear equation terms.

The ear perceives all octaves as spanning the same distance: one octave wide. Even though a sub-bass octave may span 40 Hz and a super-treble octave can span 4000 Hz, both are understood (by the ear) to span only one octave each.

This diagrams presents octaves as they appear to the ear, as equally spaced units.

There is a method to translate an exponential scale into a linear one. It causes all octaves to span the same numerical distance (as the ears perceive them). It also causes all notes to have the same "width."

[edit] Normalizing an exponential scale into a linear one

The inverse of an exponent is a logarithm.

The inverse of 2^x is log₂ x.

To normalize frequency (Hz) to a linear scale, you can use log₂(frequency).

[edit] Advantages of normalizing frequency into linear terms

In the raw frequency scale, no two steps span the same number of frequencies. Each step is slightly wider than its lower-pitched neighbor. This inflation continues to infinity.
It is difficult to calculate the step size between any note and its neighbor.
In the normalized scale, all step is equal in width. Normalized terms are linear, which makes calculations simpler.
Linear terms are easier to manipulate than exponential ones.
It is easy to convert between normalized terms and terms of raw frequency. Converting between the values is as simple as applying log₂x or 2^x.

[edit] Figure 1 supports the above assertions

The raw freq scale has unequally-spaced steps.
The raw freq scale uses an awkward equation to determine the size of step spaces.
The normalized scale has equally-spaced steps.
The normalized scale uses a simple fraction to define step size.
A simple equation translates the raw frequency terms into normalized terms, and back again.

Fig. 1
Note	Frequency (Hz)	Distance from previous note ( = f*(2^(1/12)-1) )	Normalized frequency (log₂ f)	Distance from previous note	Converting normalized frequency to raw frequency (2^{normalized freq})
A₂	110	N/A	6.781	N/A	110
A₂#	116.54	6.54	6.864	0.0833 (or 1/12)	116.54
B₂	123.47	6.93	6.948	0.0833	123.47
C₂	130.81	7.34	7.031	0.0833	130.81
C₂#	138.59	7.78	7.115	0.0833	138.59
D₂	146.83	8.24	7.198	0.0833	146.83
D₂#	155.56	8.73	7.281	.0833	155.56
E₂	164.81	9.25	7.365	0.0833	164.81
F₂	174.61	9.80	7.448	.0833	174.61
F₂#	185.00	10.4	7.531	0.0833	185.00
G₂	196.00	11.0	7.615	0.0833	196.00
G₂#	207.65	11.7	7.698	0.0833	207.65
A₃	220.00	12.3	7.781	0.0833	220.00

[edit] The harmonic series is the foundation of harmony

The harmonic series article explains the form and function of the harmonic series. Figure 2, then, reiterates the first eight harmonic identities. Each identity is followed by its common name, and an example of the specified note (assigning the fundamental = A_{110 Hz}.) Next in the figure is the equation that relates the fundamental to the given identity. You see that each identity is equal to some integer multiplied with the fundamental. Finally, a fraction is given that divides the given identity's multiple against the multiple of the nearest lower octave. Creating a fraction allows one to perceive that, if a scale were made that spans from P₁=1 and P₂=2, then the octaves would fall of 1 and 2, the Perfect Fifth on 1.5, the Major Third or 1.25, and the "Perfect Seventh" on 1.75. In this way, the Perfect Fifth is the half-point of the octave. The Major Third is the quarter-point. And the "Perfect Seventh" is the three-quarters-point. Since the harmonic identities continue beyond 8, infinitely. It would be possible to deliniate the fraction of the octave for each subsequent identity. If you do, you find the next eight identities subdivide the octave into eighths, the next sixteen subdivide it into sixteenths, etc. This may be purely academic, but it does help to remember the reality of other harmonically pure note.

Fig. 2
Harmonic Identity	Common Name	Example	Multiple of Fundamental Freq	Ratio (this identity/last octave)
1	Fundamental	A₂ - 110Hz	1x	1/1 = 1x
2	Octave	A₃ - 220 Hz	2x	2/1 = 2x (also 2/2 = 1x)
3	Perfect Fifth	E₃ - 330 Hz	3x	3/2 = 1.5x
4	Octave	A₄ - 440 Hz	4x	4/2 = 2x (also 1x)
5	Major Third	C#₄ - 550 Hz	5x	5/4 = 1.25x
6	Perfect Fifth	E₄ - 660 Hz	6x	6/4 = 1.5x
7	"Perfect Seventh"	?₄ - 770 Hz	7x	7/4 = 1.75x
8	Octave	A₅ - 880 Hz	8x	8/4 = 2x (also 1x)

$This diagram lists the first 16 harmonic identities, along with their names and frequencies. It is reveals the exponential nature of the octave and the simple fractional nature of the non-octave harmonics.$

This diagram lists the first 16 harmonic identities, along with their names and frequencies. It is reveals the exponential nature of the octave and the simple fractional nature of the non-octave harmonics.

[edit] The harmonic series can be normalized also

Since the harmonic series describes points on an exponential frequency scale and our minds/ears perceive a linear scale, it can be helpful to normalize the harmonic identities to a linear scale.

Fig 3
Harmonic Identity	Common Name	Linear Point on an Exponential Scale	Linear Point on a Normalized (linear) Scale
1	fundamental	1/1 = 1x	log₂(1.0) = 0.00
2	octave	2/1 = 2x	Log₂(2.0) = 1.00
3	perfect fifth	3/2 = 1.5x	log₂(1.5) = 0.585
4	octave	4/2 = 2x	log₂(2.0) = 1.00
5	major third	5/4 = 1.25x	log₂(1.25) = 0.322..
6	perfect fifth	6/4 = 1.5x	log₂(1.5) = 0.585..
7	"perfect seventh"	7/4 = 1.75x	log₂(1.75) = 0.807..
8	octave	8/4 = 2x	log₂(2.0) = 1.00

This diagram lists the first 16 harmonic identities normalized, along with their names, frequencies, and normalized frequencies.

Table 3 lists the normalized value of the fractions associated with each harmonic identity. Table 3 reveals why:

The Perfect Fifth (the 1/2-point of the octave) is located on the 7th step of 12-TET scale. 7/12 = 0.583... ≈ 0.585....
The Major Third (the 1/4-point of the octave) is located on the 4th step of the 12-tet scale. 4/12 = 0.333... ≈ 0.322....
The Perfect Fourth (the distance from a Perfect Fifth to it nearest upper octave) is located on the 5th step of the 12-tet. 5/12 = 0.416... ≈ 1 (the octave) - 0.585... (the perfect fifth) = 0.414....
The Minor Third (the distance from a Major Third to its nearest upper Perfect Fifth) is located on the 3rd step of the 12-tet. 3/12 = 0.25 ≈ 0.585 (the perfect fifth) - 0.322 (the major third) = 0.263....
No note on the 12-tet represents the 7th harmonic identity, because no integer divided by 12 will yield a number like 0.807....

[edit] Multiply in exponential scales, add in linear scales

Converting exponential figures into linear ones involves the log₂x function. According with the rules of logarithms, log₂(fundamental frequency)×(fraction) = log₂(fundamental frequency) + log₂(fraction).

In the exponential scale, the fractions may be multiplied against a fundamental frequency to solve for the frequency of the specific identity. For example, to find the Perfect Fifth above, say, G_{196 Hz}, you multiply 196 Hz by 1.5 resulting in 294 Hz—which is represented by D_{293.66 Hz}.

In the normalized scale, the fractions serve the same role, but solving for a value involves addition, not multiplication. This follows from the rule of logs stated above. For example, solving for the normalized value of the Perfect Fifth above G_{196 Hz} involves these steps:

Converting 196 Hz to normalized terms: log₂(196Hz) = 7.6147...
Adding the specific normalized fraction: 7.6147... + 0.5849... = 8.1997...
One can check the work by converting the solution back into exponential terms: 2^8.1997... = 294 Hz. The same value is found as when calculated via the exponential scale.

[edit] What other equal tempered scales have harmonic identities 1-8 represented?

The diagram below compares/contrasts several good equal-tempered scales. The scale units are normalized so that each step is equally spaced. Before we normalized the scale units, each step was of exponentially growing size. Normalizing the scale units makes this analysis easier. It is clear how nearly each scale approximates the exact M3, P5, and P7. (Since the P7 is seldom used in Western music, you may discount that bar from your analysis.) Note: the scale steps are the black bars separating the colored spaces.