Human ears interpret all octaves as spanning a range of pitches the same size, even though a sub-bass octave may span 40 Hz and a super-treble octave can span 4000 Hz.

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

Note	Frequency (Hz)	Distance from previous note ( = f*(2(1/12)-1) )	Log frequency (log2 f)	Distance from previous note	Converting log frequency to frequency (2log 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	0.0833	155.56
E2	164.81	9.25	7.365	0.0833	164.81
F2	174.61	9.80	7.448	0.0833	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

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.

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, frequencies, and log frequencies.

• The Perfect Fifth is located on the 7th step of 12-TET scale. 7/12 = 0.583... ≈ 0.585....
• The Major Third 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....

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

The diagram below compares/contrasts several good equal-tempered scales. The frequencies are plotted on a logarithmic scale so that each step is equally spaced. On a linear frequency scale, the steps would exponentially grow in size. It is clear how nearly each scale approximates the exact M3, P5, and P7. (The P7 is seldom used in Western music.) Note: the scale steps are the black bars separating the colored spaces.

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See also

• Physics of music
• Equal temperament
• Interval (music)
• Musical tuning
• Piano key frequencies
• Harmonic
• Musical acoustics

[

References

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External links

• Article by Adriaan Daniël Fokker analysing 31-tet and the harmonic identities
• Website for Musimathics book. Contains a table of content
• Music, Mathematics, Philosophy and tuning
• Google Scholar Seach for 'music and mathematics'
• The method for transformation of music into an image through numbers and geometrical proportions
• Twelve-Tone Musical Scale.
• Sonantometry or music as math discipline.
• Music: A Mathematical Offering by Dave Benson.
• Nicolaus Mercator use of Ratio Theory in Music at Convergence
• LucyTuning, a musical tuning, harmonic, and scalecoding system derived from pi, and the writings of John Harrison

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