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Ref: Musical scale, musical note frequencies
https://en.wikipedia.org/wiki/Diatonic_scale
In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other (i.e. separated by at least two whole steps).
The seven pitches of any diatonic scale can also be obtained by using a chain of six perfect fifths. For instance, the seven natural pitch classes that form the C-major scale can be obtained from a stack of perfect fifths starting from F:
F–C–G–D–A–E–B
Any sequence of seven successive natural notes, such as C–D–E–F–G–A–B, and any transposition thereof, is a diatonic scale. Modern musical keyboards are designed so that the white-key notes form a diatonic scale, though transpositions of this diatonic scale require one or more black keys. A diatonic scale can be also described as two tetrachords separated by a whole tone.
https://en.wikipedia.org/wiki/Twelfth_root_of_two#The_equal-tempered_chromatic_scale
A musical interval is a ratio of frequencies and the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into twelve equal parts. Each note has a frequency that is 21⁄12 times that of the one below it.[citation needed]
Applying this value successively to the tones of a chromatic scale, starting from A above middle C (known as A4) with a frequency of 440 Hz, produces the following sequence of pitches:
. . .
1.059463094359
1.122462048308
1.189207115001
1.259921049893
1.334839854168
1.414213562370
1.498307076873
1.587401051964
1.681792830503
1.781797436275
1.887748625357
2
The final A (A5: 880 Hz) is exactly twice the frequency of the lower A (A4: 440 Hz), that is, one octave higher.
https://professionalcomposers.com/note-frequency-chart/
- C = 261.63 Hz (Middle C)
- C#/Db = 277.18 Hz
- D = 293.66 Hz
- D#/Eb = 311.13 Hz
- E = 329.63 Hz
- F = 349.23 Hz
- F#/Gb = 369.99 Hz
- G = 392.00 Hz
- G#/Ab = 415.30 Hz
- A = 440.00 Hz
- A#/Bb = 466.16 Hz
- B = 493.88 Hz
See also: https://www.intmath.com/trigonometric-graphs/music.php
https://pages.mtu.edu/~suits/notefreqs.html
("Middle C" is C4 )
Other tuning choices, A4 =432 434 436 438 440 442 444 446
So, then comes the question, why there are more tuning choices other than A4 = 440 Hz?
https://en.wikipedia.org/wiki/A440_(pitch_standard)
Before standardization on 440 Hz, many countries and organizations followed the French standard since the 1860s of 435 Hz, which had also been the Austrian government's 1885 recommendation.[2] Johann Heinrich Scheibler recommended A440 as a standard in 1834 after inventing the "tonometer" to measure pitch,[3] and it was approved by the Society of German Natural Scientists and Physicians at a meeting in Stuttgart the same year.[4]
The American music industry reached an informal standard of 440 Hz in 1926, and some began using it in instrument manufacturing.
In 1936, the American Standards Association recommended that the A above middle C be tuned to 440 Hz.[5] This standard was taken up by the International Organization for Standardization in 1955 as Recommendation R 16,[6] before being formalised in 1975 as ISO 16.[7]
Hence,
A440 (also known as Stuttgart pitch[1]) is the musical pitch corresponding to an audio frequency of 440 Hz, which serves as a tuning standard for the musical note of A above middle C, or A4 in scientific pitch notation. It is standardized by the International Organization for Standardization as ISO 16. While other frequencies have been (and occasionally still are) used to tune the first A above middle C, A440 is now commonly used as a reference frequency to calibrate acoustic equipment and to tune pianos, violins, and other musical instruments.
https://en.wikipedia.org/w/index.php?title=C_(musical_note)
Historically, concert pitch has varied. For an instrument in equal temperament tuned to the A440 pitch standard widely adopted in 1939, middle C has a frequency around 261.63 Hz (for other notes see piano key frequencies). Scientific pitch was originally proposed in 1713 by French physicist Joseph Sauveur and based on the numerically convenient frequency of 256 Hz for middle C, all C's being powers of two. After the A440 pitch standard was adopted by musicians, the Acoustical Society of America published new frequency tables for scientific use. A movement to restore the older A435 standard has used the banners "Verdi tuning", "philosophical pitch" or the easily confused scientific pitch.
https://en.wikipedia.org/wiki/Scientific_pitch
Scientific pitch, also known as philosophical pitch, Sauveur pitch or Verdi tuning, is an absolute concert pitch standard which is based on middle C (C4) being set to 256 Hz rather than 261.62 Hz, making it approximately 37.6 cents lower than the common A440 pitch standard. It was first proposed in 1713 by French physicist Joseph Sauveur, promoted briefly by Italian composer Giuseppe Verdi in the 19th century, then advocated by the Schiller Institute beginning in the 1980s with reference to the composer, but naming a pitch slightly lower than Verdi's preferred 432 Hz for A, and making controversial claims regarding the effects of this pitch.
The octaves of C remain a whole number in Hz all the way down to 1 Hz in both binary and decimal counting systems.[3][4] Instead of A above middle C (A4) being set to the widely used standard of 440 Hz, scientific pitch assigns it a frequency of 430.54 Hz.[5]
Since 256 is a power of 2, only octaves (factor 2:1) and, in just tuning, higher-pitched perfect fifths (factor 3:2) of the scientific pitch standard will have a frequency of a convenient integer value. With a Verdi pitch standard of A4 = 432 Hz = 24 × 33, in just tuning all octaves (factor 2), perfect fourths (factor 4:3) and fifths (factor 3:2) will have pitch frequencies of integer numbers, but not the major thirds (factor 5:4) nor major sixths (factor 5:3) which have a prime factor 5 in their ratios. However scientific tuning implies an equal temperament tuning where the frequency ratio between each half tone in the scale is the same, being the 12th root of 2 (a factor of approximately 1.059463)1, which is not a rational number: therefore in scientific pitch only the octaves of C have a frequency of a whole number in hertz.
- The twelfth root of two was first calculated in 1584 by the Chinese mathematician and musician Zhu Zaiyu using an abacus to reach twenty four decimal places accurately,[1], and this number was proposed for the first time in relationship to musical tuning in the western world in 1580 (drafted, rewritten 1610) by Simon Stevin.[4] In 1581 Italian musician Vincenzo Galilei may be the first European to suggest twelve-tone equal temperament.[1]