Musical tuning systems

[lpetrich]

As to why we use the musical scales that we do, it is to produce overlapping harmonics, because that sounds very pleasing to us. Not quite overlap, however, sounds bad: "dissonance".

Interval	# ST	Ratio	ET Value	5th Exact	5th value	Mj3rd value	Mn3rd value
Octave	12	2	2	2	2	2	2
Fifth	7	3/2	1.49831	3/2	1.5	1.49535	1.49380
Fourth	5	4/3	1.33484	4/3	1.33333	1.33748	1.33887
Major third	4	5/4	1.25992	81/64	1.26563	1.25	1.24483
Minor third	3	6/5	1.18921	32/27	1.18519	1.19628	1.2

ET = "equal temperament" tuning, that makes all neighboring note intervals equal.
# ST = number of semitones or neighboring-note intervals. --  Interval (music)

It's easy to see how an octave makes overlaps (starred):
1, 2, 3, 4, ... and 2, 4, 6, 8, ... -> 1, 2*, 3, 4*, 5, ...

A fifth makes 1, 3/2, 2, 3*, 4, 9/2, 5, ...

A major chord, like C4-E4-G4, is 1, 5/4, 3/2, 2, 5/2, 3*, 15/4, 4, 9/2, 5*, 6*, 25/4, ...

Etc.


 Equal temperament is a compromise for making all musical keys sound OK. In past centuries, various alternatives have been used, like  Pythagorean tuning and  Meantone temperament

Pythagorean tuning makes the fifths and fourths exact, but if one wants to make all twelve fifths fit, at least one of them will be bad: a  Wolf interval

For a single wolf interval, (3/2)¹² * 2^-7 ~ 1.01364 -- that difference is enough to sound dissonant, thus the "wolf" in its name.

 Quarter-comma meantone makes the major thirds exact. That makes a fifth 5^1/4. Third-comma meantone makes the minor thirds exact. That makes a fifth (10/3)^1/3. Both of these fifths make worse wolf fifths than Pythagorean tuning, however.

Pythagorean and meantone tuning are various forms of  Just intonation. There are also versions that make some key's major chords perfect ratios, like (1, 5/4, 3/2) or minor chords perfect ratios, like (1, 6/5, 3/2).

 

[lpetrich]

For the frequency for each note,  Piano key frequencies gives the usual convention for them.

 A440 (pitch standard) - that's A4 = 440 Hz
Also called  Concert pitch
That article goes into the history of pitch conventions and how they emerged.

 

[Swammerdami]

Equal Temperament is sometimes treated as an important mathematical invention (even Galileo's father Vincenzo Galilei was involved), but its adoption may have robbed music of the quirky variations among the 12 major and 12 minor keys. (From that link see that C major supposedly reflects innocence, D major triumph, G major calm success, G minor uneasiness, and so on. "If ghosts could speak, their speech would [use] the horrible D# minor.")

One of the most famous music compositions is J.S. Bach's  The Well-Tempered Clavier which cycles through all 24 keys and exploits their different moods. But all major keys have the exact same intervals with equal temperament, and similarly for the minor keys.

I've wondered: Are pianos always tuned to equal temperament today? For a special performance or recording of, for example, The Well-Tempered Clavier, wouldn't it make sense to tune the piano as Bach did? The Wiki article discusses this a little; I'm afraid ignorance of exactly how Bach tuned his clavichord is part of the problem.

Searching my laptop, I see that 7 years ago I downloaded a list of 409 named intervals. (At least 15 of these intervals are named after famous mathematicians, showing how music and mathematics have been intertwined for millennia.) I've added the decimal value of the interval and sorted on that field. Here's the list:

Spoiler: List of Named Musical Intervals

1.0000000 1/1 unison, perfect prime
1.0000430 23232/23231 lesser harmonisma
1.0000508 19657/19656 greater harmonisma
1.0000939 10648/10647 harmonisma
1.0001020 9801/9800 kalisma, Gauss' comma
1.0001689 450359962737049600/450283905890997363 monzisma
1.0001880 250047/250000 Landscape comma
1.0002286 4375/4374 ragisma
1.0002442 4096/4095 tridecimal schisma, Sagittal schismina
1.0003306 3025/3024 Lehmerisma
1.0004166 2401/2400 Breedsma
1.0004861 2058/2057 xenisma
1.0004979 7629394531250/7625597484987 ennealimmal comma
1.0005777 1732/1731 approximation to 1 cent
1.0007770 1288/1287 triaphonisma
1.0007999 274877906944/274658203125 semithirds comma
1.0008230 1216/1215 Eratosthenes' comma
1.0011291 32805/32768 schisma
1.0012714 1575/1573 Nicola
1.0013351 750/749 ancient Chinese tempering
1.0013580 65625/65536 horwell comma
1.0013654 2200/2197 Parizek comma
1.0014005 715/714 septendecimal bridge comma
1.0014814 676/675 island comma
1.0016276 19073486328125/19042491875328 '19-tone' comma
1.0017530 4000/3993 undecimal schisma
1.0018552 540/539 Swets' comma
1.0019299 6115295232/6103515625 Vishnu comma
1.0019531 513/512 undevicesimal comma, Boethius' comma
1.0020903 19383245667680019896796723/19342813113834066795298816 Mercator's comma
1.0021997 33554432/33480783 Beta 2, septimal schisma
1.0022727 441/440 Werckmeister's undecimal septenarian schisma
1.0026041 385/384 undecimal kleisma
1.0028490 352/351 minthma
1.0030613 1224440064/1220703125 parakleisma
1.0031020 6144/6125 porwell comma
1.0033313 5120/5103 Beta 5, Garibaldi comma
1.0035200 3136/3125 middle second comma
1.0035607 1600000/1594323 kleisma - schisma
1.0037494 10976/10935 hemimage
1.0038164 118098/117649 stearnsma
1.0039215 256/255 septendecimal kleisma
1.0040459 16875/16807 small BP diesis
1.0041322 243/242 neutral third comma
1.0042265 1188/1183 kestrel comma
1.0042754 390625/388962 dimcomp comma
1.0044642 225/224 septimal kleisma
1.0046939 15625/15552 kleisma, semicomma majeur
1.0047095 640/637 huntma
1.0048605 65536/65219 orgonisma
1.0048828 1029/1024 gamelan residue
1.0050551 2187/2176 septendecimal comma
1.0056116 896/891 undecimal semicomma
1.0057142 176/175 valinorsma
1.0058283 2109375/2097152 semicomma, Fokker's comma
1.0066329 393216/390625 W�rschmidt's comma
1.0069444 145/144 29th-partial chroma
1.0075801 1728/1715 Orwell comma
1.0077696 78732/78125 medium semicomma
1.0078105 4000/3969 septimal semicomma
1.0080000 126/125 small septimal comma
1.0082304 245/243 minor BP diesis
1.0083333 121/120 undecimal seconds comma
1.0084200 50421/50000 Trimyna
1.0093688 33075/32768 mirwomo comma
1.0096021 736/729 vicesimotertial comma
1.0096153 105/104 small tridecimal comma
1.0101010 100/99 Ptolemy's comma
1.0102040 99/98 small undecimal comma
1.0102173 67108864/66430125 Misty comma, diaschisma - schisma
1.0105263 96/95 19th-partial chroma
1.0111111 91/90 medium tridecimal comma
1.0113580 2048/2025 diaschisma
1.0115288 36893488147419103232/36472996377170786403 '41-tone' comma
1.0120783 2430/2401 nuwell comma
1.0123096 3125/3087 major BP diesis
1.0125000 81/80 syntonic comma, Didymus comma
1.0127314 875/864 keema
1.0131578 77/76 approximation to 53-tone comma
1.0136432 531441/524288 Pythagorean comma, ditonic comma
1.0156250 65/64 13th-partial chroma
1.0158730 64/63 septimal comma, Archytas' comma
1.0161052 20000/19683 minimal diesis
1.0162962 686/675 senga
1.0172526 3125/3072 small diesis
1.0184012 34171875/33554432 Ampersand's comma
1.0195312 261/256 vicesimononal comma
1.0200000 51/50 17th-partial chroma
1.0203667 1594323/1562500 Unicorn comma
1.0204081 50/49 Erlich's decatonic comma, tritonic diesis
1.0205761 248/243 tricesoprimal comma
1.0206414 15625/15309 great BP diesis
1.0208333 49/48 slendro diesis, septimal 1/6-tone
1.0215737 8192/8019 undecimal minor diesis
1.0220275 48828125/47775744 Sycamore comma
1.0222222 46/45 23rd-partial chroma
1.0227272 45/44 1/5-tone
1.0240000 128/125 minor diesis, diesis
1.0251562 6561/6400 Mathieu superdiesis
1.0253906 525/512 Avicenna enharmonic diesis
1.0256410 40/39 tridecimal minor diesis
1.0283203 1053/1024 tridecimal major diesis
1.0285714 36/35 septimal diesis, 1/4-tone
1.0288065 250/243 maximal diesis
1.0292887 246/239 Meshaqah's 1/4-tone
1.0294117 35/34 septendecimal 1/4-tone
1.0297328 59049/57344 Harrison's comma
1.0299682 16875/16384 double augmentation diesis
1.0312500 33/32 undecimal comma, al-Farabi's 1/4-tone
1.0322580 32/31 Greek enharmonic 1/4-tone
1.0333333 31/30 31st-partial chroma
1.0355113 729/704 undecimal major diesis
1.0368000 648/625 major diesis
1.0370370 28/27 Archytas' 1/3-tone
1.0384615 27/26 tridecimal comma
1.0393182 134217728/129140163 Pythagorean double diminished third
1.0400000 26/25 tridecimal 1/3-tone
1.0404917 20480/19683 grave minor second
1.0409948 17496/16807 septimal major diesis
1.0416666 25/24 classic chromatic semitone, minor chroma
1.0434782 24/23 vicesimotertial minor semitone
1.0448979 256/245 septimal minor semitone
1.0476190 22/21 undecimal minor semitone
1.0500000 21/20 minor semitone
1.0526315 20/19 small undevicesimal semitone
1.0534979 256/243 limma, Pythagorean minor second
1.0546875 135/128 major chroma, major limma
1.0555555 19/18 undevicesimal semitone
1.0578512 128/121 undecimal semitone
1.0581580 413343/390625 BP small link
1.0588235 18/17 Arabic lute index finger
1.0594600 52973/50000 Mersenne's quasi-equal semitone
1.0595238 89/84 quasi-equal semitone
1.0625000 17/16 17th harmonic
1.0664062 273/256 Ibn Sina's minor second
1.0666666 16/15 minor diatonic semitone
1.0668672 83349/78125 BP minor link
1.0678710 2187/2048 apotome
1.0711836 6561/6125 BP major link
1.0714285 15/14 major diatonic semitone
1.0769230 14/13 2/3-tone
1.0800000 27/25 large limma, BP small semitone
1.0824403 1162261467/1073741824 Pythagorean-19 comma
1.0833333 13/12 tridecimal 2/3-tone
1.0843696 18225/16807 minimal BP chroma
1.0864197 88/81 2nd undecimal neutral second
1.0872483 162/149 Persian neutral second
1.0880755 4608/4235 Arabic neutral second
1.0888888 49/45 BP minor semitone
1.0904977 241/221 Meshaqah's 3/4-tone
1.0909090 12/11 3/4-tone, undecimal neutral second
1.0922666 2048/1875 double diminished third
1.0932944 375/343 BP major semitone, minor BP chroma
1.0937500 35/32 septimal neutral second
1.0973936 800/729 grave whole tone
1.0986328 1125/1024 double augmented prime
1.1000000 11/10 4/5-tone, Ptolemy's second
1.1020408 54/49 Zalzal's mujannab
1.1022927 625/567 BP great semitone, major BP chroma
1.1034482 32/29 29th subharmonic
1.1098579 65536/59049 Pythagorean diminished third
1.1111111 10/9 minor whole tone
1.1113651 21875/19683 maximal BP chroma
1.1160714 125/112 classic augmented semitone
1.1176470 19/17 quasi-meantone
1.1193415 272/243 Persian whole tone
1.1200000 28/25 middle second
1.1224489 55/49 quasi-equal major second
1.1224637 134217728/119574225 whole tone - 2 schismas, 5-limit approximation to ET whole tone
1.1250000 9/8 major whole tone
1.1363636 25/22 undecimal acute whole tone
1.1377777 256/225 diminished third
1.1390625 729/640 acute major second
1.1403486 4782969/4194304 Pythagorean double augmented prime
1.1428571 8/7 septimal whole tone
1.1458333 55/48 undecimal semi-augmented whole tone
1.1520000 144/125 classic diminished third
1.1538461 15/13 tridecimal 5/4-tone
1.1550000 231/200 5/4-tone
1.1562500 37/32 37th harmonic
1.1571428 81/70 Al-Hwarizmi's lute middle finger
1.1574074 125/108 semi-augmented whole tone
1.1666666 7/6 septimal minor third
1.1692330 16777216/14348907 Pythagorean double diminished fourth
1.1705532 2560/2187 grave minor third
1.1718750 75/64 classic augmented second
1.1739130 27/23 vicesimotertial minor third
1.1764705 20/17 septendecimal augmented second
1.1785714 33/28 undecimal minor third
1.1818181 13/11 tridecimal minor third
1.1851851 32/27 Pythagorean minor third
1.1865234 1215/1024 wide augmented second
1.1875000 19/16 19th harmonic
1.1904761 25/21 BP second, quasi-tempered minor third
1.1911764 81/68 Persian wosta
1.2000000 6/5 minor third
1.2013549 19683/16384 Pythagorean augmented second
1.2136296 4096/3375 double diminished fourth
1.2142857 17/14 supraminor third
1.2150000 243/200 acute minor third
1.2187500 39/32 39th harmonic, Zalzal wosta of Ibn Sina
1.2190476 128/105 septimal neutral third
1.2203389 72/59 Ibn Sina's neutral third
1.2222222 11/9 undecimal neutral third
1.2240000 153/125 7/4-tone
1.2244897 60/49 smaller approximation to neutral third
1.2250000 49/40 larger approximation to neutral third
1.2272727 27/22 neutral third, Zalzal wosta of al-Farabi
1.2307692 16/13 tridecimal neutral third
1.2345679 100/81 grave major third
1.2352941 21/17 submajor third
1.2359619 10125/8192 double augmented second
1.2485901 8192/6561 Pythagorean diminished fourth
1.2500000 5/4 major third
1.2592592 34/27 septendecimal major third
1.2600000 63/50 quasi-equal major third
1.2631578 24/19 smaller undevicesimal major third
1.2641975 512/405 narrow diminished fourth
1.2656250 81/64 Pythagorean major third
1.2666666 19/15 undevicesimal ditone
1.2692307 33/26 tridecimal major third
1.2698412 80/63 wide major third
1.2727272 14/11 undecimal diminished fourth or major third
1.2777777 23/18 vicesimotertial major third
1.2800000 32/25 classic diminished fourth
1.2814453 6561/5120 acute major third
1.2828922 43046721/33554432 Pythagorean double augmented second
1.2857142 9/7 septimal major third, BP third
1.2962962 35/27 9/4-tone, septimal semi-diminished fourth
1.3000000 13/10 tridecimal semi-diminished fourth
1.3020833 125/96 classic augmented third
1.3061224 64/49 2 septatones or septatonic major third
1.3090909 72/55 undecimal semi-diminished fourth
1.3125000 21/16 narrow fourth
1.3153871 2097152/1594323 Pythagorean double diminished fifth
1.3168724 320/243 grave fourth
1.3183593 675/512 wide augmented third
1.3200000 33/25 2 pentatones
1.3333333 4/3 perfect fourth
1.3348388 10935/8192 fourth + schisma, 5-limit approximation to ET fourth
1.3500000 27/20 acute fourth
1.3515243 177147/131072 Pythagorean augmented third
1.3611111 49/36 Arabic lute acute fourth
1.3636363 15/11 undecimal augmented fourth
1.3653333 512/375 double diminished fifth
1.3714285 48/35 septimal semi-augmented fourth
1.3732910 5625/4096 double augmented third
1.3740000 687/500 11/4-tone
1.3750000 11/8 undecimal semi-augmented fourth
1.3846153 18/13 tridecimal augmented fourth
1.3857576 536870912/387420489 Pythagorean double diminished sixth
1.3888888 25/18 classic augmented fourth
1.3913043 32/23 23rd subharmonic
1.4000000 7/5 septimal or Huygens' tritone, BP fourth
1.4046639 1024/729 Pythagorean diminished fifth
1.4062500 45/32 diatonic tritone
1.4117647 24/17 1st septendecimal tritone
1.4141414 140/99 quasi-equal tritone
1.4142857 99/70 2nd quasi-equal tritone
1.4166666 17/12 2nd septendecimal tritone
1.4222222 64/45 2nd tritone
1.4238281 729/512 Pythagorean tritone
1.4285714 10/7 Euler's tritone
1.4375000 23/16 23rd harmonic
1.4400000 36/25 classic diminished fifth
1.4432537 387420489/268435456 Pythagorean double augmented third
1.4444444 13/9 tridecimal diminished fifth
1.4545454 16/11 undecimal semi-diminished fifth
1.4555555 131/90 13/4-tone
1.4563555 8192/5625 double diminished sixth
1.4583333 35/24 septimal semi-diminished fifth
1.4648437 375/256 double augmented fourth
1.4666666 22/15 undecimal diminished fifth
1.4693877 72/49 Arabic lute grave fifth
1.4798105 262144/177147 Pythagorean diminished sixth
1.4814814 40/27 grave fifth
1.4927113 512/343 3 septatones or septatonic fifth
1.4980000 749/500 ancient Chinese quasi-equal fifth
1.4983081 16384/10935 fifth - schisma, 5-limit approximation to ET fifth
1.5000000 3/2 perfect fifth
1.5151515 50/33 3 pentatones
1.5170370 1024/675 narrow diminished sixth
1.5187500 243/160 acute fifth
1.5204648 1594323/1048576 Pythagorean double augmented fourth
1.5238095 32/21 wide fifth
1.5277777 55/36 undecimal semi-augmented fifth
1.5306122 75/49 BP fifth
1.5360000 192/125 classic diminished sixth
1.5384615 20/13 tridecimal semi-augmented fifth
1.5423728 91/59 15/4-tone
1.5428571 54/35 septimal semi-augmented fifth
1.5555555 14/9 septimal minor sixth
1.5589773 67108864/43046721 Pythagorean double diminished seventh
1.5607376 10240/6561 grave minor sixth
1.5625000 25/16 classic augmented fifth
1.5714285 11/7 undecimal augmented fifth
1.5750000 63/40 narrow minor sixth
1.5757575 52/33 tridecimal minor sixth
1.5789473 30/19 smaller undevicesimal minor sixth
1.5802469 128/81 Pythagorean minor sixth
1.5820312 405/256 wide augmented fifth
1.5833333 19/12 undevicesimal minor sixth
1.5873015 100/63 quasi-equal minor sixth
1.5882352 27/17 septendecimal minor sixth
1.6000000 8/5 minor sixth
1.6018066 6561/4096 Pythagorean augmented fifth
1.6181728 16384/10125 double diminished seventh
1.6190476 34/21 supraminor sixth
1.6200000 81/50 acute minor sixth
1.6250000 13/8 tridecimal neutral sixth
1.6296296 44/27 neutral sixth
1.6326530 80/49 smaller approximation to neutral sixth
1.6333333 49/30 larger approximation to neutral sixth
1.6339869 250/153 17/4-tone
1.6363636 18/11 undecimal neutral sixth
1.6406250 105/64 septimal neutral sixth
1.6410256 64/39 39th subharmonic
1.6460905 400/243 grave major sixth
1.6470588 28/17 submajor sixth
1.6479492 3375/2048 double augmented fifth
1.6647868 32768/19683 Pythagorean diminished seventh
1.6666666 5/3 major sixth, BP sixth
1.6770186 270/161 Kirnberger's sixth
1.6800000 42/25 quasi-tempered major sixth
1.6842105 32/19 19th subharmonic
1.6855967 2048/1215 narrow diminished seventh
1.6875000 27/16 Pythagorean major sixth
1.6923076 22/13 tridecimal major sixth
1.7000000 17/10 septendecimal diminished seventh
1.7059558 4096/2401 4 septatones or septatonic major sixth
1.7066666 128/75 diminished seventh
1.7085937 2187/1280 acute major sixth
1.7105230 14348907/8388608 Pythagorean double augmented fifth
1.7142857 12/7 septimal major sixth
1.7280000 216/125 semi-augmented sixth
1.7297297 64/37 37th subharmonic
1.7311827 161/93 19/4-tone
1.7333333 26/15 tridecimal semi-augmented sixth
1.7361111 125/72 classic augmented sixth
1.7500000 7/4 harmonic seventh
1.7538495 8388608/4782969 Pythagorean double diminished octave
1.7558299 1280/729 grave minor seventh
1.7578125 225/128 augmented sixth
1.7600000 44/25 undecimal grave minor seventh
1.7777777 16/9 Pythagorean minor seventh
1.7818181 98/55 quasi-equal minor seventh
1.7857142 25/14 middle minor seventh
1.8000000 9/5 just minor seventh, BP seventh
1.8020324 59049/32768 Pythagorean augmented sixth
1.8125000 29/16 29th harmonic
1.8181818 20/11 large minor seventh
1.8204444 2048/1125 double diminished octave
1.8225000 729/400 acute minor seventh
1.8285714 64/35 septimal neutral seventh
1.8310546 1875/1024 double augmented sixth
1.8333333 11/6 21/4-tone, undecimal neutral seventh
1.8409090 81/44 2nd undecimal neutral seventh
1.8461538 24/13 tridecimal neutral seventh
1.8518518 50/27 grave major seventh
1.8571428 13/7 16/3-tone
1.8666666 28/15 grave major seventh
1.8728852 4096/2187 Pythagorean diminished octave
1.8750000 15/8 classic major seventh
1.8823529 32/17 17th subharmonic
1.8876404 168/89 quasi-equal major seventh
1.8888888 17/9 septendecimal major seventh
1.8947368 36/19 smaller undevicesimal major seventh
1.8962962 256/135 octave - major chroma
1.8984375 243/128 Pythagorean major seventh
1.9000000 19/10 undevicesimal major seventh
1.9047619 40/21 acute major seventh
1.9090909 21/11 undecimal major seventh
1.9166666 23/12 vicesimotertial major seventh
1.9200000 48/25 classic diminished octave
1.9221679 19683/10240 acute major seventh
1.9243383 129140163/67108864 Pythagorean double augmented sixth
1.9285714 27/14 septimal major seventh
1.9290123 625/324 octave - major diesis
1.9375000 31/16 31st harmonic
1.9393939 64/33 33rd subharmonic
1.9418074 32768/16875 octave - double augmentation diesis
1.9428571 68/35 23/4-tone
1.9440000 243/125 octave - maximal diesis
1.9444444 35/18 septimal semi-diminished octave
1.9496638 32768/16807 5 septatones or septatonic diminished octave
1.9531250 125/64 classic augmented seventh, octave - minor diesis
1.9600000 49/25 BP eighth
1.9660800 6144/3125 octave - small diesis
1.9683000 19683/10000 octave - minimal diesis
1.9687500 63/32 octave - septimal comma
1.9730807 1048576/531441 Pythagorean diminished ninth
1.9753086 160/81 octave - syntonic comma
1.9775390 2025/1024 2 tritones
1.9868214 390625/196608 octave - W�rschmidt's comma
1.9977442 65536/32805 octave - schisma
2.0000000 2/1 octave
2.0272865 531441/262144 Pythagorean augmented seventh
2.0833333 25/12 classic augmented octave
2.1250000 17/8 septendecimal minor ninth
2.1333333 32/15 minor ninth
2.1428571 15/7 septimal minor ninth, BP ninth
2.1648806 1162261467/536870912 Pythagorean double augmented seventh
2.2000000 11/5 neutral ninth
2.2222222 20/9 small ninth
2.2500000 9/4 major ninth
2.2857142 16/7 septimal major ninth
2.3333333 7/3 minimal tenth, BP tenth
2.5200000 63/25 quasi-equal major tenth, BP eleventh
2.7777777 25/9 classic augmented eleventh, BP twelfth

 

[Swammerdami]

I watched a Youtube on chords and decided to post in this thread. Please correct any errors here; my understanding of these matters may be flawed.

Throughout this post, I consider only the Key of C-Major; i.e. the white keys on a piano.
And I am concerned ONLY with relative pitch. In the following table I assign C the pitch of 24, not because I think C is tuned to 24 Hz but because 24 is a convenient integer to work with. (I've left spacers 'bbb' for the piano's black keys to clarify the distinction between half-steps and whole-steps.)

. . C . . 24 . . 24
. . bbb
. . D . . 27 . . 27
. . bbb
. . E . . 30 . . 30.38
. . F . . 32 . . 32
. . bbb
. . G . . 36 . . 36
. . bbb
. . A . . 40 . . 40.50
. . bbb
. . B . . 45 . . 45.56
. . C . . 48 . . 48

The 3rd column in the above table is the "Pythagorean scale." I'm not sure what the scale in the 2nd column is called — "Renaissance scale" ?? Note that both those scales have a 9:8 ratio for the whole-step from F to G, and for the whole-step from A to B. But only the Pythagorean scale has a 9:8 ratio for the whole-step from G to A. Only the Renaissance scale, however, has the "perfect third" 5:4 ratio for the important transition from C to E.

If I understand correctly, these are the only scales which assign simple rational values to all the white keys. Other scales ("mean-tone," "well temperament", etc.) used in the 18th and 19th centuries were compromises — allowing an instrument to be played in any key without retuning. But, as I said, let's assume we ONLY want to play in the Key of C-Major.

The most important chords are the I, IV, V and vi chords. I show the frequency ratios for these and other chords in C Major using both the Renaissance scale and Equal Temperament. I have normalized the pitch values so that the 2nd tone of a chord is always 24 or 25. 24 is the harmonic mean of 20 and 30; 25 is the arithmetic mean.

Renaissance Scale

I. C Major	C E G	20	25	30
I. C Major	E G C, (1st Inversion)	20	24	32
I. C Major	G C E, (2nd Inversion)	18	24	30
ii. D Minor	D F A	20.25	24	30
iii. E Minor	E G B	20	24	30
IV. F Major	F A C	20	25	30
V. G Major	G B D	20	25	30
vi. A Minor	A C E	20	24	30
viio. B Diminished	B D F	20	24	28.44
vii. B Minor	B D F#	20	24	30

Equal Temperament Scale

C Augmented	C E G#	19.84	25	31.50
C Major	C E G	19.84	25	29.73
C Minor	C D# G	20.18	24	30.24
C Diminished	C D# F#	20.18	24	28.54

Note that the important chords in the Renaissance system all have the perfect values. With this beautiful rational numbers available I'm still surprised that instruments are not tuned for a specific key, at least for special concerts and recordings. (ii. -- D Minor -- is the only one of the "standard" chords that ends up off.) How often are the black keys used in melodies played in the key of C Major?

 

[steve_bank]

The Well Tempered Clavier

 

[Tharmas]

I spent a few years studying piano tuning. It was supposed to be my retirement job, but circumstances intervened and it didn’t work out.

As you mention, it used to be that music was limited to only a few keys because of the perfect way intervals were tuned.

Most classical music and a surprising variety of pop music, even blues, would be enormously boring if it stayed in the same key all the time. What Bach shows in the work referenced by Steve is that the well-tempered instrument gives an enormous power to changing keys in music.

Slightly different piano tunings can be preferred for jazz as opposed to classical, and slightly different piano tunings can be used for different composer’s music. Many concert pianists take their own tuner with them on tour. Some are reported to soften or harder the felt hammers depending on the compositions they intend to play, but that's a different story.

Meanwhile, many unfretted string instruments, like violins, cellos, etc. can be and are played with perfect intervals, depending on context.

There are computer programs which duplicate many historical tunings so you can hear what older music “really” sounded like. Today professional piano tuners make use of such programs.

Edited to add: Some composer, maybe it was Irving Berlin, I don't remember, had his piano rigged with a mechanism so that he could crank the keyboard up and down relative to the hammers and strings. That way he could compose in all the keys and still play only using the white keys. That may be apocryphal.

 

[steve_bank]

If you look at the frets on a guitar you can see the changes in steps.

Notes vs frequency. Anyone who knows music knows A-440 Hz, a common tuning fork. I can hear the tome when I think of it.

Note names of musical notes keyboard piano frequencies = octave piano keys number tone tones 88 notes frequency names of all keys on a grand piano standard concert pitch tuning German English system MIDI 88 - sengpielaudio Sengpiel Berlin

Note names concert standard pitch tuning keyboard music piano key numbers frequencies octave musical grand piano keys tone 88 notes frequency names of all keys on a piano naming note names German English MIDI - Eberhard Sengpiel sengpielaudio

www.sengpielaudio.com

 

[lpetrich]

As to alternate tunings, that is difficult for most mechanical instruments, since one has to go through the trouble of retuning them for each new tuning. The exceptions are continuous-scale instruments like the trombone, and the human voice is also continuous-scale. These ones have *no* built-in tuning.

Electronic instruments are different, because they work by running oscillators or replaying samples, and one can easily change the run/replay rates. Like have a library of alternative tunings and load in whichever one wants.

 

[steve_bank]

There is a lot to the history of scales and turnings. An orchestra can be tuned to a frequency other than A 440.

A lot of thought and experiment went into it over centuries, along with a lot of controvert. Music was serious bigness.

Its been a long time since I thought about it. Mixolyduann.

Mixolydian mode - Wikipedia

en.wikipedia.org

Ionian mode - Wikipedia

en.wikipedia.org

Ionian mode is a musical mode or, in modern usage, a diatonic scale also called the major scale.

It is the name assigned by Heinrich Glarean in 1547 to his new authentic mode on C (mode 11 in his numbering scheme), which uses the diatonic octave species from C to the C an octave higher, divided at G (as its dominant, reciting tone/reciting note or tenor) into a fourth species of perfect fifth (tone–tone–semitone–tone) plus a third species of perfect fourth (tone–tone–semitone): C D E F G + G A B C.[1] This octave species is essentially the same as the major mode of tonal music.[2]

Church music had been explained by theorists as being organised in eight musical modes: the scales on D, E, F, and G in the "greater perfect system" of "musica recta,"[3] each with their authentic and plagal counterparts.

Glarean's twelfth mode was the plagal version of the Ionian mode, called Hypoionian (under Ionian), based on the same relative scale, but with the major third as its tenor, and having a melodic range from a perfect fourth below the tonic, to a perfect fifth above it.[4]

Diatonic scale - Wikipedia

en.wikipedia.org

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 pit

Circle of fifths - Wikipedia

en.wikipedia.org

In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. If C is chosen as a starting point, the sequence is: C, G, D, A, E, B (=C♭), F♯ (=G♭), C♯ (=D♭), A♭, E♭, B♭, F. Continuing the pattern from F returns the sequence to its starting point of C. This order places the most closely related key signatures adjacent to one another. It is usually illustrated in the form of a circle

Scale (music) - Wikipedia

en.wikipedia.org

Svara - Wikipedia

en.wikipedia.org

Svara or swara (Devanagari: स्वर, generally pronounced as swar) is a Sanskrit word that connotes simultaneously a breath, a vowel, the sound of a musical note corresponding to its name, and the successive steps of the octave or saptaka. More comprehensively, it is the ancient Indian concept about the complete dimension of musical pitch.[1][2] Most of the time a svara is identified as both musical note and tone, but a tone is a precise substitute for sur, related to tunefulness. Traditionally, Indians have just seven svaras/notes with short names, e.g. saa, re/ri, ga, ma, pa, dha, ni which Indian musicians collectively designate as saptak or saptaka. It is one of the reasons why svara is considered a symbolic expression for the number seven.

 

[lpetrich]

Here are these named modes -  Mode (music) - along with -  Major scale (Ionian) -  Minor scale (Aeolian)

Mode	Scale from C	As all white keys	Major scale	Minor scale
	C# D# E# F# G# A# B#		C#	A#
	C# D# E# F# G# A# B		F#	D#
	C# D# E F# G# A# B		B	G#
	C# D# E F# G# A B		E	C#
	C# D E F# G# A B		A	F#
	C# D E F# G A B		D	B
Lydian	C D E F# G A B	F	G	E
Ionian	C D E F G A B	C	C	A
Mixolydian	C D E F G A Bb	G	F	D
Dorian	C D Eb F G A Bb	D	Bb	G
Aeolian	C D Eb F G Ab Bb	A	Eb	C
Phrygian	C Db Eb F G Ab Bb	E	Ab	F
Locrian	C Db Eb F Gb Ab Bb	B	Db	Bb
	Cb Db Eb F Gb Ab Bb		Gb	Eb
	Cb Db Eb Fb Gb Ab Bb		Cb	Ab

Notice the pattern of increasing by fifths as one goes upward. Notice also that some keys are "enharmonic": major keys Db = C#, Gb = F#, Cb = B.

 

[Cheerful Charlie]

There were local music scales. The tunings used in Paris were different from that used in say, Berlin. Eventually, due to complaints, usually singers, there were negotiations to choose a standard tuning. By the early1800's our present day orchestral tuning became standardized. After much debate and argument.