Show pageOld revisionsBacklinksBack to top This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. ====== Math behind Piano keys ====== =====Relationship between key groups===== * piano keys are made of a set of octave key groups, each octave key group made from 7 white keys and 5 in-between black keys. (total 12 keys per octave key group) * each octave key group is double the sound frequency of previous octave key groups * example, * octave key group 1, containing 7 white keys {"C1","D1","E1","F1","G1","A1","B1"}, its "C1" key is at sound frequency of 32.703Hz; plus 5 black keys means {"C","d_b","D","e_b","E","F","g_b","G","a_b","A","b_b","B"}; * the C2 key from octave key group 2 is at sound frequency of 65.406 HZ, that is 32.703x2. * the middle group (4th group), C4 is at 261.6Hz, = 65.406*2^2 * as you can see, the number is exact "times power of 2", as you can guess, C7=C1x[2^(7-1)] =32.703x64=2092.9 (about 2093Hz) * youtube: - Music and math: The genius of Beethoven - Natalya St. Clair * https://www.youtube.com/watch?v=zAxT0mRGuoY - beats and math * https://www.youtube.com/watch?v=2UphAzryVpY =====Frequency Relationship within a key group and White vs Black keys===== * study 1st octave key group, white keys {"C1","D1","E1","F1","G1","A1","B1"} * C1=**32.703Hz** * D1=36.708Hz = C1 + 4.0Hz * E1=41.203Hz = D1 + 4.5Hz * F1=43.654Hz = E1 + 2.4Hz (so F1 is "half step", aka "half incremental" from E1) * G1=48.999HZ = F1 + 5.3Hz * A1=55.000Hz = G1 + 6.0Hz * B1=61.735Hz = A1 + 6.7Hz * C2=**65.406Hz** = B1 + 3.7Hz (so C2 is mostly like "half step" from B1) * so we can see, the incremental is based of a power function curve. * based on that, "half step" incremental white keys has no in-between black keys, while those "full" incremental white keys has in-between black keys, representing the "half-step" frequency increase. * in short, all keys in piano are layout in "half step" increment, either "black to white", or "white to black", or "white to white" is a "half step". =====How Many key groups we need===== * while average human can hear from 20Hz to 8000Hz, so from C1 32.7Hz to C8 4186Hz reaching C9's 8392Hz are enough; * which means 8 sets * (7 white key+ 5 black key) = 96 keys =====How much frequency difference to make a key identifiable and photographic-like-color-depth===== * as in previous calculation, the lower the frequency, the lower the frequency difference between keys, like C1 and D1 are just 4Hz apart, C4 and D4 are 32Hz+ apart. (4Hz x2^3= 32Hz) * however, the real human hearing factor is that the effect of each key difference is about similar, like **"count extra 4 hits from 32 hits in 1 second"** has similar effect of count **"extra 400 hits from 3200 hits in 1 second"**; * it is more a ratio difference than a count difference * compare D1 to C1 (full step), 4Hz difference is roughly 1/8 of 32.703Hz (C1); D1=C1 * 1.12; * compare B1 to A1 (full step), 6.7Hz difference is roughly 1/8 of 55Hz (A1); B1=A1 * 1.12; * compare F1 to E1 (half step), 2.4Hz difference is roughly 1/16 of 41.203Hz (E1); F1=E1 * 1.06; * compare C2 to B1 (half step), 3.7Hz difference is roughly 1/16 of 61.735Hz (B1); C2=B1 * 1.06; * so, * each key of piano is half step from nearby one, that is 1.06 times higher or lower than nearby one. * each "full step" is 1.12 times higher or lower; since 1.12 = 1.06 * 1.06; * each octave is 12 keys (half step) difference, power(1.06, 12) = 2 * more accurately wrriten as **power(1.0595, 12) = 2** * so to pianist term, roughly 1/16 difference (or accurately 0.0595 difference) is about the minimum frequency difference to make identifiable * related read: * "Equations for the Frequency Table" http://www.phy.mtu.edu/~suits/NoteFreqCalcs.html * Note names and frequencies calculator: http://www.sengpielaudio.com/calculator-notenames.htm =====Frequency and Loudness===== * we often find, low frequency keys like in C1-C3 range, need to press harder than normal to make them sounds loud; while C4-C5 are easy to make it loud; while C6 and higher sound are easier to think it loud. that is because our hearing are sensitive to high frequency sound. * here is a graph of "how loud" is same "loudness" for different frequency, based on our human ear "thinks": http://en.wikipedia.org/wiki/Equal-loudness_contour * Absolute threshold of hearing: http://en.wikipedia.org/wiki/Absolute_threshold_of_hearing * and our human ear use "Phon" as loudness measurement unit, while "dB (sound pressure)" is scentific measurement unit. ===== All musical notation related images ===== * guitar notation (E2 to E6) {{musicwiki:music_guitar_note.gif|}} * octave sheet notation to piano mapping {{musicwiki:music_octave_and_sheet.gif|}} {{musicwiki:music_piano_note_sheet.gif|}} * frequency ration based scale map {{musicwiki:music_piano_freq_note.gif|}}\\ {{musicwiki:music_root_ratio.gif|}} =====Property of Sound - soundwave and wave in general===== * wave has these property (think of those waterwave) * amplitude: height of wave * wave length: distance between 2 high points * period/frequency: circle time, or circle per second * speed = wave length * frequency: how fast high point shift away * soundwave * amplitude: aka "volume, sound pressure" * wave length: * frequency: aka "pitch", range from 10Hz to 1000kHz, * (Human range from 20Hz to 10kHz, piano 20Hz to 8kHz) * speed: in air, about 300m/s; thus also means high frequency sound has short wavelength. * lightwave * frequency: around 10^14Hz, (100THz) * sound to humans * low frequency sound travels further, because of lower frequency are less absorbed when traveling through medium, and also less reflected by medium (better penetration into another medium) * high frequency are more noticeable to humans, due to human ear nature. * sound "loudness" to human are affected by amplitude, frequency, bandwidth and duration. ====== Training Learning in keyboards ====== * Jingle All the way: http://www.youtube.com/watch?v=wC62sFbfuvE * We Wish You a Merry Christmas (Christmas Carol) [Easy Piano Tutorial] http://www.youtube.com/watch?v=8yYzd985jL8 * Silent Night (Christmas Carol) [Easy Piano Tutorial] http://www.youtube.com/watch?v=MVlQkgw7OBM * Jingle Bells (Christmas Carol) [Easy Piano Tutorial] http://www.youtube.com/watch?v=rvGOgEByv0o * Twinkle, Twinkle, Little Star [Easy Piano Tutorial] http://www.youtube.com/watch?v=HbUbYE9zQ6k * Mozart version : http://www.youtube.com/watch?v=KKCsujeeu8o * Guitar version : http://www.youtube.com/watch?v=E5xr_JPGNeA * Happy Birthday to You [Easy Piano Tutorial] http://www.youtube.com/watch?v=RZIy9UdWi7E ====== Songs ====== * Japanese song * Doraemon: http://www.youtube.com/watch?v=SgEzdOao-SQ Some tutorial collection * http://www.youtube.com/user/SynthesiaMidiMusic/videos?view=0 * http://www.youtube.com/playlist?list=PL7AE83358C193E4A5&feature=plcp * ====== Reference Site ====== * https://www.pianochord.org/ ====== Reference Book ====== * Piano Adventures series by Nancy and Randall Faber * John Thompson's easiest piano course * John Thompson's Modern course for the piano * Hanon 60 exercises * Czerny op. 599 * Aural Training in Practise book 1 by Ronald Smith ref: https://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=12&t=47244 ====== 3D visualization ====== * Maya code on generating piano keys * white key on even X axis spacing, and Y axis represent half step and whole step relationship * here is mel code <code javascript> string $names[]={"C","D","E","F","G","A","B"}; string $names2[]={"C","d_b","D","e_b","E","F","g_b","G","a_b","A","b_b","B"}; // -- white keys only $total=7*5; for ($i=0;$i<$total;$i++){ int $n=$i%7; int $oct=$i/7; $tCube=`polyCube -ch on -o on -w 1 -h 1 -d 2 -cuv 4 -n ($names[$n]+"_1")`; float $s=$n; if($n>=3) $s=$s-0.5; move -a $i ($oct*6+$s) $oct $tCube[0]; } for($i=1;$i<=5;$i++){ select -r ("*_"+$i); group -n ("octGrp_"+$i); } // -- white keys and black keys $total=12*5; for ($i=0;$i<$total;$i++){ int $n=$i%12; int $oct=$i/12; $tCube=`polyCube -ch on -o on -w 1 -h 1 -d 2 -cuv 4 -n ("ns1:"+$names2[$n]+"_1")`; string $sn=$tCube[0]; move -a ($oct*6+$n*0.5+$oct) ($oct*6+$n*0.5) ($oct+10) $tCube[0]; if($n>=5) move -r 0.5 0 0 $tCube[0]; if(`gmatch $sn "*_b_*"`) move -r 0 0 -.5 $tCube[0]; } </code> musicwiki/piano.txt Last modified: 2023/01/07 15:50by ying