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Pitches, in hertz, produced by a keyboard instrument with 88 digitals, tuned to equal temperament (12-tone equally tempered octave)
Preliminary Note: On a keyboard instrument like a piano, the thing that you push down with your finger, we call a digital. Each digital is connected to a series of action parts which transfer your finger motion to a swinging "hammer" which strikes either one string, two strings tuned in unison, or three strings tuned in unison. The string or strings vibrate and produce a tone having a specific pitch. Most pianos have 88 digitals which produce 88 distinct pitches.
Computer: Human, working from pitch number 49, to which you have assigned a value of 440 hz, I have calculated 88 pitches. Of those, pitch number 40 is 261.625565301 hz.
We can extend our piano keyboard beyond the usual 88 pitches. If you entered a pitch number above 88 or below 1 (you can enter negative numbers, or, for a half-tone below pitch 1, enter 00), you won't see its frequency listed above, but I can calculate and display its frequency for you here. Pitch number 40 is 261.625565301 hz.
pitch 1 is 27.5000 pitch 2 is 29.1352 pitch 3 is 30.8677 pitch 4 is 32.7032 pitch 5 is 34.6478 pitch 6 is 36.7081 pitch 7 is 38.8909 pitch 8 is 41.2034 pitch 9 is 43.6535 pitch 10 is 46.2493 pitch 11 is 48.9994 pitch 12 is 51.9131 pitch 13 is 55.0000 pitch 14 is 58.2705 pitch 15 is 61.7354 pitch 16 is 65.4064 pitch 17 is 69.2957 pitch 18 is 73.4162 pitch 19 is 77.7817 pitch 20 is 82.4069 pitch 21 is 87.3071 pitch 22 is 92.4986 pitch 23 is 97.9989 pitch 24 is 103.826 pitch 25 is 110.000 pitch 26 is 116.541 pitch 27 is 123.471 pitch 28 is 130.813 pitch 29 is 138.591 pitch 30 is 146.832 pitch 31 is 155.563 pitch 32 is 164.814 pitch 33 is 174.614 pitch 34 is 184.997 pitch 35 is 195.998 pitch 36 is 207.652 pitch 37 is 220.000 pitch 38 is 233.082 pitch 39 is 246.942 pitch 40 is 261.626 pitch 41 is 277.183 pitch 42 is 293.665 pitch 43 is 311.127 pitch 44 is 329.628 pitch 45 is 349.228 pitch 46 is 369.994 pitch 47 is 391.995 pitch 48 is 415.305 pitch 49 is 440.000 pitch 50 is 466.164 pitch 51 is 493.883 pitch 52 is 523.251 pitch 53 is 554.365 pitch 54 is 587.330 pitch 55 is 622.254 pitch 56 is 659.255 pitch 57 is 698.456 pitch 58 is 739.989 pitch 59 is 783.991 pitch 60 is 830.609 pitch 61 is 880.000 pitch 62 is 932.328 pitch 63 is 987.767 pitch 64 is 1046.50 pitch 65 is 1108.73 pitch 66 is 1174.66 pitch 67 is 1244.51 pitch 68 is 1318.51 pitch 69 is 1396.91 pitch 70 is 1479.98 pitch 71 is 1567.98 pitch 72 is 1661.22 pitch 73 is 1760.00 pitch 74 is 1864.66 pitch 75 is 1975.53 pitch 76 is 2093.00 pitch 77 is 2217.46 pitch 78 is 2349.32 pitch 79 is 2489.02 pitch 80 is 2637.02 pitch 81 is 2793.83 pitch 82 is 2959.96 pitch 83 is 3135.96 pitch 84 is 3322.44 pitch 85 is 3520.00 pitch 86 is 3729.31 pitch 87 is 3951.07 pitch 88 is 4186.01
Computer: I work fast, don't I? Most likely I calculated and displayed all 88 frequencies in less than a second. You know how long this would have taken you, human, if you had had to do it with an electronic calculator and a typewriter or pen? Probably a good few hours. You know how long this would have taken you, if you had had to do it without any kind of electronic or mechanical computing device (not even a slide rule) to aid you? Probably several days of working 8 hours per day.
Computer Programmer: The php program I wrote, copyright 2006 by Theodore Zuckerman, calculates all the frequencies, all the pitches, on a piano tuned to equal temperament, that is, on a piano tuned so as to have each octave be a 12-tone equally-tempered octave. Each pitch is equal to the previous pitch, multiplied by the 12th root of two. It is a server-side program, so you won't see it in the source document for this page. The calculations are done on the web server computer, then just the results are sent to your computer. If you are interested in seeing a copy of the source code, please contact me. I should add that to start with one pitch, multiply it by the twelfth root of two, note the result, multiply that by the twelfth root of 2, note the result, multiply that by the twelfth root of 2, etcetera, across the entire keyboard, would result in the accumulation of imprecision. To avoid that, I calculated each pitch using a formula that related it directly to pitch 49, the standard reference pitch.
Here is a sketch of a piano keyboard with each digital numbered (from 1 to 88) and labeled with the pitch it produces. I made that sketch around 1976, before I owned a personal computer. I calculated the frequencies, one by one, with an electronic calculator. Rather than provide you with a color-corrected image, I have provided an image which allows you to see how the paper that the digitals were drawn on, has yellowed more than the paper, with the frequencies on it, that I pasted it to. I'm not sure whether this is an unnecessarily nasty-looking bit of business, or a charming bit of memorbilia that adds character to my web site.
Note that the digitals, in addition to being named with numbers from 1 to 88, are named with letter names, followed by an octave number from 0 to 8. Thus synonyms for note 49, are A49, and A4. The 4 in A4 means that this is the A in octave 4. There are 8 A's on an 88-note piano, A0, A1, A2, A3, A4, A5, A6, and A7.
Beat Rates for Every Equally Tempered Interval across the keyboard — Get your Excel Spreadsheet
My Beat Rate Calculator Spreadsheet file for Excel, which you can purchase, not only shows the fundamental frequency of every pitch from 1 to 88, but it also shows the coincident partials and their beat rates, for just about every equally tempered interval that a piano tuner might ever want to listen to, when perfecting their tuning — fifths, fourths, major thirds, minor thirds, major sixths, minor 6ths, and 10ths. You can see the coincident partials and beat rates for each interval, from one end of the piano to the other, not just for an octave of intervals in a "temperament octave." (scroll down a bit to see about getting a free sample). For example you will have 84 beat rates for 84 major thirds, progressing from A0-C1 to A7-C8, not just the 9 major thirds progressing from F3-A3 to C#4-F4. The chart will show you whether any interval is expanded or contracted. And if there is some interval that you want a chart of partials and beat rates for, that the spreadsheet file doesn't already have, I will add that on, at no extra charge. Here is how to purchase: scroll down.
Instant update of all pitches and beat rates
If you change the value of A49 (A4) from 440 hz to a lower or higher value, you will be able to see how all the fundamental pitches, and by consequence all the coincident partials, and all their beatrates, will be changed — all 7 pages of changes (each interval has its own page), each page with 73 to 85 intervals, 146 to 170 partials, and 73 to 85 beat rates — are re-calculated almost instantaneously. This could be useful if you want to tune an instrument to equal temperament, without first raising its pitch to standard pitch, or if you want to tune a historic intrument that was designed to sound and work best at a lower pitch, such as A49 = 430 hz, instead of at the contemporary standard of A49 = 440 hz.
What if...? Using Excel's Goal Seek feature, you can create a "what if" scenario: what if any pitch of any note, or any beat rate of any interval, were changed: (1) what would the value of A4 have had to be, in order to get that pitch, or that beat rate — and (2) what would be all the fundamentals, partials, and beat rates, for every interval, given the new value of A?. For example change the beat rate of the Major third, F3 to A3, from 6.929 hz to 7.0 hz, and see A4 changed from 440 hz to 444.48 hz, and at the same time, see what every fundamental will be, and what every partial, and every beat rate of every interval, will be, given a reference tone of 444.48, instead of 440. Almost instantaneously! Computers never cease to amaze me, doing even those simplest things they do. Like make and display hundreds of calculations in less than a second.
Easy to use: The spreadsheet has been carefully planned so that it is easy to read and understand. All the Excel formulas are visible so you can see how the fundamentals, partials, and beat rates are being calculated. Many of the variables are referred to by names, in addition to being referred to by their Excel row and column identifiers. This makes the formulas easier to understand. Scroll down a bit to see how to purchase.
A1 to C88: As you know, in addition to checking octaves, from end to end, you can check any interval, below and above your "temperament octave." From A1 to C88. Or using an alternate nomenclature, from A0 to C8. Checking more intervals enables you to exert more control over how far you stretch your octaves, and to get a more precisely stretched, better-sounding tuning, better-sounding piano. This is why I always check progressive 3rds, 10ths, 17ths, etcetera, far beyond the temperament octave. You could check your 10ths by using the chart for Major 3rds; just find the third that ends at the same note; however I included a chart for 10ths to make your life a little easier. You can check 11ths, 12ths, and 17ths with the charts for 4ths, 5ths, and Major 3rds or 10ths.
Normally $19.00. Order between 2011 Dec 22 and 2012 Jan 30 and you can have the beat rate spreadsheet for just $12.50
For just $19 $12.50 you can have this handy spreadsheet. You can pay with PayPal or credit card. Just click the donation button near the top of this page and let me know where to email your spreadsheet. Or email me if you have any questions.
Beat Rate Spreadsheet — Free Sample
You can get a free sample here of the beat rate calculator spreadsheet. Save the *.zip file to a directory or to your desktop, then unzip it. Open the *.xls spreadsheet file with Excel. Open the *.doc instructions with Word. Note that the free sample is a limited version of the spreadsheet. It shows the partials and beat rates for all the fifths. It shows one octave of fourths. It does not show Major 3rds, minor 3rds, 6ths, or 10ths. Also, the feature to change A49 from 440 hz, to any value you choose, and have the spreadsheet instantly recalculate all the pitches, coincident partials, and beat rates, has been disabled.
Article Coming Soon. Inequality in Dental Care
If you have money, you get to keep your teeth. If you don't have money, you
lose them.
The middle classes and the rich get to keep their teeth;
the poor get told their teeth need to be extracted — even though their teeth do not need to be extracted.
Creating a useful tooth from a decayed, broken, or infected tooth, can be much more expensive than simply removing it. Very little is being said or done about this alarming disparity
— despite the fact that keeping one's teeth past middle age
results in not only a better quality of life in one's senior years
but there is evidence that it helps people live longer.
