I understand that when western classical music started and the piano
was invented that the natural scale  doh, re, me, fah, so, la, ti, doh was
compromised and the intervals between the notes were altered to a
strict- a tone- a tone- a half tone, in order that doh could start
from anywhere.
   My question is, what were the natural intervals between
doh to re,  re to mi , mi to fah  etc.
Can you describe this in general, eg 'in the natural scale fah is
sharper than it is on the piano'
If Doh is C, can you tell me in Megahertz what each note is?
If DOh is D  Can you tell me in Megahertz what each note is?

Taken from chordwizard.com's fundamentals of music:

The Chromatic Scale

There is a magic number in music, known as the twelfth root of two,
and it has a value of approximately 1.059463. This is the number that,
when multiplied by itself twelve times, gives a result of two.

Why is this important to music?

Remember that with notes one octave apart, the higher note has double
the frequency of the lower note. The range of frequencies in between
is divided up into the twelve steps that give us all of our notes.

The twelfth root of two, when multiplied by the frequency of a note,
gives the frequency of the next note up. After doing this for twelve
notes, you end up with twice the frequency, which is the note one
octave up from the starting note. We could continue in both
directions, until we have calculated the frequency of every musical
note.

The set of all musical notes is called the Chromatic scale, a name
which comes from the Greek word chrôma, meaning color. In this sense,
chromatic scale means "notes of all colors". Remember that colors
also, are made up from different frequencies, those of light waves.

Because notes repeat in each octave, the term "chromatic scale" is
often used for just the twelve notes of an octave.

Chromatic Scale Notes

The table below shows the frequencies of the twelve notes between note
A at 440 Hz, and note A one octave up from it. For higher frequencies,
the frequency steps between notes become larger (in Hertz), but each
step makes an equal change to the pitch we perceive between notes.

Chromatic Scale

A
440.00 Hz

A#/Bb
466.16 Hz

B
493.88 Hz

C
523.25 Hz

C#/Db
554.37 Hz

D
587.33 Hz

D#/Eb
622.25 Hz

E
659.25 Hz

F
698.46 Hz

F#/Gb
739.99 Hz

G
783.99 Hz

G#/Ab
830.61 Hz

A
880.00 Hz

So you see, there's really no real meaning of "sharper" or "flatter"
in deciding note progression in a particular octave, rather, it's just
the mathematical relationship between each note, the magical twelfth
root of two which determines these relationships. As to your last two
questions, Doh and C are defined to be the same, so the relationship
is as above. As for the latter, just shift each note name by two steps
down the ladder, C becomes D, D becomes E, E becomes F#, etc. But that
really isn't what happens, you're just calling notes by a different
name, it doesn't change the relationship. For more info check out the
google links below.


Additional Links:

Fundamentals of Music:
http://www.chordwizard.com/fundamentals_music.html


Search Strategy:

music theory notes hertz:
://www.google.com/search?q=music+theory+notes+hertz


Hope I could help you on your way to understanding music theory!
skermit-ga

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Request for Answer Clarification by declan-ga on 19 May 2002 04:45 PDT

Dear Sir/Madam

I am new to this service. I am wondering if I now have to pay you $10
for this reply?
I am wondering it I will have to pay a further $10 for your reply to
this answer clarification?
I did not feel that my question was answered in any way. Apart from
the info on how the intervals are calculated by my electronic
chromatic  tuner (which is useless to me), you meerly told me the
figures for each note.  I can see this on my tuner.(A is 440 etc)
 My question was, Can you tell me the intervals in hertz of the older,
'Natural' scale which was in existance before 'chromatic' tuning.

Yours Sincerely

Declan

---

Clarification of Answer by skermit-ga on 19 May 2002 10:35 PDT

My apologies declan-ga, I misunderstood the question and took for
granted that the natural scale was the exact same as the chromatic
one. Listed below are the frequencies for the archaic natural scale
which uses whole number ratios multiplied by the base note of the
octave to achieve the frequency of the other notes. This is an
imperfect or dissonant method of composing scales and usually does not
sound right.

Tone/Maj 2nd:
9:8

Major 3rd:
5:4

Perfect 4th:
4:3

Perfect 5th:
3:2

Major 6th:
5:5

Major 7th:
15:8

Octave
2:1

So looking at the frequency chart above, let's start with the C scale.
D is found by multiplying 523.25 Hz * (9 / 8) giving you 588.66 Hz
which is near 587.33 Hz found by the chromatic magical twelfth root of
2 but not quite. E is given by 523.25 Hz * (5 / 4) giving you 654.06
Hz which is near 659.25 Hz. This works ok in the C scale, but other
scales made using the natural progression may not sound as clean. So
mathematically, the differnces between the chromatic scale and natural
scale is that the natural scale uses close approximations in whole
number ratios to find the next note, while the chromatic scale uses
more precise calculations.

Sorry for the snafu, and thank you for your patience.
skermit-ga

---

Clarification of Answer by skermit-ga on 19 May 2002 10:36 PDT

Additional Links For Natural Scale:

Frequency Ratio Chart for Natural Scale:
http://www.digital-daydreams.com/theory/tutorials/octave.asp

I appreciated the quick response to my request for clarification. I
was satisfied by the answer.

All scales represent a compromise of some sort, and there are several
non-equal-tempered scales which have been (and still are) used to give
more nearly "pure" or "perfect" intervals more of the time. To
understand why there is a problem at all, you need to understand what
a "perfect" interval is. If you sound two notes together (either
simultaneously or in close succession), then they will tend to sound
"in tune" if their vibrating frequencies are related by a simple
fraction. When this happens, they share common overtones and will tend
to reinforce each other. The "octave" is the most obvious example (a
ratio of 2). The other important perfect intervals are the fifth
(3/2), the fourth (4/3) (which is really just the inverse of the
perfect fifth: 2/(3/2)), the major third (5/4) and the minor third
(6/5). Major and minor sixths are inverses of major and minor thirds.
The direct answer to your question is obtained by extending this
concept to the major and minor second for which the simplest fractions
are 9/8 and 16/15.

However, the situation is really a bit more complicated. You can also
arrive at any interval by following a "circle of fifths." You can get
the 9/8 for a perfect second by stacking two perfect fifths (3/2 x 3/2
= 9/4), then dividing by two to move the interval back into one octave
(i.e., form a tenth, and then shift down an octave to get a second).
Do this twice to form a major third, and you get 81/64 for a "perfect"
third instead of 5/4. Herein lies the problem of tempered scales. The
"Pythagorean" scale constructed entirely out of perfect fifths gives a
third which sounds quite out of tune, since the ear prefers the
simpler 5/4 ratio to the 81/64 ratio. You can construct a "Just" scale
using the simplest combinations of perfect fifths and thirds, but it
has other problems. Both scales also end up with more than 12 notes,
because there ends up being separate best-fits for "enharmonic" notes
such as c-sharp and d-flat. The perfect "augmented unison" (interval
between c and c-sharp) is 25/24, while the perfect minor second (c to
d-flat) is 16/15. Furthermore, a Just scale has two major seconds, 9/8
and 10/9, which stacked together give a perfect third (9/8 x 10/9 =
5/4). Thus, a Just scale turns out to be awkward to use for real music
as well.

One practical solution is the equal-tempered scale described by
skermit-ga. This evens out all the intervals but makes them all
"wrong" since none of them are pure fractions, being multiples of an
irrational number (the 12th root of 2). It turns out that the fifth is
pretty good, differing from a pure fifth by just slightly over a
"just-noticeable-difference." (The equal-tempered fifth is just a hair
flat from perfect.) The major third is, however, significantly "out of
tune" (sharp).  For the last 200 years or so, though, this has been
the most popular compromise.

If your music tends to stay mostly in one key so that you only need
one of each of the enharmonics (c-sharp and not d-flat, for example),
then there are various so-called "mean-tone" scales which give the
perfect or near-perfect major thirds which are so important to chords.
If you play an instrument with infinitely variable pitch, then all
notes can also be "bent" to improve tuning of important chords.
Performers of "early music" (music written before about 1750) are more
likely to use mean-tone scales and to adjust their thirds to be
perfect whenever possible.

If you want to experiment with alternate temperaments, you can find
electronic tuners with many choices built in. The "best" choice is not
unique, and may depend both on the music you are playing and the
instrument you are playing.

Finally, I would just like to note that rather than using frequencies
to express intervals as you suggest, the most common approach in to
divide the octave into 1200 "cents." This is a logarithmic scale that
works in any "octave." One equal-tempered semi-tone is 100 cents. A
perfect fifth (ratio of 3/2) is 702 cents, 2 cents over the
equal-tempered fifth). A perfect major third is 386 cents, 14 cents
less than its equal-tempered cousin. You will often find tuning
variants described in cents instead of fractions or frequencies.

I hope that very brief introduction to the world of tempered scales
helps you!

Thank you for this clarification. Very helpfull.
Thanks especially to drDavid for volunteering such a detailed
explaination of a complex matter.

You're welcome, please feel free to use the rating button to give me a
rating now that we've completed this question to your satisfaction.
And thank you for giving me the opportunity to learn something too!

Just a minor correction to my Comment above:

Adding two fifths gives a ninth, not a tenth as I stated! Interval
arithmetic is a little weird. You have to subtract one from everything
for normal addition to work. To add two fifths (which really represent
intervals between notes which are each four steps apart) you add 4+4
to get 8, then add one again to get a ninth. It's all because the
counting starts from one (unison) instead of zero.