Author Topic: Guitar Tunings  (Read 7262 times)

satish

  • Newbie
  • *
  • Posts: 3
    • View Profile
Guitar Tunings
« on: May 18, 2009, 09:39:44 AM »

In most cases, for each separate tuning below, I have listed:
The notes of the retuned guitar's open strings, going from fat string (string 6) to thin string, (string 1) so that standard tuning is represented EADGBE.
A line or two of descriptive text.
An example chord shape for that tuning: obviously there's going to be a an awful lot of feasible chord shapes for most tunings but the fun is in finding them!
Generally you won't know what the notes are for a made-up chord shape in a strange tuning, so let it be a stimulus to creativity-
if you don't know what the notes are then you're forced to really focus on the sounds and what those sounds evoke...
TUNING: DADGBE
Used by: Ry Cooder, John Renbourn and others
Suggestions: known as 'dropped D' tuning- provides a deeper bass for songs in D
Example chord shape:
1st string 2nd fret
2nd string 3rd fret
3rd string 2nd fret
(and the rest open)


rotating dot TUNING: DADGBD
Used by: Bert Jansch, Roy Harper, many others
Suggestions: lots of folk things
Example chord shape
second string third fret
third string second fret
(and the rest open)


rotating dot TUNING: DADGAD
Used by: invented by Davy Graham, used by many folk guitarists
Suggestions: if you can be patient enough to find lots of shapes which use strings 6, 5, and 3, a 'sound' useful for playing with fiddle/mandolin etc can be had
Example chord shape
fifth string second fret
sixth string second fret
(and the rest open)


rotating dot TUNING: DADF#AD
Used by: Bert Jansch, John Martyn, Robin Williamson, Missisipi John Hurt
Suggestions: this one is used extensively for blues playing, but it can also be a 'mellow' tuning as well-
try this shape for example (which is not a D, G or A chord)...
Example chord shape
third string eighth fret
fifth string ninth fret
sixth string ninth fret
(and the rest open)


rotating dot TUNING: DGDF#AD
Used by: Davy Graham, various blues artists
Suggestions: using the open G on the 5th gives you scope for a variety of G shapes that you can't get in the 'full open D' tuning above
Example chord shape
fifth string second fret
(and the rest open)


rotating dot TUNING: DGDGGD
Used by: the author (Richard Ebbs)
Suggestions: make the most of the resonant second string by keeping it as a drone across different shapes
Example chord shape
fourth string fourth fret
fifth string fourth fret
sixth string second fret
(and the rest open)


rotating dot TUNING: DBDF#BD
Used by: the author (Richard Ebbs, once upon a time)
Suggestions: a very 'minor' tuning which can get a bit much very easily!
Example chord shape
second string second fret
third string fifth fret
fourth string fourth fret
(and the rest open)


rotating dot TUNING: CGDGCD
Used by: the author (Richard Ebbs)
Suggestions: this is a good one. There's a lot of potential here
Example chord shape
first string tenth fret
fourth string tenth fret
fifth string ninth fret
sixth string ninth fret
(and the rest open)


rotating dot TUNING: CGCGCE
Used by: John Martyn
Example chord shape
fourth string fifth fret
fifth string second fret
sixth string fifth fret
(and the rest open)


rotating dot TUNING: FGDGG#Eb
or (the same tuning a semitone lower)
rotating dot TUNING: EG#C#F#GD
Used by: the author (Richard Ebbs)
Suggestions: this was discovered by putting a Ravi Shankar (sitar) raga on the record player and twiddling the machine heads until all strings were related to the music on record- consequently some very Eastern sounds can be had
Example chord shape
first string twelfth fret (harmonic)
second string twelfth fret (harmonic)
third string twelfth fret (harmonic)
fourth string twelfth fret (harmonic)
fifth string twelfth fret (harmonic)
sixth string twelfth fret (harmonic)


rotating dot TUNING: F#ADGAC#
Used by: the author (Richard Ebbs)
Suggestions: -another 'sitar-scale-like' tuning, discovered by putting another (sitar) raga on the record player and twiddling the machine heads until all strings were related to the music on record- also good for non-sitar things like Mayonnaisse (Smashing Pumpkins)...
Example chord shape
third string second fret
fourth string second fret
sixth string third fret
(and the rest open)


rotating dot TUNING: DF#C#F#AE
Used by: the author (Richard Ebbs)
Suggestions: recently discovered when trying to work out a way to play a song by Eddi Reader called 'It's What You Do With What You've Got'
Example chord shape
first string fourth fret
second string fourth fret
fifth string third fret
sixth string fourth fret
(and the rest open)


rotating dot TUNING: DADEAE
Used by: the author (Richard Ebbs).
This sounds a bit like a Dick Gaughan tuning (but it isn't) -it's a good 'un!
Suggestions: hard to use at first but the plethora of 4ths and 5ths can make for a good sound
Example chord shape
third string second fret
(and the rest open)


rotating dot TUNING: EF#C#GBE
Used by: the author (Richard Ebbs)
Suggestions: I use it for playing the blues/gospel song 'Freedom'
Example chord shape
first string second fret
second string second fret
third string second fret
sixth string second fret
(and the rest open)


rotating dot TUNING: GBDGBD
Used by: possibly Led Zeppelin(?)
Suggestions: you can play Led Zep's 'Rain Song' in this tuning
Example chord shape
second string third fret
fourth string third fret
fifth string third fret
(and the rest open)


rotating dot TUNING: GGDGBD
Used by: Pink Floyd


rotating dot TUNING: EBDGAD
Used by: Crosby, Stills, Nash and Young


rotating dot TUNING: CGCGCE
Suggestions: this tuning is useful if you want your guitar to sound like a mandolin (if you want to play Joni Mitchell's Case Of You on a guitar, for example)
_____________________
high school musical
dermatologist

satish

  • Newbie
  • *
  • Posts: 3
    • View Profile
Re:The Physics Of Sound
« Reply #1 on: May 18, 2009, 09:44:56 AM »
Including Auditory Range, Piano Range, Loudness, Fundamentals Overtones Harmonics and Combination Tones, Beats, Scales, Physiological Effects Of Sound

rotating dot Auditory Range

The range of human hearing is from around 2Hz (2 cycles per second) to 20,000Hz (alias 2KHz) although with age one tends to lose acuity in the higher frequencies so for most adults the upper limit is around 10KHz.
The lowest frequency that has a pitch-like quality is about 20Hz.
A typical value for the extent to which an individual can distinguish pitch differences is 05-1% for frequencies between 500 and 5000Hz. (Differentiation is more difficult at low frequencies). Thus at 500Hz most individuals will be unable to tell if a note is sharp or flat by 2.5-5Hz (ie an 'allowable' pitch range for that note might be from 495Hz to 505Hz maximum.

rotating dot Piano Range

The lowest note on the piano 'A0' is at 27.5Hz, whilst the highest, C8, has a frequency of 4186Hz.
'Concert pitch' has been internationally accepted to be based on a frequency of 440Hz for A4, that is A in the fourth piano octave.

rotating dot Loudness

The quietest sounds that can be heard have a power (measured in Watts) of 10 to the -12 W/m2, whilst the loudest that can be withstood have a power of 1 W/m2. The range is therefore in the order of 10 to the 12, or one million million times.
One decibel is a leap by a factor of 10, so that 0Db is the quietest noise, and 120Db is the loudest.

rotating dot Fundamentals, Overtones, Harmonics And Combination Tones

A note played by most ordinary acoustic intruments is not 'pure', it is in fact a spectrum of frequencies which largely give the note it's characteristic quality. The most significant components of this spectrum are the overtones. Overtones may be inharmonic (ie dissonant, sounding bad) or harmonic (ie consonant, sounding good).
The 'natural overtone series' is the set of harmonics which are particularly prominent in the spectrum of frequencies for notes played by acoustic instruments, and which are also produced by producing waves in a string, where there are successive integer values for the number of 'crests' of the wave, ie 1,2,3,4,5,6,7,8 and so on.
Taking 1 crest as the 'fundamental' note, then 2 produces a frequency an octave above. 3 produces a frequency of a '12th' interval (ie the fifth above the octave). 4 produces a frequency of the second octave above. 5 produces a frequency of a major third above that. 6 produces a frequency of the fifth note in this third octave. 7 produces a frequency of the flattened (minor) seventh in this octave. 8 produces a frequency of the fourth octave. 9 produces a frequency of the second interval in the fourth octave. (et cetera...) The increments between notes becomes successively smaller the further into the series you go, such that soon there are semitone intervals, then quarter- tone intervals, then even smaller fractional intervals.
Also, there is less and less congruence between harmonics and the 'proper' frequencies of the equitempered scale the further into the series you go. (But maybe in some forms of music this is NOT a 'problem' od course).
In the first octave there are no overtones, in the second there is one, in the third there are three, and in the fourth octave there are lots. Overtones which are 'harmonic' are at frequencies equal to the fundamental frequency multiplied by AN INTEGER: eg to find the fifth harmonic of C4, multiply it's frequency of 261.63 by 5: the result is a frequency of 1308.15, close to (but NOT the same as) the frequency of 1318.5 given for E6 (the fifth harmonic of C4) in the equitempered scale.
Overtones which are inharmonic (dissonant) are non-integer multiples of the fundamental frequency.
The constituent frequencies of a note can be ascertained with a wave analyser, which uses the equations developed by Fourier.
If you filter out the fundamental frequency of say 200Hz from a note with overtones of 400Hz, 600Hz, 800Hz and 1000Hz (say, played on a piano), the brain will recreate the fundamental such that it appears to be present even when it is not there.
The ear can be far more subtle still, however, in 'creating' sounds which are not actually there in received vibrations. When a tone of frequency f causes the eardrum to vibrate, the full harmonic series of tones 2f, 3f, 4f, 5f, 6f etc is apparent to the listener, these harmonics being produced by the eardrum itself.
'Combination' tones are additional, new tones produced when two frequencies, f1 and f2 are sounded together, and these combination tones are also 'manufactured' by the ear, this time from combinations of the first frequency and the harmonics of the second, and from the second frequency and harmonics of the first. (Note 3f2 is notation for 'the third harmonic of the second frequency'). The series of combination tones will be:

2f1 - f2 2f2 - f1 3f1 - f2 3f2 - f1
3f1 - 2f2 3f2 - 2f1 4f1 - 3f2 4f2 - 3f1 and so on.

Whereas combination tones can be ascertained by subtracting one value from another, 'summation' tones, also manufactured by the ear, can be found by adding values together, ie

f1 + f2
2f1 + f2 2f2 + f1 3f1 + f2 3f2 + f1
3f1 + 2f2 3f2 + 2f1 4f1 + 3f2 4f2 + 3f1 and so on.

Interestingly, however, the resulting frequencies of these combination and summation tones turn out to be dissonances when played with the notes of the equitempered scale since the frequencies do not exactly match.
It is possible to contrive a scale from difference tones. For example, if C4 is played together with Eb4, then a difference tone of 311.1 - 261.6 = 49.5Hz is created. This value of 49.5Hz is almost the frequency of Ab1. Other notes in the 1st octave can then be contrived in a similar manner from frequencies around the fourth octave.
These sort of difference tones, however, might be called 'first-order' difference tones. A 'second-order' difference tone can be found in the difference between the frequency of a first-order difference tone and the frequency of the fundamental, so that in the example above, we have a first- order difference tone of (almost) Ab4, at 49.5Hz, and subtracting this from the frequency of the fundamental, we find 261.6 - 49.5 = 212.1 which is close to the frequency of Ab3. And so on.
Here are some examples of combination and difference tones that can be demonstrated on a piano.

C4+Eb4 produces Ab1 first-order difference tone.
C4+Eb4 produces Ab3 second-order difference tone.

rotating dot Beats

Beat frequencies are produced when two different sounds are produced which are very close to each other in frequency. In such a situation the crests and troughs of each wave are generally slightly out of phase. But because the two notes have differing frequencies, after a certain repeating interval of time the crests of one wave will be aligned with the crests of the other, when a pulse or beat appears to the listener.
Research has found that any two notes of different frequencies tend to sound good together (ie consonant) if there is an absence of beat frequencies between 8 and 50 Hz produced. Beat frequencies of 2-8Hz have been found to be pleasing, while beat frequencies above that level are generally though to be unpleasant.
The beat frequency produced by any two notes is found by subtracting the value of the higher from the lower, ie Fbeat = Fhigher - Flower. Thus beat frequencies are a subset of difference tones -the 'beat' sensation occurring when the beat frequency value is low, say from 0.5Hz (1 beat every two seconds) to say 20Hz (the lowest frequency that has a pitch-like quality) J.Askill, in 'The Physics of Musical Sounds) says 'in general beat frequencies of 2-8Hz are considered pleasing, whereas if the beat frequency is above 15-20Hz, an unpleasant or dissonant effect is produced'. Personally I am curious about the range in the middle, say 8-12Hz which is also the frequency of 'alpha' brain-waves.

rotating dot Scales

The scale used extensively in the West has 13 notes from octave to octave and 12 intervals. In order for a scale to 'work' there should be:

* a minimum of dissonance when different notes across the whole range of pitches are sounded together
* an effective mapping of the harmonics of low notes onto higher notes, and an effective mapping onto the harmonics of higher notes
* the possibility of key modulation which does not result in further frequency mismatches.

The scale which has been widely adopted to fulfil these criteria is based on mathematics, such that the ratio of the frequency of any note to the frequency of the note a semitone above is constant. This is particularly useful in the extent to which it allows key modulation. However, this 'equitempered' scale is a compromise solution, because the frequency ratios of all intervals except the octave differ slightly from the 'perfect' intervals that the human ear really expects to hear.
The 'exact' interval of a fifth, for instance, is found by multiplying the frequency of the fundamental by 3/2, the fourth is found by multiplying the fundamental frequency by 4/3, and the major third interval is found by multiplying the fundamental frequency by 5/4. (Other intervals involve slightly less obvious fractions).
The problem with a scale built on fractional values like this, however, is that the increments from note to note are not constant (eg 5/4 - 4/3 does not equal 4/3 - 3/2) which creates difficulties when the required key for a piece is different to that of the fundamental from which the scale is constructed. For example, if we move up an octave from C by adding a fifth, and then adding a fourth, then the resulting high C will have a different frequency to that arrived at if our key is F, and we try to arrive at the same high C by adding a major third and then a minor third to that fundamental F. So in the equitempered scale all semitone increments have been 'tempered' such that they are always a little flat, or a little sharp.
The constant value on which this scale is based is 1.0594630915, such that if we call this value S, then the semitone above a fundamental note is found by multiplying the frequency of the fundamental by S to the power of 1.
The second above the fundamental is found by multiplying it's frequency by S to the power of 2, and so on until the octave above the fundamental is found by multiplying it's frequency by S to the power of 12. The value of the constant S is the 12th root of 2, since in order to find the twelve equal divisions between two notes an octave apart, where the frequency of the octave is twice that of the fundamental, the 12th root of 2 is the value we are looking for.
Other divisions of the octave have been proposed, such as a 19-step octave, and a 53-step octave. The maths for these 'works' although these 'scales' may be harder to use effectively. The maths for the 53-division scale is particularly elegant in fact, and closer to a 'perfect' musical scale than the 13-note scale which we currently use. (In that case the constant value for each successive interval is found by using the 53rd root of 2, ie 1.013164143).
The 'exact' scale, built on the 'perfect' intervals that the ear expects to hear, has much to recommend it if one key is kept to. This scale, however, has fifteen intervals and fourteen notes, since in the first octave there are all the notes of the equitempered scale (at slightly different frequencies) but there is also both a 'major whole tone' and a 'minor whole tone', and both an 'augmented fourth' and a 'diminished fifth'. (In the second octave there is both an 'augmented octave' and a 'dimished ninth' and also both an'augmented eleventh' and a 'diminished twelfth'). Thus successive octaves above the fundamental differ from each other in the way that they are put together. Furthermore, however, when we look at the extent to which the frequencies of harmonics of exact-scale notes match notes higher up in the exact scale, we see that we can list the intervals octave, fifth, fourth, major third, major sixth, minor third, minor sixth in terms of increasing dissonance, so that in the case of the minor sixth, if we look at all harmonics up to the twelfth, only one 'matches'.
The mathematical elegance of the 53-division scale should make it a more appropriate scale for dealing both with key modulation, and a preoccupation with harmonics.
The 53-interval scale uses eneven 'chunks' of these 53rd-of-an-octave division to create the notes of the diatonic scale. The size of each incremental chunk is as follows:


satish

  • Newbie
  • *
  • Posts: 3
    • View Profile
Re: Classical Music
« Reply #2 on: May 18, 2009, 09:52:43 AM »
"...Roi Aloni entertains and teaches with music, stories about composers and their works, both classical and modern. His knowledge of these subjects is extensive and his delivery is clear and often very humorous. His virtuosity on the piano never ceases to amaze...”

“...Roi Aloni brings to the listener a world of knowledge in various categories of music, from classical through all its genres to Frank Sinatra and Edith Piaf. His lectures combine artistic, historical and sociological aspects while linking them to the music world. He introduces to the audience various musical styles, composers, and performers from Bach to the Beatles and in between…. Roi has tremendous charisma. The audience immediately responds to him. He presents his lectures in a fluent, articulate way building them up with fascinating approach. During the lectures Roi plays recordings of musical masterpieces and also demonstrates on the piano with his astonishing skills..."

Wayne Higgins

  • Hero Member
  • *****
  • Posts: 616
    • View Profile
    • Oenyaw
Re: Guitar Tunings
« Reply #3 on: May 20, 2009, 10:32:46 AM »
Great work.  I really enjoy experimenting with different guitar tunings.  Joni Mitchell is a master on the subject.  I have yet to see any book written to deciefer her.  Jimmy Page is also a good one at guitar tunings.   He wrote some tunings out during the tour in which he broke his finger and had to play everything with two fingers.  How he came up with the variation on "The Rain Song" baffles me.  The only one I ever came up with (which I need to write down) was a tuning with open 7th chords, wanting a tuning to play slide guitar for The Beatles "I Call Your Name."

Great information Satish.  Thanks :).
So, I'm a "Sr Member", huh?  In June it's SENIOR DISCOUNT TIME!!!
http://oenyaw.net/
http://oenyaw.blogspot.com/

IamBetaCloud

  • Newbie
  • *
  • Posts: 33
    • View Profile
    • beta cloud on myspace-
Re:The Physics Of Sound
« Reply #4 on: July 08, 2009, 10:52:46 PM »
Including Auditory Range, Piano Range, Loudness, Fundamentals Overtones Harmonics and Combination Tones, Beats, Scales, Physiological Effects Of Sound

rotating dot Auditory Range

The range of human hearing is from around 2Hz (2 cycles per second) to 20,000Hz (alias 2KHz) although with age one tends to lose acuity in the higher frequencies so for most adults the upper limit is around 10KHz.
The lowest frequency that has a pitch-like quality is about 20Hz.
A typical value for the extent to which an individual can distinguish pitch differences is 05-1% for frequencies between 500 and 5000Hz. (Differentiation is more difficult at low frequencies). Thus at 500Hz most individuals will be unable to tell if a note is sharp or flat by 2.5-5Hz (ie an 'allowable' pitch range for that note might be from 495Hz to 505Hz maximum.

rotating dot Piano Range

The lowest note on the piano 'A0' is at 27.5Hz, whilst the highest, C8, has a frequency of 4186Hz.
'Concert pitch' has been internationally accepted to be based on a frequency of 440Hz for A4, that is A in the fourth piano octave.

rotating dot Loudness

The quietest sounds that can be heard have a power (measured in Watts) of 10 to the -12 W/m2, whilst the loudest that can be withstood have a power of 1 W/m2. The range is therefore in the order of 10 to the 12, or one million million times.
One decibel is a leap by a factor of 10, so that 0Db is the quietest noise, and 120Db is the loudest.

rotating dot Fundamentals, Overtones, Harmonics And Combination Tones

A note played by most ordinary acoustic intruments is not 'pure', it is in fact a spectrum of frequencies which largely give the note it's characteristic quality. The most significant components of this spectrum are the overtones. Overtones may be inharmonic (ie dissonant, sounding bad) or harmonic (ie consonant, sounding good).
The 'natural overtone series' is the set of harmonics which are particularly prominent in the spectrum of frequencies for notes played by acoustic instruments, and which are also produced by producing waves in a string, where there are successive integer values for the number of 'crests' of the wave, ie 1,2,3,4,5,6,7,8 and so on.
Taking 1 crest as the 'fundamental' note, then 2 produces a frequency an octave above. 3 produces a frequency of a '12th' interval (ie the fifth above the octave). 4 produces a frequency of the second octave above. 5 produces a frequency of a major third above that. 6 produces a frequency of the fifth note in this third octave. 7 produces a frequency of the flattened (minor) seventh in this octave. 8 produces a frequency of the fourth octave. 9 produces a frequency of the second interval in the fourth octave. (et cetera...) The increments between notes becomes successively smaller the further into the series you go, such that soon there are semitone intervals, then quarter- tone intervals, then even smaller fractional intervals.
Also, there is less and less congruence between harmonics and the 'proper' frequencies of the equitempered scale the further into the series you go. (But maybe in some forms of music this is NOT a 'problem' od course).
In the first octave there are no overtones, in the second there is one, in the third there are three, and in the fourth octave there are lots. Overtones which are 'harmonic' are at frequencies equal to the fundamental frequency multiplied by AN INTEGER: eg to find the fifth harmonic of C4, multiply it's frequency of 261.63 by 5: the result is a frequency of 1308.15, close to (but NOT the same as) the frequency of 1318.5 given for E6 (the fifth harmonic of C4) in the equitempered scale.
Overtones which are inharmonic (dissonant) are non-integer multiples of the fundamental frequency.
The constituent frequencies of a note can be ascertained with a wave analyser, which uses the equations developed by Fourier.
If you filter out the fundamental frequency of say 200Hz from a note with overtones of 400Hz, 600Hz, 800Hz and 1000Hz (say, played on a piano), the brain will recreate the fundamental such that it appears to be present even when it is not there.
The ear can be far more subtle still, however, in 'creating' sounds which are not actually there in received vibrations. When a tone of frequency f causes the eardrum to vibrate, the full harmonic series of tones 2f, 3f, 4f, 5f, 6f etc is apparent to the listener, these harmonics being produced by the eardrum itself.
'Combination' tones are additional, new tones produced when two frequencies, f1 and f2 are sounded together, and these combination tones are also 'manufactured' by the ear, this time from combinations of the first frequency and the harmonics of the second, and from the second frequency and harmonics of the first. (Note 3f2 is notation for 'the third harmonic of the second frequency'). The series of combination tones will be:

2f1 - f2 2f2 - f1 3f1 - f2 3f2 - f1
3f1 - 2f2 3f2 - 2f1 4f1 - 3f2 4f2 - 3f1 and so on.

Whereas combination tones can be ascertained by subtracting one value from another, 'summation' tones, also manufactured by the ear, can be found by adding values together, ie

f1 + f2
2f1 + f2 2f2 + f1 3f1 + f2 3f2 + f1
3f1 + 2f2 3f2 + 2f1 4f1 + 3f2 4f2 + 3f1 and so on.

Interestingly, however, the resulting frequencies of these combination and summation tones turn out to be dissonances when played with the notes of the equitempered scale since the frequencies do not exactly match.
It is possible to contrive a scale from difference tones. For example, if C4 is played together with Eb4, then a difference tone of 311.1 - 261.6 = 49.5Hz is created. This value of 49.5Hz is almost the frequency of Ab1. Other notes in the 1st octave can then be contrived in a similar manner from frequencies around the fourth octave.
These sort of difference tones, however, might be called 'first-order' difference tones. A 'second-order' difference tone can be found in the difference between the frequency of a first-order difference tone and the frequency of the fundamental, so that in the example above, we have a first- order difference tone of (almost) Ab4, at 49.5Hz, and subtracting this from the frequency of the fundamental, we find 261.6 - 49.5 = 212.1 which is close to the frequency of Ab3. And so on.
Here are some examples of combination and difference tones that can be demonstrated on a piano.

C4+Eb4 produces Ab1 first-order difference tone.
C4+Eb4 produces Ab3 second-order difference tone.

rotating dot Beats

Beat frequencies are produced when two different sounds are produced which are very close to each other in frequency. In such a situation the crests and troughs of each wave are generally slightly out of phase. But because the two notes have differing frequencies, after a certain repeating interval of time the crests of one wave will be aligned with the crests of the other, when a pulse or beat appears to the listener.
Research has found that any two notes of different frequencies tend to sound good together (ie consonant) if there is an absence of beat frequencies between 8 and 50 Hz produced. Beat frequencies of 2-8Hz have been found to be pleasing, while beat frequencies above that level are generally though to be unpleasant.
The beat frequency produced by any two notes is found by subtracting the value of the higher from the lower, ie Fbeat = Fhigher - Flower. Thus beat frequencies are a subset of difference tones -the 'beat' sensation occurring when the beat frequency value is low, say from 0.5Hz (1 beat every two seconds) to say 20Hz (the lowest frequency that has a pitch-like quality) J.Askill, in 'The Physics of Musical Sounds) says 'in general beat frequencies of 2-8Hz are considered pleasing, whereas if the beat frequency is above 15-20Hz, an unpleasant or dissonant effect is produced'. Personally I am curious about the range in the middle, say 8-12Hz which is also the frequency of 'alpha' brain-waves.

rotating dot Scales

The scale used extensively in the West has 13 notes from octave to octave and 12 intervals. In order for a scale to 'work' there should be:

* a minimum of dissonance when different notes across the whole range of pitches are sounded together
* an effective mapping of the harmonics of low notes onto higher notes, and an effective mapping onto the harmonics of higher notes
* the possibility of key modulation which does not result in further frequency mismatches.

The scale which has been widely adopted to fulfil these criteria is based on mathematics, such that the ratio of the frequency of any note to the frequency of the note a semitone above is constant. This is particularly useful in the extent to which it allows key modulation. However, this 'equitempered' scale is a compromise solution, because the frequency ratios of all intervals except the octave differ slightly from the 'perfect' intervals that the human ear really expects to hear.
The 'exact' interval of a fifth, for instance, is found by multiplying the frequency of the fundamental by 3/2, the fourth is found by multiplying the fundamental frequency by 4/3, and the major third interval is found by multiplying the fundamental frequency by 5/4. (Other intervals involve slightly less obvious fractions).
The problem with a scale built on fractional values like this, however, is that the increments from note to note are not constant (eg 5/4 - 4/3 does not equal 4/3 - 3/2) which creates difficulties when the required key for a piece is different to that of the fundamental from which the scale is constructed. For example, if we move up an octave from C by adding a fifth, and then adding a fourth, then the resulting high C will have a different frequency to that arrived at if our key is F, and we try to arrive at the same high C by adding a major third and then a minor third to that fundamental F. So in the equitempered scale all semitone increments have been 'tempered' such that they are always a little flat, or a little sharp.
The constant value on which this scale is based is 1.0594630915, such that if we call this value S, then the semitone above a fundamental note is found by multiplying the frequency of the fundamental by S to the power of 1.
The second above the fundamental is found by multiplying it's frequency by S to the power of 2, and so on until the octave above the fundamental is found by multiplying it's frequency by S to the power of 12. The value of the constant S is the 12th root of 2, since in order to find the twelve equal divisions between two notes an octave apart, where the frequency of the octave is twice that of the fundamental, the 12th root of 2 is the value we are looking for.
Other divisions of the octave have been proposed, such as a 19-step octave, and a 53-step octave. The maths for these 'works' although these 'scales' may be harder to use effectively. The maths for the 53-division scale is particularly elegant in fact, and closer to a 'perfect' musical scale than the 13-note scale which we currently use. (In that case the constant value for each successive interval is found by using the 53rd root of 2, ie 1.013164143).
The 'exact' scale, built on the 'perfect' intervals that the ear expects to hear, has much to recommend it if one key is kept to. This scale, however, has fifteen intervals and fourteen notes, since in the first octave there are all the notes of the equitempered scale (at slightly different frequencies) but there is also both a 'major whole tone' and a 'minor whole tone', and both an 'augmented fourth' and a 'diminished fifth'. (In the second octave there is both an 'augmented octave' and a 'dimished ninth' and also both an'augmented eleventh' and a 'diminished twelfth'). Thus successive octaves above the fundamental differ from each other in the way that they are put together. Furthermore, however, when we look at the extent to which the frequencies of harmonics of exact-scale notes match notes higher up in the exact scale, we see that we can list the intervals octave, fifth, fourth, major third, major sixth, minor third, minor sixth in terms of increasing dissonance, so that in the case of the minor sixth, if we look at all harmonics up to the twelfth, only one 'matches'.
The mathematical elegance of the 53-division scale should make it a more appropriate scale for dealing both with key modulation, and a preoccupation with harmonics.
The 53-interval scale uses eneven 'chunks' of these 53rd-of-an-octave division to create the notes of the diatonic scale. The size of each incremental chunk is as follows:




Helmholtz and beyond, very nice.