| T O P I C R E V I E W |
| khanhead |
Posted - 07/26/2003 : 01:36:45
I want to play the music of India. The basic tuning for Indian music is to have each note of the scale sound the sweetest in relationship to the tonic. Each note is perfectly in tune with the tonic. How close is just intonation to that sound? Is it the same or are there still some compromises? If it's different what might the difference be? Anybody know? I'm trying to play the music of India of the guitar. Eventually I will have to have the frets customised. If interested please check out my website: www.music4guitar.com. |
| 3 L A T E S T R E P L I E S (Newest First) |
| Loren Weaver |
Posted - 08/05/2003 : 11:08:46
First a correction. I mentioned people who play in just intonation. I should have said meantone.
( Thank you Loren. There's so much to learn.
I think I understand most of what shared but...
You say "two tones of 100 and 102 Hz would produce a single tone of 101 Hz which gets louder and softer 2 times per second". Then would two tones of 100 Hz and 150 Hz produce a single tone of 125 Hz that gets louder 2 times a second? I thought I was hearing 2 seperate tones.)
Two tones of 100 Hz and 150 Hz would produce a beat frequency (getting louder and softer) 50 times per second. That's too fast to detect. You would hear 2 separate tones, as you did.
(You mentioned "stretching of the scale when tuning pianos". Does that mean different octaves are tuned differently? For example to hear a perfect major third C to E in the 4th octave would mean to lower the E 13 cents in equal temperment. Is it not the same in each octave?)
My understanding is that a piano is tuned by tuning the 4th octave and then tuning up and down by the octaves as defined by the keyboard and eliminating beats. I know that that's how my wife tunes her harp. The usual definition of an octave is two tones whose frequencies are exactly a factor of 2. This definition of the octave leads directly to the definition of a cent which is an interval defined by two tones whose frequencies have the ratio of the 1200th root of 2. The octaves on a piano keyboard, or a guitar, are not exact octaves because of the stiffness of the strings, hence the "stretching" of the octaves. A bowed violin or an organ will have exact octaves.
The data I have been able to find are for a small piano. Starting at the C above middle C and taking that as exactly in tune, and then going up, the next C is sharp by about 2 cents, the C above that by about 10 cents and the top C by about 30 cents. These results (upper half of the keyboard) will be about the same for a grand. For the lower half of the keyboard, middle C will be flat by about 1 cent, or maybe less. The C below that will be flat by about 3 cents, the next C below by almost 10 cents, and the bottom C by 30 cents. The deviations in the lower half of the keyboard will be less for a grand because of the longer strings. This inharmonicity depends on the inverse of the square of the length and also on the fourth power of the string radius (which is why the bass strings are wrapped).
(I am using a Peterson digital tuner set at just intonation and I want to train my ear to recognize an interval that is "in tune". In addition to that I want to position several frets to be more "in tune". I've already studied Indian music with a master teacher for 23 years. I can now hear the out of tuneness of the guitar in standard tuning and for Indian music it is irritating. I am hoping to find out how many Hz "out of tune" certain notes are (in just intonation or equal tempermnent). I found a wonderful freeware program (for mac OS 9) called pitchfork. It simply offers two oscillators with a readout of the frequencies and the ability to change the frequencies( by 1/100,000 of a cent). I suppose I can find each note by ear but some of the intervals (maj 7th -min 6th etc) I find very difficult. I was hoping there might be somewhere something like Peterson's just intonation chart that would show: to get a perfect major 7th in the key of C lower the 7th 2.58 cents.)
I don't think that there is such a chart or that one is possible. It depends on the root note of the interval. I've included a chart below showing the intervals of the various notes in the C major scale from C in just and equal temperament, including enharmonic equivalents. To get the interval that you want, just subtract the number of cents for the lower note from the number for the upper note. In equal temperament, that will always be a whole number times 100, which is why the equal temperament scale is so commonly used. In the just scale it will vary, depending on what you want to do. In fact, your hope to get a single number for a perfect major 7th is based on the idea that there is such an interval, which doesn't vary. That is true only in the equally tempered scale.
Eb is E flat. If you would like these in terms of frequencies, let me know.
Equal
Note Just Temp.
C 0 0
C# 71 100
Db 112 100
D 204 200
D# 275 300
Eb 316 300
E 368 400
Fb 427 400
E# 457 500
F 498 500
F# 590 600
Gb 631 600
G 702 700
G# 773 800
Ab 814 800
A 884 900
A# 977 1000
Bb 1018 1000
B 1088 1100
Cb 1129 1100
B# 1159 1100
C 1200 1200
A note about the just scale. It is based on the idea that the notes in a major triad should have the ratios 4:5:6 and the octave tones should have a ratio of 2. For the key of C, you start with the triad below C which has the C as its fifth and raise the F and A by an octave. Then you take the triad C-E -G and define those frequencies by the ratios 4:5:6. Then take the triad starting on G and give those the ratios 4:5:6. The fifth (D) is then dropped by an octave (the frequency is divided by 2). That gives you the entire diatonic scale. There are three just triads. But as soon as you start to define semitones you run into a mess. Which triads are you going to define as perfect? The note A in the diatonic scale has a frequency which is 5/3 (=1.60) times the frequency of the C. But if you define D-F#-A as a just triad, A will have the frequency of 27/16 (=1.687) times the frequency of the C, which can't be.
Loren
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| khanhead |
Posted - 07/31/2003 : 22:30:40
Thank you Loren. There's so much to learn.
I think I understand most of what shared but...
You say "two tones of 100 and 102 Hz would produce a single tone of 101 Hz which gets louder and softer 2 times per second". Then would two tones of 100 Hz and 150 Hz produce a single tone of 125 Hz that gets louder 2 times a second? I thought I was hearing 2 seperate tones.
You mentioned "stretching of the scale when tuning pianos". Does that mean different octaves are tuned differently? For example to hear a perfect major third C to E in the 4th octave would mean to lower the E 13 cents in equal termerment. Is it not the same in each octave?
I am using a Peterson digital tuner set at just intonation and I want to train my ear to recognize an interval that is "in tune". In addition to that I want to position several frets to be more "in tune". I've already studied Indian music with a master teacher for 23 years. I can now hear the out of tuneness of the guitar in standard tuning and for Indian music it is irritating. I am hoping to find out how many Hz "out of tune" certain notes are (in just intonation or equal tempermnent). I found a wonderful freeware program (for mac OS 9) called pitchfork. It simply offers two oscillators with a readout of the frequencies and the ability to change the frequencies( by 1/100,000 of a cent). I suppose I can find each note by ear but some of the intervals (maj 7th -min 6th etc) I find very difficult. I was hoping there might be somewhere something like Peterson's just intonation chart that would show: to get a perfect major 7th in the key of C lower the 7th 2.58 cents.
Thanks again for you time. P.S Have you checked out melatonin for sleep?
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| Loren Weaver |
Posted - 07/27/2003 : 17:51:22
I"ll take a whack at this one.
DISCLAIMER: It's been thirty years since I did Physics of Music, so I'm likely to remember some things incorrectly. Worse, I'm not at home and can't look anything up for another week, and I'm writing this at 1 a.m. because of severe insomnia.
DEFINITIONS:
Beat Frequency - When two tones of almost the same frequency are played together (close physical locations, equally loud, and other simplifying assumptions) what is actually heard is a single tone whose frequency is the average of the two. This single tone gets louder and softer (beats) at a frequency which is the difference of the two. So two tones of 100 and 102 Hz would produce a single tone of 101 Hz which gets louder and softer 2 times per second.
Harmonic Series - Ideally, a musical tone at a certain (fundamental) frequency is actually composed of a series of sine waves at that frequency, twice that frequency, three times that frequence, etc. So a 100 Hz tone actually consists of sine waves at 100, 200, 300, 400, 500, 600, ... If you learn the math for this you discover that these numbers actually only work for organ pipes and bowed string instruments. Guitars, pianos, harps, gut-buckets, etc. don't qualify because the tone changes contually in loudness. There is a series of frequencies produced, but the frequencies are shifted upward (that's why people talk about the stretching of the scale when tuning pianos). And the amount of the shift isn't constant. It increases as the multiplier of the fundamental gets larger.
JUST SCALE
Now, to the just scale. In my experience, it's often called the perfect scale, but not the by people who have to tune the instruments. An octave is ideally defined as two notes whose frequencies have a ratio of 2, e.g. 100 Hz and 200 Hz. This is probably what you mean by a sweet sound. The series of frequencies from the lower tone would be 100, 200, 300, 400, 500, 6,, etc. The upper tone would have the frequencies 200, 400, 600., and so on. The harmonics of the upper cooincide exactly with the harmonics of the lower. There are NO beat frequencies.
In the just scale, a perfect fifth is defined as two frequencies with a ratio of 3 to 2, e.g. 100 and 150 Hz. The lower would have the series 100, 200, 300, 400, 500, 600, etc. The upper would have the frequency series 150, 300, 450, 600, ... Some of the frequencies from the upper note coincide exactly with the frequencies of the lower. The others are two far away from the frequencies of the lower to produce a beat frequency. So again, a "sweet sound"
All of the intervals in the just scale are defined in terms of small, whole numbers, so the harmonics of the upper note in an interval coincide, as much as possible, with the harmonics of the lower note. It's "perfect" if you ignore the fact that you get two different sizes of semi-tones and a few other minor problems.
PROBLEMS: The first is the two different sizes of semi-tones.
A larger problem is that the frequencies which give you the proper intervals for one tonic are quite wrong for most other tonic notes. You can't switch keys without completely re-tuning. There is a story that J.S. Bach once was hired to play a dedicatory recital on a new organ whose maker used a tuning scale, almost just, of which Bach did not approve. He transposed the pieces he played into keys which didn't work with the tuning of the organ and, according to a contemporary source, it sounded so awful that the organ builder "ran from the church with his hands over his ears".
A worse problem is that the series of frequencies produced by a plucked or struck instrument don't have the simple numerical relationships of the harmonic series. They vary from them by amounts which depend on the square of the frequency multiplier (1, 2, 3, ...), the length of the string, the thickness (flexibility) of the string, etc. (Yet another definition - these frequencies are usually referred to as partials, in my experience, because they aren't the same frequencies as harmonics would be.) So, if you play a 100 Hz tone, it might be made up of frequencies 100, 202, 304.5, 408... Now you play the octave. In order for the fundamentals to match, you have to play 202 Hz. But that will produce a set of frequencies which might be 202 405, ... and you will get beat frequencies among the partials of the two notes which will produce an out of tune sensation.
CONCLUSION - I haven't seen a sitar up close, but I have the impression from looking at pictures that they have strings which are quite long. This greatly reduces the frequency differences between the partials (this is the reason why piano concerts are played on full size grands, rather than baby grand pianos) something you cannot reproduce on a guitar.
I don't know if Indian music is played with different tonic notes, but if it is, and you have your guitar frets adjusted, you will need several guitars. You can use one guitar for several keys, how many I don't remember, but certainly not for all.
Any tuning is going to be a compromise, even if you hang the "perfect" label on it. The people I know who play harpichords in just tuning tune one octave to the just scale and then tune up and down from there by octaves. They're compromising by accepting the stretching of the octaves imposed by the lengths of the strings.
Probably the best solution if you really want to play Indian music is to find a sitar and a good teacher, though "best" solutions can be highly impractical.
I'm afraid that I got carried away, and wrote a bunch of stuff which probably won't help too much. I'm also afraid that this sounds like "you can't get there from here". Actually, you probably can't, but you may be able to get close enough to satisfy yourself. It just depends on how much compromise you are willing to accept.
Loren Weaver
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