17 Tone Just Intonation Guitar
Posted 27 August 2013 - 08:49 PM
I designed an 18 tone just intonation guitar and had it fretted (split frets), a cheap classical. I have a new design, 17 tone, and it's SO much less cluttered (I could post the fretfind2d fretboard if anyone is interested. So I want to make this in electric. but the 18 tone was a good stepping stone.
My friend suggested using a fretless and just putting marks where the tones would be. Is that realistic, is my question? Could I play chords easily looking at the fingerboard and using marks?
Whether I go fretless or fretted, I plan on using an 8 string fretboard and only using 6 strings. Also maybe a baritone length, if they make 8 strings that way? Why? Because my hands don't fit "standard sized" guitars, even with straight frets.
I have a lot more to say about my tuning, if there's any interest. I believe I have recovered how ancient 19 tone harpsichords were originally tuned, and also how 12 tone pianos were originally tuned, and double keyboard harpsichords, organs, etc (hint: they did NOT play the same 12 tones on each keyboard).
Some people say they don't really play guitar, feigning modesty. I, on the other hand, really don't, I can barely play, because my fingers don't fit. Call it what you will, but, they say the show must go on, so I've gone ahead and made 6 youtube videos (so far) of my 18 tone guitar, (in slow motion), so people can hear the ancient tuning, and maybe it will inspire some. Here's one I just made ( I look a little frazzled because trying to play these simple chords I kept muting strings...), but you can get an idea of the tones.
I would love to talk about this just intonation tuning if anyone's interested, I think a whole Universe of new/ancient music is waiting to be written, and if you are unfretted already, all you'd need to do is hear the notes...
Posted 27 August 2013 - 09:52 PM
Posted 28 August 2013 - 01:34 PM
Have you ever wondered why an A sharp today, and a B flat, are called the same note? Answer: because originally they were two separate notes, and this has been lost. You can find images online of very old harpsichords, that played 19 tones per octave. The black keys were doubles, so that would allow one to be called a sharp, and one a flat. There were also 2 additional keys where the black keys are missing today, giving a total of 19 notes.
I once tried to calculate what the pitch ratio fractions are for the seven basic musical tones in pure intonation, guessing, and found the same fractions online, attributed to Galileo, as the perfect notes. If you put these to a common denominator, you can then put this all to whole number pitch ratios: 24 27 30 32 36 40 45 and 48 would be the octave. So if 24 beats per second (hertz)is your first note (your 1/1), then your second note would be 27 beats per second and third would be 30 ..., giving the ratios of 8 to 9 to 10, or 1 to 9/8 to 10/8 (5/4)...
When I looked at this and looked at those ancient harpsichords, the light bulb went on, there is room for two whole numbers between 24 and 27, like those two black keys, ... and I am very confident the ancient tuning was based on this pattern. So if you tune your 1/1 note to 24 hertz, or octave of, the rest of the white keys would be tuned to: 27 30 32 36 40 and 45 hertz, and if we include the black keys, the sequence goes: 24 25 26 27 28 29 30 ... hertz, where 25 and 26 are the first set of double black keys, etc. This pattern holds up till note 32, the fourth, and it is a perfect fourth, 32 to 24, or 4 to 3 pitch ratio to the first note. If you play with this simple math, you will see that after 32hertz, the following notes are open to discussion, what would they be? I believe I have it figured out, what the standard tuning was for the rest in 19 tone. My new guitar will play these same 19 notes, except for the two notes where there are not black keys today (between B and C, and between E and F). I think you can get those by bending strings of needed...
Just to give you the whole tuning here, it will be, in hertz: 24 25 26 27 28 29 30 32 33 1/3 34 2/3 36 37 1/3 38 2/3 40 42 45 and 48 hertz is the octave. This tuning is based on hertz, clocks, because, for one thing, I believe the Ancients synchronized music with clocks and the Earth's spin, but I won't go into that here.
Playing with these numbers, I realized that 12 tone pianos were really meant to be tuned, originally based on this "harmonic series" principle, but, having just 12 tones, they lend themselves to a different portion of the harmonic series. I am very confident they were originally tuned with the first C note at 32 hertz. Because this fits with the one black key pattern. The octave before 32, not played on a piano, but it would be 16 hertz, and so the first three notes, would be, on the white keys, 16 18 and 20. This leaves room for black keys in between, tuned to 17 and 19, that would cover the first five notes, if you follow what I'm saying. 16 17 18 19 20
Not coincidentally, the first C note on today's pianos is the closest note to 32 hertz in "standard tuning". Btw I think "equal temperament" tuning is a mathematical hoax. where all the music is blurred, all the nuances and crispness of sound has been ironed out. Which gets to another riddle. Playing in a different key today has no real meaning, except for a different starting point. Each "key" has the same flavor. But it used to be, each key had its own flavor, some were happy, some eerie, some sad... and all harmonically related.
Here is the original tuning, I believe, for 12 tone pianos: 32 34 36 38 40 42 45 48 51 54 57 60 and the octave is 64 hertz. So the white keys would be, the C major scale, 32 36 40 42 48 54 60 hertz. This gives the pitch ratios of 1/1 9/8 5/4 21/16 3/2 27/16 15/8 and 2/1.
The G major scale gives Galileo's tuning of 1/1 9/8 5/4 4/3 3/2 5/3 15/8.
And you have all these other beautiful scales, each with their own flavor, and based on the harmonics of the Universe. So when you hear "just intonation" only gives you one playable scale, that is not true, they are just using the wrong numbers. Here's what the D major scale in this 12 tone tuning would be: 36 40 45 48 54 60 68, which gives the pitch ratios: 1/1 10/9 5/4 4/3 3/2 5/3 17/9.
One last point here. You'll notice modern pianos look terrible, the key layout. At the left end, there is a lone black key, the rest of the keyboard has pairs or triplets of black keys. It looks off. This ugly set up allows for the first note on pianos to be the A key.
How did this come about? I believe what they have done is removed two bass keys from the left end. If you added back these two keys, one white, one black, then the first black keys would be a pair, and this fits pleasingly with the rest of the keyboard layout. And this makes the very first note the G note, which just so happens to be the closest note in standard tuning to that magical number 24 hertz. Online you can find a picture of this beautiful 1877 piano in France, that has the two extra keys, a 90 key piano instead of today's 88 keys. It looks great. Check it out, I think it's called the Frederick collection or something like that. So it looks like the note names were moved over one character from what hey originally were, so G used to be called A, and started the piano. Does this explain the song Doe a deer, fitting the D sound with what is now called C but used to be the D note?
Lastly, harpsichords with double keyboards, and organs... I believe they were originally tuned with one keyboard with the 12 tone tuning described above for pianos, and the second keyboard would be tuned offset, so 32 34 36 38 40 is do re mi on one, with black keys, and 33 35 37 39 41 hertz is do re mi on the other, allowing for, I would assume, some pretty interesting and amazing music.
hope I got the numbers right here
Posted 28 August 2013 - 06:56 PM
I know most people are used to working with fractions or cents, but using these whole numbers greatly simplifies things. Think of note G as 24, A is 27, B is 30, C is 32, D is 36, E is 40. (F is 42 on a piano, but a 19 tone harpsichord would have had the white keys set up with the first note as G, not C like pianos, so it raises a question of how they notated the 7th tone... also keep in mind flats and sharps were different keys, and also remember the note names have been moved over a character... is this getting confusing? Hopefully not.)
The guitar is oriented around the G string, the 1/1, which is tuned open to 192 hertz (an octave of 24 hertz). I plan on using a drpp D tuning, so the deepest string will be tuned to D instead of E, and will be 72 hertz (octave of 36 hertz). (This allows a perfect just interval of 2/3 with the next deepest string, the A, tuned to 108 hertz (octave of 27 hertz). E would have been 80 hertz (octave of 40 hertz), so the open strings from E to A wouldn't be a pure interval. Also the frets work out better with the deepest string tuned to D, not E. The E string, highest string, is tuned to 320 hertz (an octave of 40 hertz.) The B string is tuned to 240 hertz.)
So all open string intervals are just from one string to the next, going from deepest to highest, a perfect fifth (3/2), 3 perfect fourths (4/3), a perfect third (5/4), and a perfect fourth (4/3).
high E = 320 hertz B = 240 G = 192 D = 144 A = 108 low D = 72 hertz
So it's standard strings, but the low E is tuned to D at 72 hertz.
The 18 tone tuning on my classical has this same basic tuning, but I had tried to keep all the notes in whole hertz numbers, so I ended up adding a note. I now realize, having worked with these numbers for a while, the music will be more flowing tuned as outlined here in 17 tone.
I'm sure some will disagree but I think ancient 17 tone music was originally based on this tuning. And as I say, 19 tone was the same tuning with two passing tones added, 31 hertz and 46 1/2 hertz probably (put to appropriate octaves).
So I've outlined four ancient tunings here, 12 tone, 17 tone, 19 tone, and 24 tone (double keyboards). They are all anchored with a perfect doe ray mi 8 9 10 ratio, and all contain Galileo's 7 notes. 17/19 tone is oriented around a 1/1 of 24 hertz, and 12/24 tone are oriented around a 1/1 of 32 hertz.
Surely you can see this!
think I'm done talking
Posted 28 August 2013 - 07:19 PM
Posted 29 August 2013 - 04:04 PM
Recall, the 12 notes, starting with today's C, in beats per second, are: 32 34 36 38 40 42 45 48 51 54 57 60
So let's look at the C sharp scale, I'll list all the relationships using C sharp (34 hertz) as the 1/1, the reference pitch, and you can see all these beautiful just intervals emerge. Btw we can call C sharp the 17 scale, because it's based on the prime number, 17 (34/2):
34 36 38 40 42 45 48 51 54 57 60 64 beats per second
17/17 18/17 19/17 20/17 21/17 45/34 24/17 3/2 27/17 57/34 30/17 32/17 and the octave is 34/17
the standard 7 note major scale,from these numbers, would be:
17/17 19/17 21/17 45/34 3/2 57/34 32/17
Note how the second note approximates 18/16 (9/8), the third approximates 20/16 (5/4), the fourth approximates 44/33 (4/3), the fifth is 51/34 which is a perfect 3/2 ratio (who knew 51 is not a prime number, but 17 times 3?), the sixth approximates 55/33 (5/3) and the seventh is near 30/16 (15/8).
Or you could say the second approximates 20/18 (10/9), etc. But 19/17 is not 10/9 nor 9/8, it's its own number, and C sharp is its own scale. Based on the numbers of the Universe. Not some weird numbers with 25 decimal places... like "equal temperament" cents...
Let's look at another scale, the F sharp scale. This is based on the number 45. If we make 45 our reference note, the 1/1, then the 12 notes line up in hertz as follows: 45 48 51 54 57 60 64 68 72 76 80 84, and we get these pitch ratios:
15/15 16/15 17/15 18/15(6/5) 19/15 20/15(4/3) 64/45 68/45 24/15(8/5) 76/45 16/9 28/15
Looks like a beautiful minor scale sitting in there...
If anyone wants to do the math for the rest of the scales, I think you will be amazed at all the just intervals you will find.
The proof is in the listening. Go check it out and see what you think. I had a piano tuner come out to retune my parents' old piano a while back, to this tuning but he never showed up... and I talked to a few more, they seem perplexed...
How can this be? I posted the basics here in another forum once and the mathematician/musicians seemed unfamiliar with this tuning. There's that free scala program online that lets you tune things as you wish... which is nice, but you have to click the mouse to play notes, not the same as having a real keyboard or guitar in hand.
If you check this stuff out and agree this it the original tuning for pianos, and it's beautiful, all these amazing just scales..., then you will have to ask why no one knows about this. At least I don't seem to find anyone. Questions have answers. Good luck.
Posted 30 August 2013 - 05:13 PM
C (32 htz) 16/16 17/16 18/16 (9/8) 19/16 20/16 (5/4) 21/16 22.5/16 24/16 (3/2) 51/32 27/16 57/?32 30/16 (15/8)
Csharp (34) 17/17 18/17 19/17 20/17 21/17 22.5/17 24/17 51/34 (3/2) 27/17 59/34 30/17 32/17
D (36) 18/18 19/18 20/18 (10/9) 21/18 (7/6) 22.5/18 (5/4) 24/18 (4/3) 51/36 (17/12) 27/18 (3/2) 57/36 (19/12) 30/1 (15/8)
Dsharp (38) 19/19 20/19 21/19 22.5/19 24/19 51/38 27/19 57/38 (3/2) 30/19 32/19 34/19 36/19
E (40) 20/20 21/20 22.5/20 (9/8) 24/20 (6/5) 51/40 27/20 58/40 30/20 (3/2) 32/20 (8/5) 34/20 (17/10) 36/20 (9/5) 38/20 (19/10)
F (42) 21/21 22.5/21 (15/14) 24/21 (8/7) 51/42 27/21 (9/7) 58/42 30/21 (10/7) 32/21 34/21 36/21 (12/7) 38/21 40/21
Fsharp (45) 15/15 16/15 17/15 18/15 (6/5) 19/15 20/15 (4/3) 64/45 68/45 24/15 (8/5) 76/45 80/45 (16/9) 84/45 (28/15)
G (48 aka 24) 48/48 51/48 (17/16) 54/48 (9/8) 57/48 (19//16) 60/48 (5/4) 64/48 (4/3) 68/48 (17/12) 72/48 (3/2) 76/48 (19/12)_80/48 (5/3) 84/48 (7/4) 90/48 (15/8)
Gsharp (51) 17/17 18/17 19/17 20/17 64/51 68/51 (4/3) 24/17 76/51 80/51 28/17 30/17 32/17
A (54 aka 27) 18/18 19/18 20/18 (10/9) 64/27 34/27 24/18 (4/3) 38/27 40/27 42/27 (14/9) 45/27 (5/3) 48/27 (16/9) 51/27 (17/9)
A sharp (57) 19/19 20/19 64/57 68/57 24/19 76/57 (4/3) 80/57 28/19 30/19 32/19 102/57 108/57
B (60 aka 30) 15/15 16/15 17/15 18/15 (6/5) 19/15 20/15 (4/3) 21/15 3/2 24/15 (8/5) 17/10 9/5 19/10
If there's not a typo in there it's a miracle. My eyes are blurry. Ok don't say I never have humanity a gift. If you fell out of your chair seeing all those just intervals, you are getting it.
The original tunings were as shown in hertz. I wrote a rather long thread last year in a guitar forum, called "17 Tone Music Attuned to the Earth's Spin", arguing that 24.00 hertz is exactly an octave of one beat per 24 hours, because there is a slight shift in pitch over the 21 octaves between. See if you take the daily spin of the Earth as a deep bass note, and keep increasing it by octaves, the first audible tone is 24 beats per second and change. That should be pretty interesting right there, that that number 24 comes up again. You can look at the math I posted if you want to, but I believe whoever designed clocks had an extremely advanced understanding of harmonics, and so hertz are based on seconds, and seconds are precisely in synch with the Earth rhythm.
And no I'm not going to do the scales for 19 tone music. Ok, so anyone that gets it, check this stuff out, and retune, and share share share, so we can get the Earth back in synch. Hopefully I'm done typing here.
Posted 30 August 2013 - 05:18 PM
Posted 05 September 2013 - 03:05 AM
The video sounded okay for a non-guitarist />/>
Playing your 17 tone system on fretless is of course possible just more difficult, especially larger chords, but the instrument is more flexible and simple.
Concerning instrument size, i play like a guitarist but i prefer the size of a bass, so i string up basses with higher strings to create baritone guitars, but there are also advantages to using a bass as a microtonal guitar: the longer scale opens up the gaps between the micro frets and enables closer pitched frets to be playable, also since fret installing error is constant a longer scale makes fret pitch accuracy better.
On fretless a longer scale improves sustain and pitch accuracy, fretless basses are cheaply available but fretless guitars are rare.
You can find affordable 6 string fretted basses in short scale, i think Rondo music do one, cheap but okay. There are also a few affordable 6 string fretless basses for micro-fretting.
However defretting is easy i've just done my 2nd defret today.
I recommend using fretfind2D to see your tone system in an open tuning with intervals low to high 5th 4th 5th 4th 5th. A good one on guitar is CGCGCG. With this open tuning more frets will line up with the possibility of using full-width frets instead, being much cheaper and easier to do DIY. For your baritone or bass length instrument GDGDGD would be ideal.
Brice short scale bass
Posted 05 September 2013 - 02:05 PM
It looks very playable to me (especially after using my current 18 tone setup)
Hey Mat I hope you check out those 12 tone just scales I listed just above your post. I'm 99% certain that is the original 12 tone tuning for 12 tone harpsichords and therefore what follows, pianos... I've been playing it on that free scala program, and it is really beautiful, all the scales. Just pick what flavor you want your song in, what key?
The fretfind2d fretboard for that 12 tone with drop d tuning is very nice looking btw.
Posted 05 September 2013 - 02:15 PM
17 tone just scale
And set the frets for 17 frets, and the string tunings are:
I know you are familiar with that program because I figured it out reading your posts once
Posted 17 July 2018 - 06:22 PM
Greetings all. I first created this thread 5 years ago.
I have a youtube channel, Tom M Culhane. I have 15 short vids posted at the moment, to let people hear how the ancient musical tunings I've recovered sound.
I've learned some key things the last 5 years. Let me clarify and correct previous things I've said here.
First, the 12 tone just intonation tuning I describe above, I am very confident, was the original tuning used in Europe in antiquity for 12 tone instruments. It was "lost" during the centuries before the Renaissance, when the establishment destroyed most all written documents. There were 3 other ancient just intonation tunings that were also "lost" at this time: 17 tone, 19 tone and 24 tone.
The ancients had an advanced understanding of harmonic laws. The music was synchronized with The Clock, which is oriented to the Earth's spin. Musical notes were set to hertz or fractions of hertz.
If you want to hear how a piano sounds tuned to the ancient 12 tone tuning, listen to my vid, Sarabande. My elderly mom is playing in slow motion. A piano tuner had just tuned the old piano here a week earlier. I helped her tune it with my digital piano. I bought special software (full of bugs unfortunately). The tuning can all be done by ear though, once you have an initial reference pitch, if you know the numbers. It's all "beatless" as piano tuners would say.
Other piano tuners had told me you can't tune a piano to pure just intonation because while you will have one or two keys that sound beautiful, the rest of the keys will be unplayable. I proved them wrong. They just don't know the correct numbers. All the keys of the ancient 12 tone tuning are beautiful, and in tune with the laws of harmonics. The 12 numbers again, in hertz, are:
32 34 36 38 40 42 45 48 51 54 57 60 (and 64 would be the octave...).
32 36 40 42 48 54 60 are the 7 white keys. 34 38 45 51 57 are the 5 black keys.
The piano tuning should be started in the center octave of the piano, using the appropriate octave numbers. You can use a meter for the center octave but the rest should be tuned by ear to give the best sound. (the notes will "spread" slightly over the register due to the stiffness of piano strings, which cause overtones to vibrate a bit sharp- that's per a man who fixes pianos I talked to).
Some ancient harpsichords and organs had a second keyboard. These were intended to be tuned each to different notes (why would you make two keyboards to play the same exact sound??). So these instruments played a total of 24 tones per octave. The tuning: one keyboard would use the numbers I've just written here and the second keyboard would be tuned to the notes between the notes:
33 35 37 39 41 43.5 46.5 49.5 52.5 55.5 58.5 62
Ancient composers must have written masterpieces in this tuning. I can barely play keyboard like I can barely play guitar but check out my vid, Mountains, to hear a minute of 24 tone music I play on my digital piano using special software.
19 tone music was another of the 4 main tunings systems used in antiquity. Search "19 tone Roman harpsichord" and you will find an image of a 19 tone reconstructed Roman harpsichord. It looks like a modern piano layout except the black keys are double keys (making sharps and flats two separate notes) and there are single black keys between B and C and between E and F (which are empty spaces on the modern piano), giving 19 notes total per octave. Here are the numbers for the ancient 19 tone tuning:
24 25 26 27 28 29 30 31 32 33 1/3 34 2/3 36 37 1/3 38 2/3 40 42 43 1/2 45 46 1/2 (octave at 48...)
Ok so the 7 white keys would be 24 27 30 32 36 40 45 and the black keys are the rest.
I have three vids posted in ancient 19 tone on the digital piano: Flow, Lonely and Plantation Earth.
The last of the 4 major ancient tunings was 17 tone. The keyboard would look like a modern piano layout except the black keys would be doubles. The tuning is as follows:
48 50 52 54 56 58 60 63 66 69 72 75 78 81 84 87 91 (and the octave begins at 96 hertz).
The white keys there would be 48 54 60 63 72 81 91.
An alternative tuning for 17 tone is to use 88 hertz instead of 87.
My vid "Yule Party" is in ancient 17 tone.
Ok, I began this thread 5 years ago talking about making a 17 tone just intonation guitar. It was a bad idea I've concluded, the way I was thinking. A good way to do it is make a 19 tone guitar using the 19 tone tuning I've described here. Then the frets will be fairly straight. You could use fretlets and paint lines connecting them so visually it would look like 19 frets per octave, and the chords shapes would be familiar to you.
But I never did try to make a fretted guitar in this way. I took Mat's advice above and bought a fretless 6 string bass and put thinner strings on it to play in the baritone range. I like it. I use it to find different sounds and play slow motion sonic massages for myself. I don't even try to play chords fretless at this point. Being fretless is very educational for me, letting me hear the notes and harmonics. I normally tune the open strings thickest to thinnest in hertz to: 54 72 96 128 160 216.
Those are all notes found in the ancient 12 tone and 19 tone tunings: 27 36 24 32 40 27. I consider this baritone fretless guitar to be like an electric viola da gamba. Search images for viola da gamba and marin marais composer and look at the one where he's seated playing. Legend has it he's a distant relative of Robert Plant. That viola da gamba is one of the most awesome instruments ever btw. Played with a bow or pick, it is obviously the true precursor of the modern electric guitar. You can play chords up to 5 strings at a time. And the frets are movable, and I believe they used triangular beads with a bit of wax (my own theory) to let you in effect create fretlets for fine tuning notes. Anyway I have 3 vids posted with my baritone fretless guitar: Dr Seuss, Sonic Massage and Sonic Massage 2, just me hitting a few notes. The rest of my vids are in ancient 12 tone.
Ok that should conclude this 5 year update of my guitar project and my recovery of what I believe were the original musical tunings used in Europe in antiquity.
Posted 18 July 2018 - 09:53 AM
WoW Tom, you are pretty prolific!
I've tried marking dots for 19TeT on a fretless but the problem is if you don't know how 19 sounds you are going to be a little off in intonation.
Way back the tuning for middle C was always 256 Hz it was called scientific scale or drawing room scale. You can still get tuning forks at antique places with the C at 256 as they were widely used in schools and teaching establishments.
My Uncle discussed piano tuning with me in the 60s, Showing me how the key of C played then the key of F which sounded better but F# was very different. Back then piano tuners started from C=256 and worked by ear through the various notes, I guess it was close to JI - there were no electronic tuners back then. As pianos got older they would be tuned to themselves, gradually going flat from 256. Sounds like the skill was lost as modern pianos play the same in all keys.
PS on my PC it says the videos are not available.
Author of the book "Fretless Guitar The Definitive Guide" fretlessguitar.co.uk
Posted 18 July 2018 - 01:39 PM
The vids should all be there, if you search Tom M Culhane on youtube and go to my page. Hopefully. 15 vids at the moment.
Very interesting what you say about antique tuning forks set to middle C at 256 hertz. That's exactly the right number (reduces by octaves to 256>128>64>32 hertz for the first C note). And the ancient 12 tone tuning I've described here can all be done easily by ear if you have that middle C note to start with.
I tried to show an "authority" on ancient tunings the 12 tone tuning here, and that the piano keyboard pattern (which comes from earlier instruments such as harpsichords) is clearly intended to be tuned to this just intonation whole hertz number tuning. But he didn't have time to listen, insisting "this is not a historical tuning". It is in fact THE historical tuning.
If you look at the left end of the standard 88 key piano today, the left end looks terrible, with that lone black key. I claim the design has been altered to hide the tuning intended for these types of instruments. If you were to add two extra bass keys, one white one black, to the design, you will have a beautiful looking keyboard with 90 keys. Search 1877 Paris Frederick Collection and you should find a beautiful old piano made this way, with 90 keys. Now if you look up the modern equal temperament tuning in hertz for the beginning white keys, it would be:
Those numbers wouldn't mean anything to someone today, but if you take Galileo's 7 pure tones for just intonation: 1/1 9/8 5/4 4/3 3/2 5/3 15/8, you'll notice the smallest common denominator is 24, and these 7 tones can be put to whole number proportions as:
24 27 30 32 36 40 45
So you see how the modern tuning is a blurry version of these numbers in hertz? The tuning originally was intended to be set to just intonation, but has been made blurry. Two keys were chopped off to hide this and the note names were moved over a character. So G really should be called A, the first note of music. 24
Note: I didn't list the equal temperament tuning number for the seventh key because the piano white keys do not stay on the G scale, the 7th note in G is a black key. This raises the question, why is the piano keyboard oriented to the C scale? The answer further supports my claims.
If the piano were a double black key instrument instead of a single black key instrument, then the keyboard white keys would stay in G (which should be called A as I say). That's because if you want to tune to whole number just intonation, this is the perfect key of music to orient things to because whole numbers easily drop right in for the black keys to start the tuning. In other words if you begin the first three white keys as 24 27 30 in hertz, then the double black keys can be tuned the first pair to 25 and 26 and the second pair to 28 and 29 hertz, and then the rest of the tuning goes from there. Hope that's clear.
So the 19 tone reconstructed Roman harpsichord I mentioned in the previous post, that you can look at online, which has double black keys, it would begin the white keys on 24 hertz or an octave of that, and the white keys would stay in that key: 24 27 30 32 36 40 45
But the modern piano is a single black key instrument so there's a better key of music to use to orient the tuning. Note the 7 pure tones, the second group of three numbers has the same proportions as the first three numbers. 32 36 40. The key of music that begins at 32 hertz is the perfect key to use to tune a 12 tone instrument because single whole numbers drop right in for the black keys: 34 and 38 hertz. This is easy to see if we move things over one octave to the left. The tuning would begin: 16 17 18 19 20 hertz where 17 and 19 are black keys.
The idea is very simple. The tunings were based on the simplest whole numbers because that is how things vibrate in the real world. The Ancients knew the math.
Posted 19 July 2018 - 01:54 PM
Here is a bit more info on these whole number just intonation tunings:
Apparently music teachers today teach that the basic just intonation scale (1/1 9/8 5/4 4/3 3/2 5/3 15/8) sounds terrible because there is a "wolf fifth" in it, and so we must alter the notes to fix things.
I claim there is nothing wrong with the laws of harmonics and if you try to change these notes, in 12 tone tuning, you will lose all the keys to music. I recommend people just play these notes and decide if they sound terrible, or, as I claim, beautiful. Most people (outside of forums like this) have never even listened to them.
Interestingly, on my baritone guitar, the "wolf fifth" sounds nothing like a wolf but distinctly like another animal: the lamb. How did the lamb get turned into a wolf? Is someone messing with your head?
When I tune my baritone fretless guitar, as mentioned earlier, I generally tune the 6 strings to 54 72 96 128 160 216 hertz. That's the pattern on a modern guitar of a fourth a fourth a fourth a third and a fourth. Except this is just intonation. So the first four intervals sound very smooth as they are the proportions 4 to 3 for the fourths, and 5 to 4 for the third. But the last fourth interval is a "wolf note". It creates a vibrato sound. And like I say it sounds like a lamb, very cool. Now why is it when singers and string players use vibrato, people love it, but if my guitar plays vibrato we're supposed to think this sounds terrible?
Now the people that are trying to fix the laws of harmonics (which have nothing wrong with them as I say) speak in terms of "the fifth", "the fourth", "the third", "the whole tone", etc., as if these are singular things. But in reality there are various types of fifths, fourths, thirds, whole tones... For example one whole tone is the pitch ratio 9 to 8. Another is 10 to 9. Another is 8 to 7. (All these btw are found in the 12 tone whole number just tuning I've outlined here). Now these obviously have different sounds. Some notes played together give smooth sounds, some give vibrato sounds. Indeed it is this irregularity that gives us the keys to music. And as I claim the keys have been lost for centuries.
These vibrato sounds tie in with the idea of "difference tones". There is a youtube vid of microtonal guitarist Jon Catler playing a fretless guitar where he's playing two high pitched notes and it generates bass notes you can hear. Why is that? As you may know, when two notes are played together they generate additional notes, the sum and also the difference between the frequencies. So if you play a 100 hertz note with a 140 hertz note, this will generate a 40 hertz difference tone, for example. Notice that with the whole number tunings I have given here, the differences between the notes are also in whole numbers. So this creates harmonies on different levels, some audible as tones, some as vibrato beats, etc. Other tuning systems would not have these harmonies.
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Tom M Culhane