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Topic: JI tuning - Warning! math content |
Karlis Abolins
From:
(near) Seattle, WA, USA
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Posted 17 Mar 2013 11:34 am |
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I wanted to try out a Just Intonation tuning on my pedal steel as an experiment. I thought it would be an easy thing to do. I thought that all I had to do was to choose the JI temperament on my Peterson VSII tuner and tune. I discovered that the Just Intonation temperament (and all of the other temperaments as well) on the Peterson is based on the Root = C. This means that the ratios of the notes are all relative to the note C. My pedal steel tuning is an E9 tuning using a Root = E. I needed to create a Just Intonation tuning for the scale of E.
After some research, I found the ratios for a Just Intonation tuning:
Root 1:1
major semitone 16:15
major tone 9:8
minor 3rd 6:5
major 3rd 5:4
4th 4:3
diminished 5th 64:45
5th 3:2 (typo corrected. thanks Chas.)
minor 6th 8:5
major 6th 5:3
minor 7th 9:5
major 7th 15:8
octave 2:1
These ratios are for the ascending scale (there are different ratios for the descending scale).
There are 1200 cents in a musical octave. The Equal Temperament tuning spaces the notes exactly 100 cents apart so the cents for an Equal Temperament tuning are:
Root 0
major semitone 100
major tone 200
minor 3rd 300
major 3rd 400
4th 500
diminished 5th 600
5th 700
minor 6th 800
major 6th 900
minor 7th 1000
major 7th 1100
octave 1200
Knowing this and using the Equal Temperament on the Peterson tuner, I should be able to put in the cents offset for Just Intonation. All I need is the cents difference between the JI and EQU values.
Further research got me the formula for calculating JI cents using the ratios above.
For any ratio n/p, the number of cents in the interval is log (n/p) x 1200/log 2.
Using this formula, I derived the following cents values:
Root 0
major semitone 111.7313
major tone 203.91
minor 3rd 315.6413
major 3rd 386.3137
4th 498.045
diminished 5th 609.7763
5th 701.955
minor 6th 813.6863
major 6th 884.3587
minor 7th 1017.596
major 7th 1088.269
octave 1200
I then subtracted the corresponding EQU cents value from the JI cents value to give me the following offsets:
Root 0
major semitone 11.7313
major tone 3.91
minor 3rd 15.6413
major 3rd -13.686
4th -1.955
diminished 5th 9.7763
5th 1.955
minor 6th 13.6863
major 6th -15.641
minor 7th 17.596
major 7th -11.731
octave 0
I rounded off to the nearest tenth of a cent (the Peterson tuner doesn't support a higher resolution) and entered the values into the VSII using the EQU temperament as a starting point for each note in the E scale:
E 0
F -29.3 see below
F# 3.9
F# -17.6 "C" pedal F# - see below
G 15.6
G# -13.7
A -2.0
A# 9.8
B 2
C 13.7
C# -15.6
D 17.6
D# -11.7
E 0
I made one final adjustment. I adjusted the F note (used with the A and F pedals) so that it would be in the correct relationship to C# as a major third. Major 3rd (-13.686) plus major 6th (-15.641) equals -29.327.
(edited on 3/28 to add offsets for alternate F# for the "C" pedal) I adjusted the F# on the "C" pedal so that it would be in the correct relationship to the A on the "B" pedal as root of minor 3rd. The A (-1.955) minus minor 3rd (15.641) equals -17.596.
The final step was retuning my pedal steel. The result is very musical with even the A F pedal combination sounding acceptable. My pedal steel has a very small cabinet drop so the tuning is acceptable. A pedal steel with more cabinet drop would have to be adjusted to compensate when the pedals are down.
Karlis
Last edited by Karlis Abolins on 28 Mar 2013 3:53 am; edited 2 times in total |
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chris ivey
From:
california (deceased)
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Posted 17 Mar 2013 11:49 am
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glad you're enjoying math class!
once you've played long enough you'll learn to cut corners, round off numbers, use your ears and skip school. then the fun begins! |
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Bud Angelotti
From:
Larryville, NJ, USA
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Posted 17 Mar 2013 11:55 am
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Some serious woodshedding going on 
Keep having fun with it!
_________________
Just 'cause I look stupid, don't mean I'm not. |
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Lane Gray
From:
Topeka, KS
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Posted 17 Mar 2013 1:34 pm
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My approach came the other way around. I tuned by ear meticulously, and made a note of each tone's deviation
_________________
2 pedal steels, a lapStrat, and an 8-string Dobro (and 3 ukes)
More amps than guitars, and not many effects |
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Dickie Whitley
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Posted 17 Mar 2013 1:54 pm
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...when I tried that formula, all I got was a division by zero error....
Update: Tried this formula in Excel, but apprently still wrong - LOG(A1/B1* (1200/LOG(2,10))) |
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Karlis Abolins
From:
(near) Seattle, WA, USA
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Posted 18 Mar 2013 5:53 am
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Dickie,
Try this in excel: =LOG(A1/B1,10)*(1200/LOG(2,10))
You failed to specify the base for the first log function and didn't close it before multiplying by the second half of the equation. I had the same problem until I caught the error.
Chris,
I spent 40 years as a computer programmer/analyst. It is second nature for me to look at a problem and figure out the solution whether it is computer related or not. I wish I had spent those years learning how to play the pedal steel professionally instead. It wasn't an option for me at the time and now I analyze everything instead of approaching a situation organically. I am retired now and working on the organic thing (using my analytical talents).
Karlis |
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chas smith R.I.P.
From:
Encino, CA, USA
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Posted 18 Mar 2013 11:05 am
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| Karlis, I love this stuff. Typo, P-5 is 3:2. You might also look at 7:5 for the dim-5, even though the traditional is 45:32 (64:45 is its reciprocal), and 16:9 for b7. To confuse things a bit, the JI scale usually has (2) maj 2nds, 9:8 and 10:9. The 9:8 will be in tune with the 3:2 while the 10:9 will be in tune with the 5:3. |
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chris ivey
From:
california (deceased)
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Posted 18 Mar 2013 11:08 am
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oh...i spent 40 years playing music anywhere i could.
course..you probably made alot more money. |
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Dale Rottacker
From:
Walla Walla Washington, USA
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Ken Metcalf
From:
San Antonio Texas USA
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Posted 18 Mar 2013 11:50 am
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I thought tuning JI was tuning by ear.. and a person with a Peterson tuner would just tune by ear and save the offsets with the Peterson tuner for tuning in noisy situations?
I am not getting the point of using Excel.
_________________
MSA 12 String E9th/B6th Universal.
Little Walter PF-89.
Bunch of stomp boxes |
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Ken Metcalf
From:
San Antonio Texas USA
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Posted 18 Mar 2013 11:55 am
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I don't think the presets are based on C..
They are based on Equal Temperament.
_________________
MSA 12 String E9th/B6th Universal.
Little Walter PF-89.
Bunch of stomp boxes |
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Karlis Abolins
From:
(near) Seattle, WA, USA
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Posted 18 Mar 2013 12:11 pm
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Chas,
Thanks for catching the typo. I don't have the diminished 5 on my guitar but I included the values for all positions for clarity. I also don't have a 7b . The 9:8 2nd works with the 3:2 for me although it would be interesting to try the 10:9 to see what it sounds like. I am getting the new Peterson tuners which allow editing the presets on the PC with a lot more presets available. I will be able to experiment quickly with these kinds of alternatives with the new tuner.
Ken,
The JI temperament is quite accurate musically. Tuning by ear is an approximation but a valid approach. I used excel to calculate the exact values that I needed for a JI tuning based upon E rather than the JI preset in the Peterson tuner.
Interestingly, I read an article recently that explained the standard 6 string guitar tuning. It goes back a long way but it is a G pythagorean tuning with the low E and A added at a much more recent date. With the top 4 strings, you can create a G major chord and an E minor chord. I may try a G JI tuning on my 6 string just to see what it sounds like.
Karlis |
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Karlis Abolins
From:
(near) Seattle, WA, USA
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Posted 18 Mar 2013 12:22 pm
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Ken,
The Peterson tuner has a JI preset along with other historical temperaments. The JI on the Peterson is based on the C note. The Equal temperament preset is also included on the Peterson and is the default. With EQU, every note is 100 cents higher than the previous note.
Karlis |
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chris ivey
From:
california (deceased)
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Posted 18 Mar 2013 12:28 pm
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i hope the people you play music with are as synched up with you mathmatically or it may all be for nothing.
and i assume your gigs will be in a temperature controlled cubicle. |
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b0b
From:
Cloverdale, CA, USA
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Karlis Abolins
From:
(near) Seattle, WA, USA
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Posted 18 Mar 2013 1:22 pm
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Chris,
The geezers I play with rarely are all in tune together. I try to use my bar placement to adjust my notes relative to everyone else, as I am sure you do as well. I did buy one of my bandmates a stompbox tuner so he could be more in tune than he was in the past. Usually, if everyone tunes up before we start, it sounds OK.
My guitar is fairly stable temperature wise within normal limits. Usually when I go from my music room to the garage loft that we practice in I don't have to adjust anything much. Even if I retune, my pedals are usually right on (or CEFGW).
Karlis |
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Kevin Hatton
From:
Buffalo, N.Y.
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Posted 18 Mar 2013 1:27 pm
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Karlis Abolins
From:
(near) Seattle, WA, USA
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Posted 18 Mar 2013 1:30 pm
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b0b,
Thanks for the link. I read that article quite some time ago. I think I need to wrap my head around the two F# issue and sort it out.
Karlis |
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chris ivey
From:
california (deceased)
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Posted 18 Mar 2013 1:52 pm
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| karlis...i'll be quiet after this, and i can appreciate your diligence. my point is, of course, when playing with sort-of-tuned geezers it makes tuning to a thousanth-of-a-whatever makes it all moot and, in my opinion, ridiculous. however, continue on if you enjoy it. |
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Bob Hoffnar
From:
Austin, Tx
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Posted 18 Mar 2013 2:06 pm
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Karlis,
I'm not sure if this will help with the math but in practice I sometimes tune the pedal 3 F#(4th st) to the pedal 3 C#(5th st) and I tune the first string F# to the open B on the 5th string. They are slightly different but it works for me in some situations.
Buddy Charleton had a compensator on his first string to do the slight change that Bob Lee is talking about.
I recently played a piece for a microtonal composer that used some very serious math. After spending a week working at that level of precise intonation my perception of pitch really improved.
_________________
Bob |
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Georg Sørtun
From:
Mandal, Agder, Norway
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Posted 18 Mar 2013 2:07 pm
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As for the two-pitch notes in JI tuning...
...I found the answers I needed here.
I also found that I have for decades tuned using the JI principles as described in that article - included two-pitch notes, but not the same two-pitch notes as in the pictured table. |
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b0b
From:
Cloverdale, CA, USA
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Posted 18 Mar 2013 6:54 pm
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What Bob Hoffnar says about the tuning of the 3rd pedal F# vs the 1st string F# is what most players do, I think. You can't set up 2 different pitches for F# in the Peterson tuner, though. You have to tune one of them (probably the pedaled one) by ear.
For some seriously beautiful JI music, listen to Lou Harrison. His compositions are documented with exact pitch ratios for each note on the staff.
_________________
-𝕓𝕆𝕓- (admin) - Robert P. Lee - Recordings - Breathe - D6th - Video |
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chas smith R.I.P.
From:
Encino, CA, USA
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Posted 18 Mar 2013 7:46 pm
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| Quote: |
| I sometimes tune the pedal 3 F#(4th st) to the pedal 3 C#(5th st) and I tune the first string F# to the open B on the 5th string. |
This is the having two maj 2nds issue that I mentioned above. F# is the maj 2nd and tuned to the 9:8 it will be in tune with the 3:2 because the 9 is divisible by 3 and 8 is divisible by 2 in whole numbers. The C# (maj 6th) is 5:3 and the 10:9 maj 2nd is easily divisible by 5:3. |
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Bob Hoffnar
From:
Austin, Tx
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Posted 18 Mar 2013 9:10 pm
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It's interesting how the more simple the relationship between notes/ratios are the more natural and pleasing the sound is. I started tuning my F#'s differently and by ear just because I liked the sound. Later I find there is a simple mathematical reason for it. Pretty cool.
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Bob |
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Earnest Bovine
From:
Los Angeles CA USA
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Posted 18 Mar 2013 9:36 pm
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| Bob Hoffnar wrote: |
| It's interesting how the more simple the relationship between notes/ratios are the more natural and pleasing the sound is. I started tuning my F#'s differently and by ear just because I liked the sound. Later I find there is a simple mathematical reason for it. Pretty cool. |
Besides the math, the physical reasoning is pretty simple too. A wave that is 3 times as fast as another (i.e. one third the wavelength of the other) will line up with the other wave the same way in every cycle. If you change one slightly, it will align for a while, then not align for while, generating audible beats. People tend to prefer beatless, or slower-beating, combinations, altho contrast makes music more interesting of course. I'm not sure Iggy Pop prefers consonant, beatless intervals for example. You can see moving illustrations here & there on various web sites.
Also, remember that strings vibrate at many frequencies combined. The A string on a guitar wiggles at 110, 220, 330, 440, 550 Hz etc. So the E string at 329.26 Hz or whatever it is (I forget just now) is beating against the 3rd harmonic of the A string.
I don't think it makes any sense to define "just intervals" with high numbers such as 11/9, because the 11th and 9th harmonics are so tiny, too high to hear, and die away instantly. You do not hear the beats in any case.
Inharmonicity is the difference between the ideal string, and a real string. For example the harmonics of your guitar's A string may be 221, 334, 447, 560 instead of the exact multiples listed above (I have exaggerated my numbers). Note that even the octave is more than a doubling of frequency. This is the reason that piano tuners stretch the tuning of high notes, where piano strings are very short.
Inharmonicity is much greater on shorter strings and fatter strings, which is why it sounds so weird to play high notes on the low C string of C6 neck. |
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