M3 & JI - How much “color”?

Discussion in 'Music Theory, Lessons & Techniques' started by ElRay, Feb 10, 2019.

  1. ElRay

    ElRay Mostly Harmless

    Messages:
    3,515
    Likes Received:
    683
    Joined:
    Nov 6, 2006
    Location:
    NoIL
    I was going to post this in the temperament thread, but it a bit specific. My lure to other temperaments is different “tone color” in different keys. I ideally, I’d have squiggly frets all over the place to have one key “perfect” and let the other keys be “off”. But that’s not practical for a DIYer.

    Adjustable single string frets aren’t either. (I need to find that link)

    So, the idea is to tune the guitar in all Major 3rds (M3) and have frets that come close to JI. Then, assuming I play across more than up/down each position will have its own temperament. But, will this be playable?
    1. In first position, the 5th two stings up would be a JI M3+m3. Is that a playable 5th?
    2. Playing in 5th position, across strings, I’d stiil have a JI M3, but the M3 on the same string would be the difference between a JI P4 and JI M6.
    It seems like there’s potential, but depending on how different the intervals are, this could be a hot mess.
     
  2. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,291
    Likes Received:
    967
    Joined:
    Mar 5, 2007
    Location:
    UK
    I'm planning to start a new thread about my method for using a normal guitar to play in Just Intonation scales by tuning to thirds. Anyway here it is quickly if it helps although it's not what you are suggesting:

    JImajor7st_new.png

    To play a Just Intonation major scale:
    • Use a series of gauges from the sequence found in http://sevenstring.org/threads/retu...al-beginners-guide.161530/page-2#post-4919443
    • Tune the strings to alternating major and minor 12TET thirds, every 2 strings is a 12TET fifth.
    • For each string pair a major third apart: Slightly detune (14 cents) the higher string to create a JI major third. Tune the 4th harmonic of the higher to the 5th harmonic of the lower, or tune the interval by ear.
    • The tuning will then become alternating JI major and (almost exact) JI minor thirds. Strings 1, 3, 5, 7 will still be 12TET and will be tuned in 12TET fifths. Strings 2, 4, 6 will be microtonal and be 12TET - 14 cents.
    • Play the patten above, red dots are the tonic notes. The 3rd, 6th, 7th are played on the microtonal strings. The tonic, 4th, 5th are played on the 12TET strings.
    • Since this uses 12TET frets the pattern can be modulated to other JI major keys, many JI-fretted guitars can't do this.
     
    ElRay likes this.
  3. Winspear

    Winspear Tom Winspear Vendor

    Messages:
    10,969
    Likes Received:
    1,544
    Joined:
    Oct 23, 2009
    Location:
    Yorkshire, U.K
    You should get Scala software. It allows you to input a scale and then view the modes of it, so you can see how each key would sound in a temperament.
    But if I'm understanding your question correctly, yes, 5/4 (386c) and 6/5 (315c) make a perfect fifth, 3/2. The harmonic series stacks out 6/5/4/3/2 just like that ! So it works quite perfectly.
    You'll likely be looking at a temperament which has had plenty of ground covered and is well documented in terms of key colours.
     
  4. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,291
    Likes Received:
    967
    Joined:
    Mar 5, 2007
    Location:
    UK
    JI or 12TET major thirds?
    Exact JI fretting or modified?
    Differing 'colour' in different keys sounds like 'Well Temperament' https://www.kylegann.com/histune.html#hist3
     
    Winspear likes this.
  5. Winspear

    Winspear Tom Winspear Vendor

    Messages:
    10,969
    Likes Received:
    1,544
    Joined:
    Oct 23, 2009
    Location:
    Yorkshire, U.K
    Well temperament would definitely be a good contender. For a true JI scale how about this:

    Definitely grab Scala.
    And head here; https://en.xen.wiki/w/Gallery_of_12-tone_just_intonation_scales
    The first one on that list, https://en.xen.wiki/w/12highschool1
    I'll analyse the keys in Scala.
    Copy and paste this part into the input of scala;

    21/20

    9/8

    6/5

    5/4

    4/3

    7/5

    3/2

    8/5

    5/3

    7/4

    15/8

    2

    And it will show you this;

    0: 1/1 0.000000 unison, perfect prime
    1: 21/20 84.467193 minor semitone
    2: 9/8 203.910002 major whole tone
    3: 6/5 315.641287 minor third
    4: 5/4 386.313714 major third
    5: 4/3 498.044999 perfect fourth
    6: 7/5 582.512193 septimal or Huygens' tritone, BP fourth
    7: 3/2 701.955001 perfect fifth
    8: 8/5 813.686286 minor sixth
    9: 5/3 884.358713 major sixth, BP sixth
    10: 7/4 968.825906 harmonic seventh
    11: 15/8 1088.268715 classic major seventh
    12: 2/1 1200.000000 octave

    Now go to Analyse>Show interval attribute matrix to see the modes. Not diving into it right now but at a glance, the vast majority of keys have a good fifth and a useable third of at least one quality.

    1/1 : 84.467193 203.910002 315.641287 386.313714 498.044999 582.512193 701.955001
    813.686286 884.358713 968.825906 1088.268715 1200.000000

    21/20: 119.442808 231.174094 301.846520 413.577806 498.044999 617.487807 729.219093
    799.891520 884.358713 1003.801521 1115.532807 1200.000000

    9/8 : 111.731285 182.403712 294.134997 378.602191 498.044999 609.776284 680.448711
    764.915905 884.358713 996.089998 1080.557192 1200.000000

    6/5 : 70.672427 182.403712 266.870906 386.313714 498.044999 568.717426 653.184619
    772.627428 884.358713 968.825906 1088.268715 1200.000000

    5/4 : 111.731285 196.198479 315.641287 427.372572 498.044999 582.512193 701.955001
    813.686286 898.153480 1017.596288 1129.327573 1200.000000

    4/3 : 84.467193 203.910002 315.641287 386.313714 470.780907 590.223716 701.955001
    786.422194 905.865003 1017.596288 1088.268715 1200.000000

    7/5 : 119.442808 231.174094 301.846520 386.313714 505.756522 617.487807 701.955001
    821.397809 933.129094 1003.801521 1115.532807 1200.000000

    3/2 : 111.731285 182.403712 266.870906 386.313714 498.044999 582.512193 701.955001
    813.686286 884.358713 996.089998 1080.557192 1200.000000

    8/5 : 70.672427 155.139620 274.582429 386.313714 470.780907 590.223716 701.955001
    772.627428 884.358713 968.825906 1088.268715 1200.000000

    5/3 : 84.467193 203.910002 315.641287 400.108480 519.551289 631.282574 701.955001
    813.686286 898.153480 1017.596288 1129.327573 1200.000000

    7/4 : 119.442808 231.174094 315.641287 435.084095 546.815381 617.487807 729.219093
    813.686286 933.129094 1044.860380 1115.532807 1200.000000

    15/8 : 111.731285 196.198479 315.641287 427.372572 498.044999 609.776284 694.243478
    813.686286 925.417571 996.089998 1080.557192 1200.000000
    2/1

    You can play all of these via scalas on screen keyboard, using shift key to build chords etc. Or input with a MIDI keyboard.
     
  6. Winspear

    Winspear Tom Winspear Vendor

    Messages:
    10,969
    Likes Received:
    1,544
    Joined:
    Oct 23, 2009
    Location:
    Yorkshire, U.K
    Keep in mind in a JI tuning the 3rds between your strings will be different (some 427c instead of 386c)
     
  7. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,291
    Likes Received:
    967
    Joined:
    Mar 5, 2007
    Location:
    UK
    I assume the open tuning is all JI major thirds (5/4 386c).
    The 'fifth 2 strings up' is the open note? Whether that is a JI fifth (3/2 702c) depends on where fret 1 is placed.
    386 * 2 = 772c so fret 1 would have to be at 70c.
    A tuning of stacked JI major thirds quickly leads into very dissonant and complex intervals, that are not much used in JI scales.

    A JI major third (5/4 386c) plus a JI minor third (6/5 316c) is the JI fifth (3/2 702c).
    When you add JI intervals you multiply the fractions:
    5/4 * 6/5 = 30/20 = 3/2.

    The JI major triad is actually 3 pitches with frequencies in ratio 4:5:6. The lower interval is 5/4, the upper is 6/5 and the interval from lowest to highest pitch is 6/4 = 3/2.

    As Winspear suggests, if you set up a guitar somehow to play an exact JI major scale, every mode of that scale will be a differing JI scale with it's own 'colour' (this is the common complaint of Just Intonation, you can't modulate to other keys without altering many of the intervals). To acheive 'key colour' it's not essential to use Well Temperament which is meant to remain close and similar to 12TET.

    From your post it seems you are considering DIY microtonal fretting but using frets that go all the way across the fretboard. If these frets are placed in JI positions then various open tunings give you various scale possibilities. The simplest open tuning would be alternating JI fifths and JI fourths, so tonic-fifth-tonic-fifth ... CGCGCG...
    See this guitar project which may be similar to what you are considering http://jsnow.bootlegether.net/cbg/justintonation.html
     
    Last edited: Feb 10, 2019
  8. Bobro

    Bobro SS.org Regular

    Messages:
    111
    Likes Received:
    44
    Joined:
    Jun 26, 2017
    You are in luck if "pure major thirds at 5:4 everywhere you go" is what you are after.

    Western music, for several centuries, was based exactly on this ideal, and the tuning they came up with is called "1/4 Comma Meantone". In !/4 comma meantone, the pure fifth is flattened down by one fourth of a comma (the difference between a major third created by 4 pure fifths, which is the Pythagorean ditone of 81:64, very sharp of pure and even sharper than our current 12-tET M3, and a pure 5:4 major third. So, four fifths in a row and you're at the M3. 12-tET does this too, but the M# of 12-tET is very sharp of pure.

    1/4 Comma Meantone is a theoretically infinite tuning system, but only a handful of people on earth can hear the difference between 1/4 Comma Meantone and 31 equal divisions of the octave ("31-EDO). 31 EDO is relatively easily fretted on a guitar- Metatonal Music makes and sells 31-EDO guitar necks all the time, and their necks are awesome and well priced. I have one of their necks, in a JI system close to 17-EDO, best musical instrument investment I've ever made, even with the 27% import tariff I got shafted with, by the country I live in.

    Couple of things about 1/4 Comma Meantone, and therefore 31 EDO. The fifths are very flat of what we are used to today, about 697 cents, so, 3 cents flat of today's 12-tET fifths and 5 cents flat of pure 3:2 fifths, This bugs some people (but very few). Forget power chords, they are too flabby to have any "power" at all. And trying to modulate will immediately teach you all about "the rules" of a few centuries ago, preparing dissonances and false relations and all that, and "correct" ways of modulating (no sudden leaps to far away places), because C# and Db are two distinct pitches, for example. Best to listen to lots of Gesualdo done in historically correct tuning. He had all that stuff totally down and made wild modulating music in 1/4 Comma meantone. Don't know if his composing got better or worse after he murdered his wife for cheating on him.

    So there's your solution, very simple really. Best find a way of listening to the fifth first, though- it might be too soggy, or too quaint and "yee anciente" sounding for you, like the Three Musketeers are going to bust down the door and start drinking ale and grabbing the buttocks of the serving wenches whenever you play. If you are a fan of powdered wigs, duelling, and epic cleavage about to pop right out of the neckline, it is the real life historically accurate Bomb and only sensible choice.
     
  9. ElRay

    ElRay Mostly Harmless

    Messages:
    3,515
    Likes Received:
    683
    Joined:
    Nov 6, 2006
    Location:
    NoIL
    Thanks all! There's a lot to digest.

    The original idea was to have JI intervals along each string, so I could have straight frets across the fret board. I know that a "C" on one string will not match a "C" on another string, but that's the point, unless things dissolve into a (unusable) hot mess.

    The thought I had was that each string would be a M3 away from the previous, so every 4th string would be an octave. I forgot that 3 JI M3rds do not make an octave.

    I do like the idea of alternating JI M3rds and JI m3rds. That way, three adjacent string would be either a major or minor chord. I'd lose a bit of range, but it would make the whole guitar more playable.

    And yes, the goal is to have different temperaments, so the major chord played in 1st position on the top three strings would have a different "color" than the same major chord played at the 7th fret (If I did alternating M3/m3) on the 3rd, 4th & 5th strings.
     
    Bobro likes this.
  10. Bobro

    Bobro SS.org Regular

    Messages:
    111
    Likes Received:
    44
    Joined:
    Jun 26, 2017
    I would suggest doing what I did, before investing in a custom fixed-fret neck. Get a Turkish Saz. A baglama saz, which is about guitar sized, is the best for this, though I used a cura saz, the little one. They are dirt cheap and sound great. I got mine in Instanbul for 70 bucks, my Turkish musician friend said I paid way too much! So much for my haggling skills, lol.
    Anyway, the frets are tied on, so you can tie them wherever you want, and so experiment with any straight-fret tuning you desire. I just scooched my frets around, because it was already fretted with 17 tones to the octave, not all that far from the 17-tone JI system I finally settled on. After playing this for a couple of years (they really sound good!), I was totally immersed in the tuning and ready to commit to a fixed-fret electric guitar neck from Metatonal Music (highly recommended, they rule!).
     
    ElRay likes this.
  11. Winspear

    Winspear Tom Winspear Vendor

    Messages:
    10,969
    Likes Received:
    1,544
    Joined:
    Oct 23, 2009
    Location:
    Yorkshire, U.K
    I want to say outside of EDOs, straight frets right across would be quite a mess tuning wise yes, else it would be more popular. I can't get my head around how it would work too well for harmonic use - drones and melody would be ok, as hinted at above regarding eastern instruments. I may be completely wrong, having a look at the link posted above - perhaps its quite functional even if very confusing tuningwise. Even tuning CGCGCG style in a JI system only results in some straight frets (the 4ths/5ths). I'd need to sit down and think about an application that had so many different notes combined as you are suggesting. Mapping it out in Excel would be a good idea, it doesn't take too long to put together a fretboard map with cents so you can analyse tuning and fingerings.
     
    Last edited: Feb 21, 2019
  12. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,291
    Likes Received:
    967
    Joined:
    Mar 5, 2007
    Location:
    UK
    Straight fret JI is possible, you just add as many frets as is necessary to get the tonal system you require (there's space for more than 12 frets per octave) and use a repeating open tuning like CGCGCG or CFGCFG or CEGCEG or whatever.

    Partial frets are no more difficult than full width, cut a full width slot and hammer in a short fret.

    GuitPicWeb.jpg
     
    ElRay likes this.
  13. ElRay

    ElRay Mostly Harmless

    Messages:
    3,515
    Likes Received:
    683
    Joined:
    Nov 6, 2006
    Location:
    NoIL
    :agreed::yesway:
    I never thought of that. :facepalm: Also, you could fill the unneeded space with thin wood strips, sawdust&glue, etc.
     
  14. Bobro

    Bobro SS.org Regular

    Messages:
    111
    Likes Received:
    44
    Joined:
    Jun 26, 2017
    Strictly speaking, my guitar is not exactly to the JI tuning it should be, and would require partial, or wavy, frets. The small chain of pure fifths at the center of the tuning (basically, the black keys on a standard piano) extends one pure fifth beyond what it "should", but this does not make a darn bit of difference in real life practice.
     

Share This Page

  1. This site uses cookies to help personalise content, tailor your experience and to keep you logged in if you register.
    By continuing to use this site, you are consenting to our use of cookies.