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Circle of thirds variation of Richter tuning

Posted: Sun Mar 11, 2018 2:27 pm
by karl.nilsson
The circle of fifths (https://en.wikipedia.org/wiki/Circle_of_fifths) is a very well-known concept in music. Slightly less known is the circle of thirds, or ladder of thirds.

If you pick every second note from a diatonic scale, there will be an interval of a third between each adjacent note.

In the following I will use examples from the C scale exclusively, for the sake of simplicity. It's all adaptable to any other scale, of course.

Example of a scale of a ladder of thirds: C E G B D F A C E G... etc.

Note that the thirds will vary between major and minor thirds. Any consecutive 3 notes will form a triad, either a major, a minor or a diminished chord. In this span of about 2 octaves, there will be 3 major, 3 minor and 1 diminished chord. Any 4 consecutive notes will form a tetrad, that is with the extra major or minor 7th on top.

The regular Richter tuning have parts which use this phenomenon. That's why chords such as C major, G major, d minor are available. The Paddy Richter tuning offers a-minor as well.

Here's a suggestion of a tuning, that I'm sure that many have thought of or even implemented.

Circle 3rds layout.png
Layout of circle of thirds tuning
Circle 3rds layout.png (19.3 KiB) Viewed 1661 times

As you can see, the first six holes are unaltered, as they offer good chord possibilities. Blow 4 to 6 and draw 2 to 6 are third ladders. Holes 7 to 10 are reversed between blow and draw, except that the new draw 10 is B instead of C. This way we have third ladders all the way from blow 4 to 10 and from draw 2 to 10.

As mentioned, ANY adjacent 3 or 4 holes will form a chord as in this table:

Circle 3rds chords.png
Chords available from circle of thirds tuning
Circle 3rds chords.png (47.82 KiB) Viewed 1661 times

The letters represent the chord formed from a blow or draw in the corresponding hole and the next 2 or 3 to the right.
  • C, G, F are major chords
  • am, dm, em are minor chords
  • B with a circle is a diminished chord, that is with a minor third and a diminished 5th
  • M7 is a chord with a major 7th
  • 7 is a chord with a minor 7th
  • B with a ø7 is a "half-diminished" chord with a diminished 5th and a minor 7th
Those diminished B chords are not very useful. But all the others are.

As an extra bonus there is no reversal of breath, and you can bend all draws. (Well, still not much on holes 5 and 7)

You can play all your regular melodies (unless you need the high C), and you can of course retune the blow 3 to a Paddy Richter A, extending the ladder of thirds, but the 3 bottom blow chords will be a-minor rather than C-major.

Haven't had one made yet, but I'll test it soon from an old harp.

Re: Circle of thirds variation of Richter tuning

Posted: Sun Mar 11, 2018 8:01 pm
by Kniri
Hello
Very interesting !
see my tuning in the circle of 4th and 5ths

Layout "Kniri 221"
G/B/D/F/A/C/Eb/G/Bb/Db-----  blow
A/C/ E/G/Bb/D/F/Ab/C/Eb-----  draw

8 Triads (blow)    G      B      D   //  F      A      C  //  Eb      G      Bb  //   Db
                     G -major            F-major           Eb-major
 
                                       G      B  //   D      F      A   //  C      Eb      G  //    Bb      Db
                                                                 D-minor          C-minor               
 
                                           G   //  B     D      F  //   A      C      Eb   //  G      Bb      Db
                                                       B dim/ G7       A dim/ F7           G dim/ Eb7
 
8 triads (draw)       A  //   C      E      G  //  Bb     D      F  //    Ab     C        Eb
                             C-major             Bb-major            Ab-major
                                           A      C      E   //   G     Bb     D   //  F       Ab     C   //    Eb
                                              A-minor              G-minor              F-minor
                                            A      C  //  E      G     Bb  //  D      F       Ab  //  C        Eb
                                                             E dim/ C7          D dim/ Bb7
www.kniri.ch (BluesAccordion)

Re: Circle of thirds variation of Richter tuning

Posted: Mon Mar 12, 2018 12:03 am
by winslowyerxa
This is a variation on spiral tuning, except that the spiral doesn't start until Hole 4.

Re: Circle of thirds variation of Richter tuning

Posted: Mon Mar 12, 2018 8:47 am
by karl.nilsson
Absolutely correct, Winslow. Spiral tuning goes all the way from first to last hole, read more about it here: http://www.bluesharmonica.de/english/sp ... uning.html

Nothing new under the sun, eh...

Kind of beautiful, though, isn't it?

Re: Circle of thirds variation of Richter tuning

Posted: Mon Mar 12, 2018 10:33 am
by karl.nilsson
As stated, this tuning, that could be called ladder of thirds, circle of thirds, spiral tuning, circular tuning, etc., is not new as I suspected from the beginning. It comes all natural, of course.

However, I'd like to expand a bit on the fine-tuning for intonation.

This paragraph is pretty basic for many musicians: When you want to form a triad such as C E G, you take the C note, then you put an E, a major third (5/4 = 386 cents), on top of that. Then you put a G, a minor third (6/5 = 316 cents) on top of that, which will result in a perfect fifth above the C. Which is 5/4 * 6/5 = 30/20 = 3/2 = 702 cents. You may continue with a perfect 4th (4/3) on top of that to reach the C one octave above the first C: 3/2 * 4/3 = 12/6 = 2, exactly the doubled frequency in relation to the first C. We have here a perfectly smooth and very nice sounding C major triad with an extra C on top (C E G C).

Now it gets more complicated: Things are different when it comes to a ladder of thirds. When we put 7 thirds in a row, we will go from the bottom C to the top C like this: C E G B D F A C. The top C is 2 octaves above the bottom C. All intervals are minor thirds or major thirds. All fifths, but one, are perfect (3/2). The fifth from B to F is a diminished fifth (kind of tritone), 2 minor thirds, one on top of the other (6/5 * 6/5 = 36/25 = 1.44 = 631 cents).

Adding all the thirds up, going from the bottom C to the top C, 2 octaves up, will put an offset by about 21 cents higher than the pure 2 octaves, almost a quarter note.

Now if you want all the triads and tetrads to sound clear, sweet and beatless (except for that tritone B - F), and you don't play together with other musical instruments, then you could use the following table for intonation (all cent values in this text above and below are rounded to integers). Otherwise, playing in an ensemble, you would probably be better off with a 12TET intonation, or a compromise tuning in between. That's it, folks.

Circle 3rds cents off.png
Circle of thirds, just tuning
Circle 3rds cents off.png (31.57 KiB) Viewed 1625 times

Re: Circle of thirds variation of Richter tuning

Posted: Mon Mar 12, 2018 11:12 am
by karl.nilsson
Hi Kniri!

Interesting table you have there. Very good for chords playing. That table is a bit hard to read though, due to formatting, so I took the liberty to paste the text in a text editor, and take a screenshot. Here you go:

kniri 221.png
kniri 221.png (41.28 KiB) Viewed 1620 times

Re: Circle of thirds variation of Richter tuning

Posted: Tue Mar 13, 2018 9:47 pm
by IaNerd
Karl: What are your thoughts on changing one or both dm chords to Dmaj, which in 2nd position G here would give a V chord?

Re: Circle of thirds variation of Richter tuning

Posted: Tue Mar 13, 2018 10:46 pm
by karl.nilsson
I've been thinking about that actually, and I would change both F's to F#, which would turn the key of the whole scale from C to G major. The chords would be: G major (I), a minor (ii), b minor (iii), C major (IV), D major (V), e minor (vi) and finally the dreadful f# dim (vii).
It's like the Melody Maker, only here in part circular tuning.

I would change both F's rather than one, just because I love the pure diatonic scale, with all the seven possible modal scales,with 1 diminished, 3 major and 3 minor chords.

Another thought is having the last note as C instead of B, changing the chord on the last 3 draws to C instead of e minor. It wouldn't break anything, and that last C note might be useful; the B note could be reached with a bend on the C if necessary. If that makes sense.

Re: Circle of thirds variation of Richter tuning

Posted: Wed Mar 14, 2018 9:58 pm
by karl.nilsson
I was thinking of this "reversed" circular tuning suggested by IaNerd, that is having the basic layout of a C harp, but retuning F to F#, thereby converting it to a G harp, with the tonic chord at draw holes 1-4, typical for 2nd position playing, so I started to configure the chord layout. It came out a little different than I had anticipated.

In the first example, there is the traditional Richter layout on the first 4 holes. That is, you'll get the G, tonic (I) chord on 1-4 draw and 6-8 blow, and C, the subdominant (IV) chord on 1-6 blow, and also on 7-9 draw. The D(7), dominant (V) here appears on holes 8-10 blow and on 4-6(-7) draw. And only one dim chord, which is great, because it's best avoided.

So far so good, chordwise. On the melody side though, I feel it a bit awkward to miss out the second scale note (A) between low G and B. And if we "paddify" the scale (A at blow 3), no more C chords at the low blow notes as you can see in second example.

Third example, just for fun. Circular all the way, except for the 10 draw, raised to C, as in the previous example, too. Just to get some extra C (IV) chord area. Lots of chords, but harder to find, and melodies could be nice, yet another learning curve though.

If anyone out there would like to try out one of these, or who already used circular tuning in some form, I'd be happy to hear about the results, pros and cons. I think I'll start out with my original 1st position Richter + Circular and maybe move on from there.

Circle 3rds chords G 2nd pos.png
Circle 3rds chords G 2nd pos.png (129.94 KiB) Viewed 1535 times

Re: Circle of thirds variation of Richter tuning

Posted: Wed Mar 21, 2018 10:25 pm
by karl.nilsson
I just felt I needed to present clearer diagrams of some ideas, and also compare to other existing designs.

So here are diagrams of standard tunings (and the chords that go with them) such as Richter, Paddy Richter, Melody Maker and some variations with circular tunings in different keys.

Might give ideas on which standard tuning to start with, and then apply some changes. Here goes:

Common and uncommon harp tunings.png
Common and uncommon harp tunings
Common and uncommon harp tunings.png (84.79 KiB) Viewed 1378 times

Note: "Reversed" here means that the first note of the home key and the lower tonic chord is on draw rather than blow. Blow: upper row, draw: lower row.