EC tuning mystery solved....I think

[RogerT]

Firstly, thanks Alex Holden for that brilliant little summation on temperament, and also Don's post, with the 1848 ad, which implies that Alex may be onto something.

[Matthew Heumann]

As a non-academic musician, I'm overwhelmed with all these revelations. All I can say as an English player (for 51 years) is that its awfully damn nice to have the duplicate notes when encountering arrangements where going from the natural to the accidental is extremely awkward given the other buttons being pushed. Sometimes its so much easier for instance to be able to use the G# rather than the Ab or D# rather than the Eb button. I've never given the temperament issue any thought because for me it was just plain brilliant to offer alternate fingering. Although recently I had a discussion that suggested that many of the Salvation Army bands played in Bb a lot, so replacing a low G# with a F would be quite logical.

Bottom line, its easier to have extra accidentals than extra fingers..................................

[Little John]

On 9/1/2024 at 9:53 AM, Theo said:

There is another possible explanation for the enharmonic notes on an EC.  If they were not there there would be gaps in the layout.  So perhaps it's just a convenient way of making a neat and more consistent pattern of buttons?

 

True, the gaps would look odd so filling them in is more aesthetically pleasing, but I think there's a more important reason for including them.

 

The English system was devised with a logical consistency of fingering in mind: each accidental (sharp or flat) should be adjacent to its natural. This allows scales to be played alternating left-right for each note in turn. As there are seven naturals to the octave, that leaves space for seven accidentals, where a piano has only five; thus allowing for the "duplication" of two accidentals. Wheatstone chose to make these D#/Eb and G#/Ab, giving four sharps and three flats.

 

If, say, the D# and Ab were omitted in theory (and assuming equal temperament tuning) the English concertina would still be fully chromatic; but the alternating left-right fingering would be lost for a couple of keys; for example E major, in which the passage C# D# E would have to be played as C# Eb E - all three consecutive notes on the same hand - thus breaking the fingering pattern.

 

The English layout also means that a simple chord (a triad) can be formed on just one side of the instrument (i.e. all three notes are the same side, whether that's right or left), but again with (for example) the Ab missing an F minor chord would be split between the hands as F G# C - again breaking the pattern.

 

The truth is that, if the English concertina had only 12 notes to the octave, then despite being fully chromatic it would be possible to play in only 6 keys while retaining the pattern of alternating hands for scales and the same hand for triads. Adding the extra two accidentals increases that to 8 keys; that is those between 3 flats and 4 sharps. You can still play in C# major (seven sharps) if you want but the consistent fingering patterns will be well and truly broken.

 

Now if the English is tuned to a mean tone temperament the keys you can play in without breaking the consistent fingering patterns are exactly the same as those you can play in without encountering the wolf fifth. This is no coincidence. So if you stick to keys that retain the consistent fingering pattern you are also sticking to the keys that sound good in mean tone tuning, and vice versa.

 

It's speculation, of course, but my bet would be that Wheatstone settled on the consistent fingering patterns first and then realised the ability to use mean tone tuning was an added bonus.

 

 

[4to5to6]

I’ve always wondered about this.  Thanks.

 

The Pythagorean Comma:

 

 

As a former piano tuner, temperaments are a fascinating topic although the topic can be very controversial.

 

Why oh why couldn't all the intervals just be pure in all keys?  Is this Heaven? 

 

I highly recommend the book: How Equal Temperament Ruined Harmony: And Why You Should Care by Ross W. Duffin

.

[March Hare]

 

I just bought a Wheatstone EC treble which is in its original tuning from 1849 (see my earlier post and image).  After all these years of not understanding the purpose of the extra accidentals, temperament etc, just playing it gave me an inkling of how brilliant yet simple the design is.  I will try to share what I think I have grasped so far.  Please forgive mistakes, approximations etc - it’s a tricky and controversial topic - but I’ll have a go.  It seems to be all about the thirds......

 

1. My starting ideas:

• Pure (just) tuning is based on whole-number ratios of frequencies eg. if A4 is 440Hz then A5 is 880Hz and is an octave higher.
• E5 is a pure fifth above A4, ratio 3:2 so E5 is 660Hz.   
• C# is a pure third above A4, ratio 5:4 so C# is 550Hz.
• If you play these three notes together you hear an A major chord with pure intervals. so the chord sounds sweet.   
• I have lived with equal temperament all my life - but now that I have a concertina playing pure thirds I can hear the difference.

 

2. With my treble EC in my hands, I will focus on the range from C4 to B4 (12 notes, each a semitone from the next, ie C, C#, D, D# etc):

• Middle C  (C4, lowest C on the LH by the thumbstrap) is 261.63Hz.  Then C5 is 523.26Hz (doubled).
• G4 should be 3/2 x 261.63 = 392.45Hz and E4 should be 327.04Hz for pure intervals. My early Wheatstone is tuned like this, as far as I can tell.
• On my ET treble, G4 is 392.00Hz (so that’s near enough)
• BUT:  in ET E4 is 329.63Hz - quite a bit sharper.

 

Rounding off the numbers, it is the third that is sharp/out of tune in ET:

              ET          PURE

   C4     262         262

   E4     330         327

   G4     392         392

(You can buy a 10-hole harmonica or blues harp in ET or pure tuning, so there is lots of discussion online about how they might sound different.)

 

3. So why not tune E4 to give a pure major third?

• Yes, you can do that.  But you get into trouble if you continue the pattern.  From C to E is four semitone steps.  So a major third (four semitones) above E4 is G#.  This should be 5/4 x 327.04 = 408.8Hz.  Then another 4 semitones completes the octave to C5:  5/4 x 408.8 = 511Hz.  For an octave this should be 523Hz (see above).
• So three pure major thirds do not equal an octave.
• The octave has to be pure, so you have to do something about the thirds.

 

4. Charles Wheatstone’s wonderful design solves this problem so neatly.

• From C4 to B4 you need 12 buttons (C, C#, D, D# and so on to B).  The EC keyboard has 14, so only two spares.
• Start on C4, then major third to E4, then major third to G# - all pure.  Now, swap sides and play Ab4 to C5 - this can also be pure because Ab is sharper than G#.  If you start at C5, 523.26 x 4/5 makes Ab5 418.26Hz.   So these major thirds and the octave are pure.
• Another good outcome is that if you play any note, then the major third above it is always on the same side of the instrument, so you can always use two fingers of one hand.
• Now start on B3: B to D#, swap sides for Eb to G, then G to B.  All pure.
• Every note has a pure major third above it except for : C# (there is no E#) and F# (there is no A#).

 

So with this beautifully simple keyboard and only two spare buttons per octave, the great man gives us ten pure major thirds out of twelve, which means ten out of twelve pure major chords:  C, D, Eb, E, F, G, Ab, A, Bb, B.

 

Compare this with a concertina in equal temperament, where every major third and every major chord is slightly out of tune.

 

Personally I have always been happy with ET, as I grew up with it, I’m old, my ear isn’t that good, and would never change my Edeophone.  But that isn’t the point.  I am just trying to understand the layout.

 

5.  I have made some very simple measurements to put on a diagram (attached).

My phone has a tuning app which names the note it hears and has a scale in semitones.   I believe that a major third in ET is 1/7 of a semitone wider than a pure third.  So I can estimate from my app how many 1/7ths sharp or flat a note is compared with ET.  If the note is 1/7 sharp I have marked it +1, or -1 for 1/7 flat etc.   These are my rough measurements (in 1/7 semitones).  Bear in mind that the instrument has probably not been tuned for a century or so and I am correcting for it all being in high pitch.

 

(See layout diagram)

 

All the major thirds are reduced by 1/7 of a semitone (except for the F# and C# as explained above) to make them pure.

All of the octaves are pure and most of the fifths are pure.

And all this on a superbly logical keyboard with only two spare buttons per octave.

 

6.   Thank you for your patience (if anyone has read this) and please forgive my errors.

I hope that it is of interest to someone - I get a lot of pleasure and insight from reading the posts on this excellent forum.

PS. If anyone would like me to post a recording of the thirds and chords I am happy to do that.

Edited October 1 by March Hare

[March Hare]

The layout diagram is attached below:

Edited October 1 by March Hare

[March Hare]

     I hope this is clearer.

Edited October 1 by March Hare

[March Hare]

                          LHS                                                                         RHS

 

                                      D     D#                                                                 C     C#

                                      0      -2                                                                  0      -2

        Bb       B                                                                      Ab     A

        +1       -1                                                                     +1      -1

                                      G      G#                                                                F      F#

                                      0       -2                                                                 0      -1

        Eb       E                                                                      D#      D

        +1       -1                                                                     -2       0

                                      C      C#

                                      0       -2

 

I hope that this is the same, but the display (desktop, iPad etc) can alter the spacing.

Edited October 1 by March Hare

[Little John]

On 9/30/2024 at 5:38 PM, March Hare said:

I have lived with equal temperament all my life - but now that I have a concertina playing pure thirds I can hear the difference.

 

That was my experience, too a few years ago. I'm a convert to mean tone tuning now. Evangelical, even!

 

The most important thing to grasp about tuning temperaments is that they are all compromises. As you have observed, the big compromise with ET is that the thirds are sharp by about 14 cents. (An ET semitone is 100 cents. The 1/7 you mention is approximately 14/100 - 14 cents.) Pure intonation overcomes this, but the big compromise is that it only works in one key. You could tune a concertina to have a pure C major scale, but any other key will be wrong.

 

The solution applied to concertinas (both English and Anglo) in the 19th century was quarter comma mean tone tuning*. This makes the major thirds pure. The fifths are little flat but that doesn't seem offensive. The main compromise is that it gives six good keys**, but six somewhat nasty sounding ones.

 

[* if you stack four pure fifths (3:2) you get 81:16, which is the major third. Bring it down two octaves gives 81:64, which is approximately 1.266. This is greater than the 5:4 (=1.25) of the pure major third. The difference is called the syntonic comma. Since you've stacked four pure fifths to get this, the solution is to take a quarter of the syntonic comma off each fifth. The resulting major third is then pure, and the tuning is quarter comma mean tone.]

 

[** Eight in the case of the English concertina, because of the two extra accidentals.]

[March Hare]

Thank you Little John, that's brilliant.

 

It sent me back to my keyboard chart of (rough) measurements of the early Wheatstone.  I went through all 12 major keys note by note and recorded the spread of differences (0, -1, +1 etc) for each one.

C, Eb, E, G. look like the 'best' keys (range of 1).  

D, F, A, Bb may be 'OK' keys (range of 2).  

C#, F#, Ab, B look worst (range of 3).

 

Now, picking up the concertina, playing in the best keys sounds nice, for tunes and chords.

The OK keys also sound fine, tunes and chords.  Song accompaniments work very well in all these keys for me.

For the worst keys, C#, F#, Ab, B, at least one of the three major chords in that key is unusable - so these keys are ruled out.  (I can't play fluently in these keys anyway and I don't use them for songs.)

 

So my empirical data (rough as it is) fits your 'eight good keys' exactly, and my (amateur) ear confirms this.

 

Nothing earth-shattering here, but it does seem to work.

 

Thanks again - it pleases me.  And I have learned a lot from this.

 

PS. And I take your point that I may be in danger of being drawn into meantone tuning.......