1/4-comma meantone calculation for Anglos

[David Barnert]

43 minutes ago, seanc said:

I am really not familiar with dealing in cents.

 

Notes can be described in terms of frequency or pitch. They have a logarithmic relationship to each other: Every doubling of frequency is an octave, every 12 semitones is an octave.

 

A cent is 1/100th of an equal-tempered semitone.

 

An octave has a frequency ratio of 2 to 1.

 

A semitone is 1/12th of an octave, or a frequency ratio of the 12th root of 2 to 1.

 

A cent is 1/1200th of an octave, or a frequency ratio of the 1200th root of 2 to 1, or approximately 1.0005777895 to 1.

 

As far as how many cents humans can detect, see

https://en.wikipedia.org/wiki/Cent_(music)#Human_perception

 

47 minutes ago, seanc said:

If you are 5.6 or 8.4 cents.. Does that translate into Your X note is 5.6% or 8.4% off? 

 

Given the above, “percent off” has no real meaning unless it is clear whether you are talking about frequency or pitch.

 

49 minutes ago, seanc said:

Also, is that a constant? would your G note always be 8.4 off? or does it become 4.2 down an octave and 16.8 up an octave?

 

You’re confusing frequency and pitch again. Assuming the “8.4” is cents (a measure of pitch), it will remain constant in all octaves. The difference in absolute frequency changes by factors of 2 in different octaves, although the frequency ratios remain constant.

[Little John]

58 minutes ago, seanc said:

If you are 5.6 or 8.4 cents.. Does that translate into Your X note is 5.6% or 8.4% off? 

 

There are 100 cents to a semitone, so it's reasonable to think that a note that is, say 8.4 cents flat is flat by 8.4% of a semitone.

 

1 hour ago, seanc said:

how this equates to what you hear..  Making that 8.4 your tone center and trying to play with others. Does that make it more noticable/ more out of tune?

 

Not sure what you're saying here. What matters is the individual note. If one note in MT differs from that in ET by 8.4 cents, it's not likely to be noticeable, but MT tuning differs from ET by varying amounts according to how far round the 'circle of fifths' they are from the point at which they agree. Look at the table I posted earlier.

 

The practical point is this: if the mean tone tuning agrees with ET at A (i.e. 440 Hz) then it's unlikely any note will be so far out as to be noticeable. I've played my 1/5-comma duet in various ensembles and it's never been a problem.

[David Barnert]

Think of it this way: How an instrument is tuned (ET vs MT, etc.) doesn’t so much make individual notes sound in or out of tune, rather it makes intervals sound pure or mushy. Once you’ve heard a major third tuned to a real 5:4 ratio, equal temperament will always sound like a cheap imitation.

[adrian brown]

1 hour ago, Little John said:

 

I agree with everything in this post except for that last paragraph, quoted above. 440 refers to the frequency in Hz (or cycles/second). 5.6, 8.4 etc. in the table refers to cents, i.e. hundredths of a semitone. They don't mix.

 

The first row shows that if G (in meantone tuning) is tuned to be exactly the same frequency as G in Equal Temperament tuning, then A will be 5.6 cents flatter than the A in ET. If you tune all the notes up 5.6 cents to make the A in meantone match the A in ET you simply end up with the second block in the table.

 

The purpose of the table is to illustrate that matching A to the A in ET minimises the discrepancy between mean-tone tuning and equal temperament in the keys from two flats to four sharps.

 

Me too Little John!

 

I have been using 1/4 comma meantone on most of my Anglos for many years and they are all tuned from the root A-440Hz. I have Anglos in CG, BbF, GD FC and baritone CG sizes, so if I had not done it this way, they would have been out of tune with each other. With each size, the wolf is simply shifted around the circle of fifths so it falls on the same button combinations on each instrument and I have one enharmonic note which sounds different on the press and draw (eb press - d# draw on a CG) effectively giving me one extra key possibility.

 

As Sean pointed out, if you use a different root on each row, the instrument will be out of tune with itself, so it's a really, really bad idea.

 

I have played many times with other instruments in ET and never had a problem, although you can hear the difference, particularly if you play with another concertina. In a group of more concertinas it's unlikely to be problematic because it's unlikely the concertinas themselves are all perfectly in tune and the blend of the different instruments will hide many discrepancies.

 

Some years ago, I made a rather convoluted video to illustrate the difference between MT and ET, by playing the same pieces on both. (https://youtu.be/HT9fStxlbBQ) I had always meant to follow it up with another showing two or more concertinas playing together. I've somehow never got around to it because of the editing involved, but if there is a feeling this could be interesting, I might try and knock one up.

 

Adrian

 

[Steve Schulteis]

2 minutes ago, adrian brown said:

I have been using 1/4 comma meantone on most of my Anglos for many years and they are all tuned from the root A-440Hz. I have Anglos in CG, BbF, GD FC and baritone CG sizes, so if I had not done it this way, they would have been out of tune with each other. With each size, the wolf is simply shifted around the circle of fifths so it falls on the same button combinations on each instrument and I have one enharmonic note which sounds different on the press and draw (eb press - d# draw on a CG) effectively giving me one extra key possibility.

 

Where do you choose to put the wolf?

[adrian brown]

2 minutes ago, Steve Schulteis said:

 

Where do you choose to put the wolf?

Probably best to simply share my calculations.

 

A.

Meantone temperament calculations for anglo concertina.pdf

[seanc]

37 minutes ago, David Barnert said:

Think of it this way: How an instrument is tuned (ET vs MT, etc.) doesn’t so much make individual notes sound in or out of tune, rather it makes intervals sound pure or mushy. Once you’ve heard a major third tuned to a real 5:4 ratio, equal temperament will always sound like a cheap imitation.

I agree that the Pure intervals always sound better/ right. I am primarily a fretless bass player. And like a violin or voice. When given a root, your brain and ear just pretty much forces to to play that "right" 3rd, or 7th in relation to the root. 

 

but on an instrument that you can't just slide around on, it's different. Assuming a C tone center. In order to make that 3rd (maj or min) sound "right" you have shifted the pitch. On a minor third (Eb), you are generally flattening from where the ET Eb is. And on a major 3d, you are generally sharp from where an ET E is. (same with 7ths).

 

The issue comes in, is that since you have shapened your Maj 3 (E). When you play your E in relation to other ET instruments, you will be considerably sharp.

 

 

 

[Little John]

3 hours ago, seanc said:

... And on a major 3d, you are generally sharp from where an ET E is. (same with 7ths).

 

The issue comes in, is that since you have shapened your Maj 3 (E). When you play your E in relation to other ET instruments, you will be considerably sharp.

 

No. you've got this back to front. In 1/4 or 1/5th comma MT tuning the major third will be flatter than ET; but 'pure' in 1/4 comma and close to pure in 1/5 comma.

 

4 hours ago, seanc said:

And on a major 3d, you are generally sharp from where an ET E is. (same with 7ths).

 

Again, wrong. Interesting to note that one-row cajun melodeon players often have the 3rd and the 7th tuned down 14 cents to make them more musical.

[Clive Thorne]

6 hours ago, adrian brown said:

Some years ago, I made a rather convoluted video to illustrate the difference between MT and ET, by playing the same pieces on both. (https://youtu.be/HT9fStxlbBQ)

 

Adrian

 

I've just listened to this. Maybe I am "cloth eared", but to me the biggest difference was the base tone of the instruments The metal ended ET instrument having more "attack" and edge than the wooden ended MT one. I prefered the tone of the metal ended ET one, and to be honest, did not notice much difference that I would attribute to the tuning.

As I say maybe I am cloth eared, or maybe I am just used to playing equal temperament (Metal ended).

 

Lovely playing by the way, irrespective of temperament!

[David Barnert]

3 hours ago, Little John said:

Interesting to note that one-row cajun melodeon players often have the 3rd and the 7th tuned down 14 cents to make them more musical.

 

And recorder players (I am one) learn to put one or two fingers down (or near the holes) when playing the 3rd of a chord to flatten it a little when playing early music.

 

Here’s the math: A pure major 3rd has a frequency ratio of 5/4, or 1.25. An equal tempered major 3rd is 4 semitones, or 1/3 of an octave, or a frequency ratio of the cube root of 2, which is very close to 1.26. So a pure major 3rd is smaller than an equal tempered major 3rd. By about 14  cents.

[adrian brown]

11 hours ago, Clive Thorne said:

I've just listened to this. Maybe I am "cloth eared", but to me the biggest difference was the base tone of the instruments The metal ended ET instrument having more "attack" and edge than the wooden ended MT one. I prefered the tone of the metal ended ET one, and to be honest, did not notice much difference that I would attribute to the tuning.

As I say maybe I am cloth eared, or maybe I am just used to playing equal temperament (Metal ended).

 

Lovely playing by the way, irrespective of temperament!

Thanks Clive,

 

I think that outside of the obvious clashes I highlight in the video, that occur when you push to the limit of meantone tuning, there's probably much less in it than a lot of people think. You can certainly get used to either, or both and it certainly shouldn't be something that gets in the way of the music! As I pointed out in the video, it rather depends on the sort of repertoire you want to play. In folk and pop music where parallel fifths are quite acceptable, the cleaner fifths you get with ET is probably a better option. In Renaissance and Baroque music where thirds were a consonant interval, and parallel fifths an aberration, MT is the obvious choice.

 

Adrian

[Luke Hillman]

Thanks to everyone who's responded so far, particularly @Little John for the easily comprehensible explanations. It seems like the best practice is to either use A = 440 as the reference note for the whole keyboard, or to shift everything so A is 440 Hz whatever the reference note.

 

@Steve Schulteis, the temperament calculation video is a great explanation, thanks for the link.

 

On 3/1/2023 at 11:27 AM, adrian brown said:

[...] I have one enharmonic note which sounds different on the press and draw (eb press - d# draw on a CG) effectively giving me one extra key possibility.

 

Some years ago, I made a rather convoluted video to illustrate the difference between MT and ET, by playing the same pieces on both. (https://youtu.be/HT9fStxlbBQ) I had always meant to follow it up with another showing two or more concertinas playing together. I've somehow never got around to it because of the editing involved, but if there is a feeling this could be interesting, I might try and knock one up.

 

Aha, I was wondering how folks were handling the different enharmonics!

 

And, Adrian, some time ago, I actually downloaded your video and chopped it up so I could overlay your examples onto each other and hear the two temperaments simultaneously—it was the only way my untrained ear could hear the difference! I didn't save it, alas, but it definitely helped me understand what was going on.

Edited March 3, 2023 by Luke Hillman

[Little John]

15 hours ago, Luke Hillman said:

Aha, I was wondering how folks were handling the different enharmonics!

 

By coincidence, or probably following the same logic as @adrian brown, I too have a button which plays D# on the push and Eb on the pull. That's on a Crane duet.

[bax]

Wally Carroll made me one in 1/5 comma meantone with the wolf at G#/D# and meeting A=440 at D. It's lovely, plays well with others, and the major thirds are juuust enough prettier for me to notice. I'll see if I can tack on a pic with the details.

 

... The wolf interval is not available on the instrument anywhere

 

Edited March 7, 2023 by bax

[Little John]

8 hours ago, bax said:

Wally Carroll made me one in 1/5 comma meantone

 

This is interesting. It shows that meeting ET at D rather than A gives the same spread of deviations. The difference is that the note with the biggest discrepancy becomes G# rather than Eb.

 

8 hours ago, bax said:

... with the wolf at G#/D# ...

 

... The wolf interval is not available on the instrument anywhere

 

These two cannot both be true! In fact the table is misleading. The note labelled A# is in fact Bb. In ET they are the same pitch but in mean-tone tuning they are noticeably different. Likewise the note labelled D# is in fact Eb. The wolf fifth is at G# - Eb - considerably wide of a perfect fifth. The instrument doesn't have a D#, but if it had then G# - D# would be an (almost) perfect fifth.

 

 

Edited March 7, 2023 by Little John
Clarity

[adrian brown]

13 hours ago, bax said:

Wally Carroll made me one in 1/5 comma meantone

 

Without wishing to get too pedantic, I feel I should point out that in this example, the instrument is not at A=440Hz, but at A=439.41Hz, making the whole instrument slightly flat of concert pitch. If this can be considered negligible, it begs the question of why you wouldn't simply use the A reference in the first place?

 

4 hours ago, Little John said:

 

This is interesting. It shows that meeting ET at D rather than A gives the same spread of deviations. The difference is that the note with the biggest discrepancy becomes G# rather than Eb.

 

 

These two cannot both be true! In fact the table is misleading. The note labelled A# is in fact Bb. In ET they are the same pitch but in mean-tone tuning they are noticeably different. Likewise the note labelled D# is in fact Eb. The wolf fifth is at G# - Eb - considerably wide of a perfect fifth. The instrument doesn't have a D#, but if it had then G# - D# would be an (almost) perfect fifth.

 

 

 

I agree Little John, but again I feel I should point out that you can put the wolf anywhere, independently of where you set your reference note. In this example, D is the reference and so G# is directly opposite in the circle of fifths, rather than the Eb, if you start from A. However, there's nothing to stop you going only 2 fifths in one direction and 10 in the other. You can use a reference D and still have Eb as the biggest discrepancy if you so wish.

 

Lastly, where you decide to put the wolf and by extension which notes you would like enharmonic alternatives for, depends first on which chords you regularly use. In my own case, I opted for enharmonic Eb/D#s because I was fed up with having to choose between a clean C minor or B major chord, and wanted them both...

 

Adrian

[Little John]

1 hour ago, adrian brown said:

... I feel I should point out that you can put the wolf anywhere, ...

 

Quite so, but the table indicates that in this case the wolf is actually G# - Eb; despite Eb and Bb being wrongly labelled as D# and A#.

 

1 hour ago, adrian brown said:

... I opted for enharmonic Eb/D#s because I was fed up with having to choose between a clean C minor or B major chord, and wanted them both...

 

I, too, have an enharmonic Eb/D# button although if I were to opt for just one it would be the D#; because a C minor chord with a D# is surprisingly acceptable; whereas a B major chord with an Eb is simply unbearable!