Tuning a keyboard instrument.

[Hugh-AR]

I'm sure that all of us who play keyboards are aware that the notes we play have certain 'frequencies', and the frequencies form the melodies and harmonies of what we hear. When tuning an orchestra, the note the various instruments are tuned to is A = 440Hz, and our keyboards are set to A = 440Hz so we are 'in tune' with everybody else.

An 'octave' (eg. C to top C) is split into 12 'steps up' which gives us our white and black notes. But I wonder how many of us are aware of how these notes are related to each other mathematically, and how we have ended up with these 12 'intervals'? Our keyboard notes are tuned to what is called 'Equal Temperament', but looking back at the history of music and the early composers, the mathematics of how we hear music has been a nightmare for them. Pythagoras was the first person to analyse the relationship between the frequencies of the various notes and work out a 'tuning system' for them. His system sounded fine so long as you played everything in the one basic key, but sounded all 'out of tune' if you tried changing key or playing harmonies that were not the simple harmonies of the key you were playing in.

So put your mathematical hat on, make sure you have half an hour to spare, and take a look at this. It's fascinating!

The Mathematical Problem with Music, and How to Solve It

[Ron]

This is right up your street Hugh with your background in Mathematics. Sorry, but I am not so talented unfortunately

Ron

[Hugh-AR]

Ron,

Den has done a Topic called Other Tuning Scales available on the Tyros. Click the below to read what he has said about the tuning of our keyboards.

Do a right-click to open this up in a New Tab.
http://www.tierce-de-picardie.co.uk/viewtopic.php?f=479&t=10394

Below are some comments from me on the age old problem of tuning our pianos/keyboards so they are 'pleasant' to listen to.

I thought at first that for EQUAL TEMPERAMENT tuning you would just split the difference in an octave into 12 equal portions. With A = 440Hz and the octave higher being 880Hz, splitting the difference (440 Hz) into 12 equal portions (440 ÷ 12) would be adding 36.667 to each semitone. But this is not the case. The ear and brain perceive sound as ratios between one note and the next, so we need a 'multiplier' to get from one note to the next. Our EQUAL TEMPERAMENT tuning is attained by multiplying the previous frequency by 1.059464 (approx). This is the number he mentions in the video as 'the twelfth root of 2'.

From Wikipedia:

There is a magic number in western music, known as the twelfth root of two, and it has a value of approximately 1.0595. This is the number that, when multiplied by itself twelve times, gives a result of two. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries.

Take a look at what the actual frequencies of our notes are.
Click the direct LINK underneath the chart to see it more clearly.



Do a right-click to open this up in a New Tab.
https://mixbutton.com/wp-content/uploads/2021/08/Music-Note-To-Frequency-Chart-01-2-1024x516.jpg

A is 440Hz
A# is 440 x 1.059464 = 466.16
B is 466.16 x 1.059464 = 493.88
C = 523.25 etc.

Keep multiplying by that ratio and you end up with top A as 880.01

Tuning our keyboard/piano/organ in this way (EQUAL TEMPERAMENT) means that the ratio between notes remains the same whatever note we start on. So whatever note you start with (ie. whatever key you play in), all things being equal, what you play will always sound the same (albeit 'higher' or 'lower').

The problem with tuning in this way is that the ear (brain) only regards sounds as being 'pleasant' to listen to when the notes played are in 'perfect ratios' (which clearly 1.059464 is not).

As he says in his narrative, the FIFTH note (E in the scale of A) should be in the 'perfect' ratio of 3:2 .. which makes the 'perfect frequency for the ear' = 660Hz. Our E note using EQUAL TEMPERAMENT is 659.25 .. so a little 'out of tune'.

Out of interest, in Den's topic above, where he plays a C scale and then a C chord .. although I can't hear the 'scale' notes as not being quite 'true', I can hear his C CHORD as being 'out of tune'.



I think my ear would prefer to hear that fifth as a perfect 3:2 ratio. What he was talking about in that video was that if you get one octave of 12 notes in 'perfect ratios', then if you play in a different key everything sounds out of tune because the ratios between the notes will no longer be a 'perfect ratio' .. or even approximate to a 'perfect ratio'.

One of the reasons why I was so fascinated with this mathematical explanation in the video above is that this is how Don Wherly (DonW) 'tunes' organ pipes (he was an organ tuner/repairer). He says he starts with a tuning fork of A , then does the octave (880Hz) .. and then goes up in 5ths or down in 4ths of the scale with the note he has just tuned as the 'root'. This all has to be done 'by ear' (same as a piano tuner) as apart from the frequency at the start with the tuning fork, he has to 'listen' to the pitch of the pipe. And in particular, he is listening for 'beats' when he plays two notes together. These 'beats' occur when one of the pipes is not on a 'perfect ratio'. But you can't tune an organ to 'perfect ratios' across the board, and he knows what beats to expect, and leaves the note slightly 'out of tune'. He also has a C tuning fork and can start the process with that.

[Hugh-AR]

Well, isn't it strange how when you are just dabbling about looking at previous posts in the Forum (and not looking for anything in particular) you come across something relevant to what you are discussing!

In this case I have come across a post put up by SysExJohn in April 2015 on this very subject .. the tuning of your keyboard! It's called Tuning and temperament and is well worth a look at as he has his own explanation about it.

Do a right-click to open this up in a New Tab.
http://www.tierce-de-picardie.co.uk/viewtopic.php?f=322&t=3783

This is his last statement in his discourse:

So, I hate to tell you this, but your keyboard is completely out of tune except for the octaves.

Hugh

[Ron]

It would come as a great surprise to myself and possibly others that this is why there are variations in the recordings posted on this Forum.

Ron

[SysExJohn]

Thanks for the reference Hugh.

It's interesting, at least it is to me, that it was probably only later in the 20th century that we actually started using E.T.
When an academic started testing Broadwood pianos, that were supposed to be in Equal Temperament, he found that they were not. They were close but favoured more consonant thirds, as far as I recall.

In Bach's time ET was known about (gleichswebenden Temperatur) and was disliked by most as every key signature sounded exactly the same.
They had just come from a system where the thirds were always in perfect tune, disfavouring fifths slightly (I believe).
They were working out new ways to tune their harpsichords and organs.
Pianos weren't commercially available until 1746 just a few years before Bach's death. It doesn't appear that he ever owned one.

Did you know that there were two pitches to tune to?
One for the orchestra and another, a semitone higher as I recall, for the organ, Kammerton and Chorton.

But, yes, except for octaves all intervals in an instrument tuned to ET are slightly out of tune, the thirds the most.
The fifth is almost perfect.

If you're interested, there's a web site put up by Bradley Lehman about how to tune a harpsichord to Bach's "squiggle tuning"!
The squiggle being the ornament at the top of his Wohltemperirte Clavier.

JohnG.
aka SysExJohn.

[Hugh-AR]

Thanks for that, John. So probably the reason why I could hear that C chord in the post above as being 'out of tune' was because of the 'third' rather than the 'fifth'.

I have done a Google Search for 'What frequency is a perfect third' and it has come up with:

Major third frequency.
The major third has a frequency ratio of 5:4. Playing a 375 Hz note over a 300 Hz note gives you an interval of a major third. The major third is a central note of the diatonic major scale, and the primary note that distinguishes the feel of "major" from "minor".

So with A as the 'root' (440Hz), the 'third' (C#) would have a frequency of 440 x 5/4 = 550Hz. Looking at the Frequency Chart, C# in our Equal Temperament tuning system has a frequency of 554.37Hz.

So the 'third' is 4.37Hz higher than it should be to sound perfectly 'in tune'.

The perfect 'fifth' is in the ratio of 3:2, so E should be 660Hz. With Equal Temperament tuning it has ended up as 659.25.

So the 'fifth' is 0.75Hz lower than it should be to sound perfectly 'in tune'.

This confirms to me that it was more the third that made me think that chord was 'out of tune' .. but the two together are in opposite directions, so no wonder it sounded to me the way it did.

Now this is all very unfortunate. As you know, I like listening to the harmony in a song, but now I know those harmonies are 'out of tune' I shall probably be aware of that every time I listen to something. Reminds me of a guest we had staying at our hotel who was the pianist for an orchestra that were playing at our local Playhouse Theatre. He asked me to play something for him on my AR80 organ, so I started playing a Glen Miller number .. At Last. He stopped me in my tracks and said I was playing it in the wrong key so he couldn't listen to it. I was playing it in A, and he said it should be played in Bb. Well, I knew that I was actually playing it in G, so he had that wrong for starters. But then I realised that although I was playing it in G, I had transposed my organ 'up' a couple of semitones .. so was effectively playing it in A. So I 'upped' it another semitone and played it again .. and he was happy.

Obviously he had 'perfect pitch', and this, rather than being an asset to him, meant that he couldn't listen to music that was played in the wrong key. I don't have perfect pitch .. but I don't like listening to music that is 'out of tune'.

Hugh

[SysExJohn]

Hi Hugh,

Yup! You've now got it and all you've written about thirds and fifths etc. is quite correct.
Of course, with fixed pitch instruments like piano and organ but also with most woodwind and brass and the dreaded accordion and harp we are stuck with the tuning faults of E.T.
Most people accomodate it, i.e. get to think that ET thirds are in tune.
But some of us can hear it, especially when the tones are quite 'pure' and held for a while. That is, it's less perceptible when it's a piano rather than an organ being played.

Trombones and the violin family are able to move hand/finger position slightly to get perfect harmonies.
A good string quartet can sound absolutely wonderful, pure hamonies no matter what key they're playing in.
Really good wind players can change their embouchure ever-so-slightly to alter the tuning of the note they're playing.
e.g. the clarinet opening of rhapsody in blue.

And this is what the old temperaments were about, each key had its own 'flavour' for want of a better word.

I sometimes mess about with the Scala tables within the orchestral VSTs that I use to get a more authentic sound to e.g. baroque music.

Regarding perfect pitch, my parents were told when I was just six that I had perfect pitch, but that's never stopped me liking a piece played in a non-original key.
I just find it interesting, not wrong, but I always sing it in the key I first learned it in.
I was a treble chorister until I was 14 and was often asked to sing solo pieces ... then my voice broke!

Pythogaras, apparently, invented an instrument called a monochord.
A single string where you could slide a bridge underneath at various points, and that's how he worked out pure tuning ratios.

I find it fascinating.

JohnG.
P.S. There's a book I found interesting published a few years ago by Ross W. Duffin called "How Equal Temperament Ruined Harmony (and why you should care)".

[Hugh-AR]

With Equal Temperament tuning, although the octaves are exactly in tune (Ratio 2:1), all the other notes are 'slightly out of tune' (and some more than others). When notes are 'out of tune' mathematically (ie. they are not in perfect ratios) not only does the ear perceive them as such, but you get 'beats' sounding when you play the two notes together. On the plus side, whatever discrepancies' there are, they are the same whatever key you are playing in. Or another way of putting it, because we want our modern instruments like pianos/organs to work in all keys, we divide the dissonance up between all keys equally.

There are some who think the sound we get from Equal Temperament tuning lacks 'colour', and Yamaha deliberately tuned their AR Series organs (AR80 and AR100) at different pitches in certain areas to make them sound 'richer'. The only part of the organ that was tuned to Standard A 440Hz pitch (zero) was the Lower Manual, and in their instructions they tell you that if you are wanting to 'tune' orchestral instruments from the organ you must choose the Clarinet from Lower 1 (with no vibrato) and play the A 440Hz on the Lower Manual. Below is a chart showing how Yamaha organised these changes of pitch.

The Pitch of the Yamaha AR .. (taken from the Owner's Manual)

When choosing VOICES for playing the melody, the upper manual has three areas where you can get your Voice from.
You can choose eg. a trumpet from the LEAD VOICE section (VOICES chosen from this section are MONO so only play one note at a time .. if you play two notes it will only play the higher one), which would then be pitched 6% higher than concert pitch. Or you could choose your trumpet from the Upper Voice 1 section, which would be 2.4% higher than concert pitch; or from Lower Voice 1 section which would be 1.2% higher. We are talking about the same trumpet 'Voice' here! Note: You can't actually choose a trumpet that's playing at concert pitch!

When playing chords on the lower manual, if you wanted them to be 'in tune' with concert pitch you would have to choose eg. strings or choir from Lower Voice 1. If you chose them from Lower Voice 2 they would be 1.2% 'flat.

And your bass pedals are tuned to be 3.6% 'flat'.

Yamaha's idea of making the AR80 organ sound more 'colourful' certainly worked with me, as after listening to several organs including the Roland, I was so struck by the AR80 that I bought one (and still have it). When playing it, I would always choose my upper voice from the LEAD section, as that always sounded 'richer' to me when playing the melody.

I have a few examples of this for you to listen to, played by AR Group Members who play the organ a lot better than I do!
So click the below .. and tell me if you think the solo instruments sound 'richer' because of what Yamaha have done.

Mark Burbridge playing Trumpet Voluntary on a Yamaha AR80 organ.



Don Wherly playing Meditation (a violin piece) on a Yamaha AR100 organ.



I asked Andy Gilbert (andyg) .. who is a music teacher, has his own Music School, and used to work for Kawai .. what he thought about all this, and his comment is below.

[andyg]

Deliberate detuning that's fixed, as found in the AR100, is something I'm personally not really keen on. Yes, in theory it should add to the ensemble effect, but it's not the same effect as the multiple, constantly varying pitch differences in a natural ensemble. What I find is that it often creates unwanted phasing artefacts between voices likes strings and choir, and unwanted beats between the non-ensemble type voices. If you're playing drawbar organ sounds, for example, and hold down a static chord, melody note(s) and pedal, you'll hear the beats clearly, and there really shouldn't be any on a Hammond. Much though I love the AR100, this aspect of it niggles me!

It's something we tried at Kawai in the early 1980s and ditched because of the issues above.

[Hugh-AR]

Having discovered how I could alter the pitch for R1, R2 and R3 on my Tyros 4, I thought I would take a leaf out of Yamaha's book and see how much of a difference it would make with the sound if I 'upped' the pitch of a couple of instruments by +6.

In this piece (It's Raining In My Heart) I have set RIGHT 1 to +6 ...



.. and used the voices Violin (for the Verse), and Golden Trumpet (for the Chorus). Both (in turn) are in R1.

First time round I have set the Tuning in R1 to zero (ie. these instruments are playing in Concert Pitch).
Then second time round (at 01:25) they are both on +6.
Now +6 is not enough to make these instruments sound 'out of tune' .. but do you reckon they sound 'brighter' as Yamaha suggest?



Hugh

[Ron]

Yes Hugh, definitely a brighter sound on the violin bearing in mind I am listening on my iPad and not using headphones.

Ron

[Hugh-AR]

Another interesting discovery on YouTube.

Johann Sebastian Bach's Toccata and Fugue in D Minor BWV 565 played by organist Hans-André Stamm on the Trost-Organ of the Stadtkirche in Waltershausen, Germany.



This is one of the comments under the video:

To those complaining that the pitch is off ... this Trost organ was built in 1730 when Bach was very much active as a composer. There was NO pitch standard across Europe at that time, and even the organ in the adjacent village might be tuned differently. The tuning "standard" of A = 440 Hz is a MODERN convention ... this organ is actually tuned at A = 466.8 Hz. Mr. Stamm is being absolutely AUTHENTIC. Stop criticizing this amazing organ!

I wonder if our TDP Members Don Wherly (organ tuner) or JohnG (who said, "my parents were told when I was just six that I had perfect pitch") had noticed this?

Well, I can hear the difference! So long as I can listen to extracts of them both .. one after the other. So here's a version played on an organ tuned to A = 440 Hz. Flit from one to the other and see if you can hear the difference.

Toccata and Fugue in D Minor (Best Version Ever)

The Trost-Organ video sounds a lot 'brighter'! It would do, as this one just above sounds a bit 'flat' to me!

Hugh

[SysExJohn]

In Bach's time there were two tuning fundamentals, one for the choir and another for the orchestra, known repectively as "Chorton" and "Kammerton" (choir tone and chamber tone).

If you care to visit bach-digital.de you will find many (most?) of Bach's existing scores in their original manuscript form. (Beware it's mostly in German.)
(N.B. you have to be pretty obsessed (or a bit of a nerd) to spend time analysing handwritten scores.)

At that site, if you care to look through some of his cantatas, you will find amongst the 'parts' sheets a score for organ written in Bach's own hand (rather than one of his copyists) in a different key to those of the orchestra.
This highllights the tuning difference.
From memory, I think his Kammerton A was 415Hz. I can't remember (and to lazy to work it out now) whether that puts the organ and orchestra a tone or a semitone apart.

I did some experimentation more than ten years ago with Das Wohltemperirte Clavier preludes and fugues, tuning to A=440 or 415 and different temperaments. I could hear the difference quite clearly.
I'll see if I can find the files and post some for people to listen to. ... Don't hold your breath, will you?

JohnG.