A survey of tuning systems in music
Tuning systems in music: a gentle introduction
To make music, you need an instrument or, of course, your voice. Both have to be tuned, in that specific pitches (or notes) should be produced at given time. How does it work? We will work out details in depth here, but a little bit of mathematics cannot be skipped.
Let us focus, for instance, on a guitar string (the image on the left rather depicts what happens within an organ pipe, instead): neglect for simplicity if it is in tune or not, but suppose it has be tightened enough to produce a note.
The question naturally arises about how to produce different notes by means of the same string.
The answer is naturally twofold: either you need a neck and frets, or you can pluck your string lightly holding it at suitable points, which correspond to the overtones of the string.
Where do scales come from?  Notsoeasy a question
In this introduction, we will not try to get a historical perspective about this question.
It can be easily seen that scales can be got by comparison of first overtones. In the following, we arbitrarily chose an octave and make all sounds fit into it by raising or lowering them by an integer number of octaves (this is referred to as taking notes modulo octaves).
To fix the ideas, let us tune a guitar string to E. Second overtone is E as well: this produces in fact no new note, since we get rid of the fact it is an octave higher than the fundamental E. Third overtone is B. Let us lower it by an octave: this produces a B which is a fifth higher than the fundamental E.
Now, we can envisage to tune a string to B: iterating the above argumentation will produce a F# as its third overtone and so on. Synthesising, we can generate new notes by multiplying by 3 (and making the new notes fit into the initial octave). This device is known as Pythagorean tuning.
Let us know return to our E guitar string: once we have iterated the above process four times, we get close to the fifth overtone of E, which is G#, but 81/80 higher (see the natural third vs. the tempered third positions in the illustration above).
This means that getting four times by the third overtone equals getting the fifth overtone and multiplying further by 81/80 (modulo octaves).
Alternatively, we could conclude that four fifths equal an untuned third.
Analogously, the following chains of fifths approximate integers number of octaves (ratios are displayed on the right):
Twelve fifths
(312/219, approx. 74/73 larger)
Nineteen fifths
(319/230; approx. 105/97 larger)
Thirtyone fifths
(331/249; approx. 79/72 larger)
Fiftythree fifths
(353/284; approx. 1.0021 larger)
Thus, when generating octave scales, we can use fifths (resp. thirds) which are slightly different from the third (resp. fith) overtone, to minimise, in different ways and according to different defintions, untuned notes. Higher overtones can be used too, in a similar fashion. For instance, ten fifths equal an untuned seventh (approx. 36/35 higher).
What has to be tuned?
So now: where should frets be placed on the neck? To produce a note which is an octave higher, answer is easy: you need to divide the length of the string by two.
But what about other notes? In common western practice, we have a scale of twelve notes, got by making twelve fifths exactly fit into one octave.
These notes are often referred to as being equally spaced. Yet, it is apparent that guitar frets are not equally spaced at all on the neck.
In fact, in this case, equally spaced means that: to produce a note a step (or semitone) higher, you need to divide the length of the freely vibrating string string by 21/12= 1.0594...
Iterating division twelve times will yield division by 212/12=2, that is to say, a sound an octave higher.
As explained in the preceding section, guitar strings admit points at which an audible sound can be produced by plucking the string but holding it fixed at those points by lightly placing a finger just above: these point are known as harmonics or overtones too, by metonymy.
Harmonic points on the string do not correspond in general to frets: for instance, an overtone is found at half length of the string, as well as the twelveth fret. Both produce a note which is an octave higher. Another overtone is found near (but distinct from) the fourth fret, producing a note which is a natural major third (plus two octaves) higher. The fourth fret produces a note a tempered major third higher instead: these are macroscopically different. This difference is a measure of the deviation of 12equal tempered system from perfectly tuned intervals.
Pipe instruments, as well as the human voice, are governed by similar physical principles.
In a more general setting, however, it should be kept into account that different tensions and lenghts come into play.
The theory of musical tuning studies these issues in detail. We take a short survey of some tuning systems.
Equal temperaments
Equal temperaments divide the octave into sequences of steps such that adjacent notes always have the same frequency ratios. In western music contemporary practice, the most common temperament is the so called 12 equal temperament, dividing the octave into twelve equal parts. A tempered major third corresponds to the ratio 1.259921, which is sensibly higher than 1.25 (the ratio corresponding to the fifth overtone divided by four, or a natural major third). A Tempered perfect fifth corresponds to the ratio 1.498307, slightly lower that 1.5, the one of the third overtone divided by two, or a natural perfect fifth.
Other equal temperaments are possible: the most common ones divide the octave into 19 or 31 parts: these yield much better approximations of natural major third and perfect fifth. 31equal temperament also gives a fine approximation of the seventh overtone as 25/31 of an octave.
A 24 (or quartertone) equal temperament is also used.
Quartercomma meantone tuning
Instead of dividing octave into equal semitones, you can try to make four fifths fit into a natural third (raised by two octaves). If you imagine to start from C, this ideally corresponds to the following steps: C  G  D  A  E. Doing so, you get a narrowed fifth, given by the ratio 51/4=1.4953...<1.5; from this ratio, you can generate all missing notes by iterating multiplication.
This temperament is known as quartercomma meantone, since natural perfect fifths are lowered by one quarter of a syntonic comma (corresponding to the ratio 81/80).
Note that, differently from 12 equal scale, C# is obtained differently from Db, and so on: thus they will have different tunings.
In Nicola Vicentino's Archicembalo (a renaissance harpischord, a detail of whose keyboard is sketched in the above picture), this is implemented by breaking the usual black keys into two parts each one, and adding two black keys more, between E and F, as well as B and C (note however that in harpsichords the colours of the keys are generally reversed with respect to those of pianos).
An example of a quartercomma meantone scale  With 21 tones
Hereabove an example of a quartercomma meantone scale with 21 tones. Notes have been graphed by means of a pie chart. Ratios have been approximated by decimal expansions, with the exceptions of C,E,G#,B#, Ab, Fb.
Three types of intervals between consecutive notes are possible:
• chromatic semitone (1.045, yellow);
• narrow chromatic semitone (1.024, lilac);
• very narrow chromatic semitone (1.02, red).
Just intonation
Just intonation is a collective name for scales set up in such a way that the frequencies of notes are related by ratios of small integer numbers. This means that these ratios essentially correspond to the ones between overtones and fundamentals or their multiplicative reciprocals. In such a context, abstractions like undertones have been created: an undertone of a note X is a new note Y such that X would be an overtone of Y.
As an example of modern just intonation scale, we record in the following section Fokker's second alternate septimal tuning restricted to the 21 notes which are natural, sharp or flat. This just intonation system features the seventh overtone too.
Fokker's second alternate septimal tuning  Restricted to 21 tones
Pythagorean tuning
The Pythagorean tuning is generated by a natural perfect fifth, i.e. a 3:1 ratio. Normalising sounds so that they fit into an octave, starting from C again, and picking 21 notes, we get the scale depicted above.
The resulting ratios have been shown in the following ways:
• fractions if numerators and denominators both have at most four digits;
• decimal numbers if they cannot be represented by four digit fractions.
Note that C# is higher than Db (and so on) and that major third is definitely "offkey", bearing a ratio of 1.265625, which is much higher than both 12tone major third (1.25992...) and the natural one (1.25).
Three types of intervals between consecutive notes are possible:
• diatonic semitone (1.054, green);
• chromatic semitone (1.04, yellow);
• narrow chromatic semitone (1.013, apricot).
Ok, I would like to listen to some samples...
Compare different tuning systems applied to baroque music!
VoilÃ !
In the following sections, you will be able to find:
• some youtube videos of the author's, featuring MIDI organ interpretations Bach's Chorale "Jesu bleibet meine Freude" from Cantata BWV 147, Gamba sonata BWV 1029 as well as Paradies's Toccata, Pachelbel's Canon and Bach's Prelude and Fugue #24 from Well Tempered Clavier, Book 2;
• Some mp3 samples from Amazon: James Pressler playing MIDI organ interpretations of Bach pieces and MIDI harpsichord ones of Purcell's. Also, various performances of Bach's "Jesu bleibet meine Freude" and Gamba Sonata BWV 1029 are proposed.
Compare and enjoy!
Pachelbel's Canon and Gigue
Pachelbel's Canon in D, in four different tunings:
• 00:01  Just intonation centered on D (with septimal C);
• 04:17  31 equal temperament (septimal C used in the canon);
• 08:38  Fokker's 31 just scale centered on D;
• 12:58  Septimal meantone (31 56^0.1 fifths);
played at midi organ.
Tuned with Scala, implemented with Timidity++, soundfont: EngChamberOrgan.SF2.
Registration: 1st, 2nd, 3rd voices: Grand Cornet; Bass: Full Organ. Tempo: close to baroque performances such as Jordi Savall's or Musica Antiqua Köln's.
Which is your favourite tuning for Pachelbel's Canon and Gigue?
Jesu bleibet meine Freude BWV 147
Bach: Chorale, Jesu joy of man's desiring, "Jesu bleibet meine Freude", from cantata BWV 147;
A simple baroque MIDI organ interpretation, in five tunings:
• 00:00  Fokker's 31 just scale centered on G
• 02:45  31tet with septimal notes
• 05:24  Meantone (31 quarter comma fifths)
• 08:03  Kirnberger III (Letter to Forkel)
• 10:44  19 tet.
Tuned with Scala;
Implemented with Timidity++
Soundfont: English Chamber Organ (EngChamberOrgan.SF2);
Public domain score from Cantorion.org.
Which is your favourite tuning for "Jesu bleibet meine Freude"?
Which is your favourite tuning for
Pier Domenico Paradies' Allegro, from Harspichord sonata in A (also known as "Toccata")
Comparison of eight different tuning systems over Pier Domenico Paradies' Allegro, from Harspichord sonata in A (also known as "Toccata").
The usual repetition of the main theme at the beginning of the Allegro is not prescribed by the original score and has been dismissed here, following most recent harpsichord performances.
Tunings:
• 00:01  Meantone, 31 quarter comma fifths (without septimal notes);
• 01:48  Fokker 31 just scale centered on A, with septimal notes;
• 03:36  Just intonation, 4 layers of pythagorean fifths, three layers at distances of 5*, 7*, 25* from a base layer;
• 05:26  31tet, with septimal notes;
• 07:14  Agricola's monochord (12 notes)
• 09:04  Kirnberger III (letter to Forkel, 12 notes)
• 10:54  19 tet
• 12:45  Septimal meantone (31 56^0.1 fifths) with septimal notes.
Played at midi organ. Tuned with Scala. Implemented with Timidity++
Soundfont: English Chamber Organ (EngChamberOrgan.SF2);
Public domain score from Petrucci music library
Which is your favourite tuning for Paradies' Toccata?
What is your favourite tuning for Paradies' Toccata?
Bach's kleines harmonisches Labyrinth  Played into different tunings
You can listen here to some samples from Bach's kleine harmonisches Labyrinth BWV 591, played at MIDI organ by James Pressler. Featured temperaments are:
• Equal
• Mean tone
• Pythagorean
• Kirnberger
• Werckmeister I
• Werckmeister III
• Young I
• Young II
Bach's Gamba sonata BWV 1029
These videos feature the first movement from Bach's Gamba sonata BWV 1029, played at MIDI organ.
Each one is differently tuned: in the following order, we have: 19 equal, 31 equal, Kirnberger III, meantone and the customary 12 equal temperament.
Tuned with Scala;
Implemented with Timidity++
Soundfont: Jeux Organ Soundfont (Jeux14.sf2);
Public domain score from Petrucci Music library.
Enjoy!
Which is your favourite tuning for Bach's Gamba sonata?
Which is your favourite tuning for Bach's gamba sonata (videos above)?
Bach, Well tempered Clavier, Book 2: Prelude and fugue # 24
Bach's Prelude and fugue # 24 from Well tempered Clavier, Book 2, in four tunings: Lehman squiggle temperament, Kirnberger III (letter to Forkel), 31 equal temperament, Meantone tuning (31 quarter comma fifths with C in 15th position). Meantone also available with graphic rendering.
Which is your favourite tuning for Bach's prelude and fugue #24 from WTC, book 2?
What about your favourite tuning in general?
Which is your favourite temperament?
Purcell Suite in Gminor
You can listen here to some samples from Purcell's suite in Gminor Z 661, played at MIDI harpsichord by James Pressler. Featured temperaments are:
• Equal
• Werckmeister
• Kirnberger
• Valotti
• Young
• Meantone
• Pythagorean
Purple Star
This lens was awarded a Purple Star on 2012February24. Many thanks!
Any comments or questions?  Many thanks to all those who dropped a line!
Interesting but I'm too old to learn. I'll just keep fiddling.
About the experpt from Bach's kleines harmonisches Labyrinth. *Very* interesting example because tuning makes a great difference to the musical effect.
(Btw, the button for Werckmeisteri did not work, and I found no button for Kirnberger.)
Meantone sounded, to me, mostly wrong.
12ET, besides its being imperfect, also striked me as inadequate to present the musical thought behind this piece.
Pythagorean went lucky for a while, only.
Werckmeisteriii and Youngii sounded similar to me. They were quite smooth except just after the middle of the example, where I felt a discontinuity  just before the cadence which leads to a major chord. I felt that chord as a stopping point, so that I felt surprise when the music moved on. These effects might, conceivably, have been intended by Bach, but I doubt it.
Youngi had no discontinuity; it felt like continuously unfolding though at the middle it gave me more tension than Wiii and Yii. Next, the strength of the cadence was reduced so that the major chord felt weaker than in Wiii and Yii, so that the continuation was not surprising.
There are quite a few places in this article where "fifth" is misspelled as "fifht," and that keeps on triggering my OCD tendencies.
very interesting
Oh, been playing 31 all week and I can make any power chord you want. Frets are like mile markers, it's not like you're limited to the frets as fretted near where you want to be. You can adjust things.
11, 13, 16, and things like Catlers Ultra plus that gives you all the 12TET + some just intonation are all also valid and somehow slipped this writers mind. 31 and 32 are close but they aren't the same for that matter and people have gone past 64 tone. They all do certain things well, and as primarily a guitarist and steel player, I can fudge things to where they work for me. Amazing you left out all those other methods of tuning, since this seemed vaguely guitar oriented. Pianos that aren't synths and horns are more complicated to play with different scale forms than guitars either fretted or electric.
Wow, good stuff. It's all over my head.
Cool! Thanks for sharing.
Fascinating! I've had instruments tuned before, but never knew what all was involved!
@Padaneis: "octaves are taken for granted" is a weird way of saying it  sure, if you assume the number of fifths you'll stack will produce an octave, that's ok, but that means you're really dumping the problem on the fifths and not talking about it. The only reasonably small numbers of fifths that get closer to the octave than 12 are 53 fifths (which is 3 cents off) and
Basically:
1200 log((3/2)^12)/log 2 mod 1200 â 23.5
((1 200 log((3 / 2)^31)) / log(2)) mod 1 200 = 160.6
((1 200 log((3 / 2)^19)) / log(2)) mod 1 200 = 137.1
((1 200 log((3 / 2)^17)) / log(2)) mod 1 200 = 1 133.23501 (67 cents flat)
((1 200 log((3 / 2)^41)) / log(2)) mod 1 200 = 1 180.15 (20 cents flat)
((1 200 log((3 / 2)^106)) / log(2)) mod 1 200 = 7.23
((1 200 log((3 / 2)^53)) / log(2)) mod 1 200 = 3.615
But yeah, I realize this is but a primer  I do think big errors in a primer might put people off later down the line when they realize that hey, 31 fifths don't make a good octave unless you detune them quite a bit more than you need to with twelve fifths.


There's some mathsy errors on there: 31 and 19 fifths give worse octaves than twelve (converse, 19tet and 31tet give worse fifths than 12tet). 31 fifths on top of each other is about one and a half semitones sharp of an octave.
However, 31 5/4 major thirds (or minor thirds) come closer to an octave than four minor thirds or three major ones do, and this is the case for 19 major or minor thirds as well.
31 has some other nice benefits to it, such as approximating some ratios with 7s in them very well. Since the major chord should approximate something along the lines of [4/4, 5/4, 6/4]* some pitch, 31tet throws in a nice extension of it: you get [4/4, 5/4, 6/4, 7/4]. A really lovely chord, often heard in barbershop singing. In 31tet, though, the 6/4 (= 3/2) is somewhat sacrificed, and power chords on a 31tet guitar would just about be unusable. The major/minor thirds help make chords sound reasonable and nice (compare to how out of tune our thirds are). If you omit the fifth, even full voicings of 11 or 13chords (including alterations) can sound great even through distortion.
Major chords, as I already mentioned, are built as 4/4, 5/4, 6/4. Minor chords go the other way about this: 6/6, 6/5, 6/4  the "edges", 6/4 and 4/4 = 6/6 are identical, but the middle is different. 31tet adds in a nice additional two thirds very close to 7/6 (the ratio looks somewhat related to 3/2, 4/3, 5/4, 6/5, ... no?) and what's left between it and the fifth  9/7  a couple of supermajor and subminor thirds, to give extra dark or extra sparkly minormajor chords.
31tet also has two very lovely tritones, corresponding to 7/5 and 10/7, giving us both Ì~minor and ~major analogues for dimchords:
5/5 6/5 7/5 being the ascending enumerator (major) version, and
7/7 7/6 7/5 being the descending denominator.
Certainly versions with 10/7 also can be constructed, but either the series involved is less "obviously" a simple series, or they are easier to parse as inversions of simpler series with 7/5 in them.
Very interesting account of this controversial matter.
Fascinating! I play guitar and sing but never thought about different tunings. There is quite a lot to take in and I shall be popping back to study each of the sections in depth. Enjoyed an overview and feel you have done a great job in putting across a difficult and possibly academic subject. Blessed
A well constructed page. I really enjoyed reading this. See you around the galaxy...
Fascinating!
Very interesting lens! I really enjoy exploring other musical worlds beyond the 12tone equal tempered scale. (Composing music is one of my hobbies, and I have played around with Middle Eastern and other nonWestern scales and modes.) I really like the mathematical background you presented for all the different tunings, and listening to your audio samples is a real earopener!
Quite a scientific explanation, a comprehensive review.
Very fascinating! I've added this to my "Music is FUNdaMENTAL" lens. *Squidoo Angel Blessed! =D
Who would have thought that such a technical (to me!) and specialised field as tuning could be presented in such a beautiful and fascinating way? All praise to your skill as a lensmaster! A really lovely page  even if I can't understand it all :).
Whew, thank you for this gentle introduction, you sure did all your homework here!
Nice lens. Great info!
Wow, this is incredibly detailed, I must reread another time. Thanks for the education. Stay well, Rose
Good article and great examples.
Thanks for a very interesting article. This sounds like the sort of place I might finally find an answer to a question that's been troubling me for a while: What on earth is the matter with Alicia Keys' "Empire State Of Mind"? You can hear it at http://www.youtube.com/watch?v=4kjmIXr6MoI
The worst bit seems to be around 01:55 but I can't put my finger on what the problem is. Is she just flat? Are the vocals (consistently or inconsistently) flatter than the piano? Is the singer tending to just intonation (perhaps if the vocals were recorded in isolation) while the piano is (presumably) using equal temperament? It bugs me that nobody else I've asked can hear anything wrong!
Wow. How interesting. I need to come back and study this lens in more detail.
I find this page very interesting, though I think it needs several readings to really understand it all. Since I love music and am intrigued, I'll be back, I'm sure.
Perfectly tuned. Some tunes just flew over my head and the others are echoing between the ears. :)
My philistine ears have trouble telling the difference between the different tunings, but I've always found this to be a fascinating topic. I work in gospel choir music, where we love doing lots of modulations, so equal temperament is what works for us.
A lot of this went way over my hear, but a very good detailed lens, and blessed by an angel.
I cannot hold a note (rock band is a struggle for me). I've spent some time in recording studios but I still fail to understand a lot about tuning.
I think you are a genius or something~this lens is mega smart. Congrats!
Geat information
Well done. I agree with Rossi that it's a bit cold, but...is it really off topic?
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