Medieval and Renaissance Tuning

When is a Note Not a Note?

Keep constantly in mind that enharmonic notes – like A# and Bb – just happen to have identical tones in Well Temperments and 12 Tone Equal Temperament. Enharmonic notes have different pitches in all other tuning systems! They're still called "enharmonic" for convenience, but that doesn't mean the two notes have the same tone.

Even though most musicians don't know about it, modern musical notation with lines and staff is still written so that the correct note will be sounded when played on an instrument tuned to one of these non Equal Tempered systems. There is, however, one small exception: in Meantone systems the minor 7-ths in chords should almost always be played as augmented 6-ths instead.
 

Different Keys and Modes

All of the tables on this page (except for this example) are in the key of C. To tune an instrument to a major key different from C major, just relabel the note columns in the table: replace C D E F G A B with the notes of the major scale you want. Detune the tonic note by the same amount as given for C in the table, the second note in the scale by the same amount as D, the third the same amount as E, and so on.

To tune to a minor key or a modal key, tune to the matching major key (i.e. for A minor, just tune to C major, for E minor, tune to G major, and so on). If the minor or mode doesn't have an equivalent major (harmonic minor, for example) use the rule for sharps and flats, below.

So for example, this is the 1/4 Comma Meantone table for the key of Eb major, or C minor:
 

None of the numbers change, just the labels on the columns.
 

Meantone Rule for Sharps and Flats

This rule is for Meantone systems; it does not work in the Ptolemaic/Just system, and is not needed in Well Temperaments but for all others here, (even Lucy Tuning and Pythagorean) it applies.

Flats or sharps don't need any adjustment if they are part of the key signature that the instrument has been tuned to.

So for sharps or flats that are part of the key signature already tuned to (like the note Ab in the key of Eb), do nothing.

For accidental sharps (ones that aren't in the key signature) detune the the note from it's Equal Tempered equivalent by the amount of the note that it starts from, minus 7 times the amount from the column labeled G in the table. For flats, add 7 times the amount in the "G" column. Double sharps and double flats shift double the amount. (If you've relabeled the columns for a different key than C, use the 5-th column.)

For keys different than C, remember that in a flat key signature a natural acts like a sharp (so subtract 7 times the 5-th column). In a sharp key signature a natural acts like a flat (so add 7 times the 5-th column). The exception is when the sharp or the flat or the natural is just reminding you to go back to what's in the key signature that you tuned to. (And of course, for that you would do nothing.)
 

1/6 Comma Meantone Temperament in C (again)
C	D	E	F	G	A	B
+5	0	–5	+7.5	+2.5	–2.5	–7.5

For example: In the key of C, in 1/6 Comma Meantone Temperament, to get a Bb, the starting note is B, which detunes by –7.5 cents. The extra detuning for flats 7 times G, or 7 x +2.5 = +17.5 cents, so the detuning for Bb is –7.5 + 17.5 = +10 cents. The tuning meter is Equal Tempered, so it will "falsely" show Bb as A#. So that's what you read when you tune the note: to get a Meantone Bb you tune the meter's Equal Tempered A# to +10 cents.

To get a real A# in that system, you start with the note A, whose detuning is –2.5 cents. For sharps you subtract 17.5 cents, so the detuning for A# is –2.5 – 17.5 = –20 cents. The tuning meter will get the note name right this time and really read A# when you're tuning A#, which you tune it 20 cents flat.

If you want to have a single table that handles all notes at once, there's a recipe at the bottom that describes how to use a generic template to use to make a tuning table full of flats and sharps for any of the Meantone systems. Also, the sections below on the individual systems each has a row for the sharps and a row for the flats with the arithmatic already worked out. For double sharps and double flats you still will have to consult the generic template.
 

A COMMENT FROM A READER:

References to 1/6 comma meantone should read 1/5 comma. The fifth flatted by 2.5 cents is 1/5 whereas the fifth flatted by 1.5 cents is 1/6. Here is a list of characteristic fifth intervals for meantone tunings with fifths altered in .5 cent increments:

• -5.0 1/3 Salinas
• -4.5 2/7
• -4.0 4/11
• -3.5 1/4 comma meantone
• -3.0 2/9
• -2.5 1/5
• -2.0 2/11 Silbermann
• -1.5 1/6
• -1.0 1/7
• -.5 1/9
• -0 1/11 (Equal Temperament)

Tuner settings listed on this web page for 1/6 comma indicate a fifth interval 2.5 cents flatter than the root tone (A=0, then E=-2.5). That is 4.5 cents flat to the perfect fifth interval, which is two cents sharper (A=0, then E=+2). The mean is calculated relative to the perfect fifth. The 4.5 is divided by the 22 cent syntonic comma or 4.5 over 22. 4.5 divides into 22 approximately by 5, so the fraction is reduced and simplified so that the comma is characterized as 1/5.

The exercise yields 3.5 over 22 for the -1.5 interval. The fraction is approximately equal to 1/6 of the 22 cents comma, the comma being the tuning discrepancy between Just Major and Just Minor Intonation on a single note that must serve both the ii (re-fa-la) and the V (sol-ti-re) chords in a single diatonic scale. The note is the rogue, requiring two different tunings 22 cents apart (the syntonic comma). Historic pipe organs had separate pipes as one way to deal with the variable tuning issue.

The 1/5 comma tuning would yield sweeter chord intervals than the 1/6, having narrower major thirds and wider minor thirds. The major third is pure in 1/4 comma, so the larger the fraction the more pure the major third.

One an instrument well suited to the temperament (piano, harpsichord, zither, autoharp; all with fixed tuning on a string) there is no real advantage in using anything but 1/4 comma unless catering to a personal sense of when a fifth is too narrow, i.e. the fifth tone sounds flat in a chord. Thus I personally use 2/9 on a diatonic autoharp, sacrificing a little of the purity in the major thirds. Something like Salinas is way too flat for me and out of the question. I don't believe it was ever widely adopted. I don't use Just Intonation, because the ii chord is out of tune in Just Major and the V chord is out in Just Minor. Meantone solves that as is precisely its purpose.

The smaller fractions strike me as only a way to stay close to equal temperament, for whatever reason. One of those reasons is a modern fretted instrument that is best suited to the uniform intervals in equal temperament. The same goes for the hammered dulcimer because of the physics imposed by the bridge that bisects the string lengths, different notes on each side yet only one way (string) to tune for both. Then there is a need to stay close to the 1/11 comma tuning (Equal Temperament), 1/6 perhaps being a good limit for some of those instrument candidates. The complete array of fractions is actually academic.

Your site is a trip. Thanks for doing it.

Bob Lewis
Autoharp Works
2312 Liberty Highway
Six Mile, South Carolina 29682
http://www.autoharpworks.com

Ptolemaic Intonation
 

Ptolemaic Intonation (also called "Just Intonation")
	C	D	E	F	G	A	B
|–|	+6	+10	–8	+4	+8	–10	–6
b		+18	+22			+20	+10

This is the mother of all tuning systems, at least in modern music theory books. It is also sometimes called "Just Intonation," but people in the know tend to avoid that name because there are other "Just" scales that don't use this exact tuning. Ptolemaic Intonation is one of two systems described by ancient writers – both dating from the classic Greeks. The other system is Pythagorean.

This table has an added row with some flats in it, but remains shifted the same as the table with no flats at the top.

Ptolemaic is the "original" system that describes the diatonic notes, based on the overtone series of the tonic note. Although this was known to the ancient Greeks, and it has been used in music theory classes for at least 2000 years, I doubt that practical musicians actually used it without tempering it a bit. (I'll bet that most musicians used some kind of tempered take-off on the Pythagorean system, below.)

The frequencies of (almost) any pair of notes played in this temperament forms a small whole number ratio, making a "pure" resonance. The exact match to the resonant ratio is where the word "Just" comes in. Other "Just" tunings exactly match slightly different ratios, and they're "Just" but not "Ptolemaic." Some of the resonance ratios for Ptolemaic are listed in the not-a-tuning-table here:
 

If you prefer, you can write the ratios as Y/X instead of X:Y.

The flat notes in the bottom row are an extrapolation, based on every one of the diatonic notes in the top row having a counterpart that complements it to make an octave (hence Db pairs with B,  Eb with A,  Ab with E, and  Bb with D). Or to put it another way, complementary notes have ratios that are opposite, to within a factor of 2, so 4:5 has complement 5:4, or rather 5:8. This "complement" is called "inverse" in music theory.

There are other candidates for some of these ratios. For example: D could also be 9:10,  B could be 9:16, and Bb could be 4:7.

All three major triads, C:E:G, F:A:C, and G:B:D, are purely resonant in ratios of 4:5:6 (e.g. F:A:C). Or if the major 5-th is put on the bottom (like D:G:B) in ratios of 3:4:5. Two of the minor triads, Am (A:C:E) and Em (E:G:B) are also tuned to pure resonance in ratios of 5:6 (A:C and E:G) and 4:5 (C:E and G:B). But not Dm (D:F:A).

There's a problem with the D in the Dm chord, and that's where Ptolemaic Intonation starts to fall apart.

D is off resonance with the other two notes in the D minor triad by a factor of 80:81, which is called the "Comma of Didymus." To make the D minor chord match Am and Em, the frequency of C to the frequency of D should be in the ratio of 9:10, instead of the 8:9 that it needs to be in order to fit into the G major chord. So either all the major chords can be in tune or the Dm chord can be in tune, but not both. The Ptolemaic system (usually) sacrifices Dm for the sake of G. Even this system isn't perfect.

Something else that some people find grating in Ptolemaic Intonation is that the sound of the the whole steps is different as you go up the scale; C:D, F:G, and A:B all make a ratio of 8:9 but inbetween them D:E and G:A have a ratio of 9:10.

The other systems have (relatively) easy, natural extensions into sharps and flats, but Ptolemaic does not. The few flats listed in the table above may be wrong for the chord you want. To get a sharp or flat in the Ptolemaic system, you have to decide what chord you're trying to get, then pitch the particular sharp or flat to fit in that chord with as many of the notes you have recently played as you can. Sometimes, a resonant chord can not fit with the prior notes. The problem of the D in the Dm triad is the prime example.

Notice that even in Ptolemaic Intonation the "lost chord" is still lost. The "lost chord," or at least one of the lost chords, is a resonance of 4:7. That ratio produces a note that is close to a minor 7-th (like G:F,  D:C,  A:G,  B:A  etc.), and is what modern music is trying for and not getting in 7-th chords (like G:B:D:F). Even with all the exact resonances that Ptolemaic Intonation does give, the ratio you get for a minor 7-th is either 9:16 or 5:9 instead of 4:7. Both 1/4 Comma Meantone and 1/6 Comma Meantone come close to recovering this resonance with an augmented 6-th.

Another use of the Ptolemaic table is to critique Equal Temperament, but for that purpose it's better to shift the whole table to make the tonic (C) tune to zero. (This is just the table above, with 6 cents subtracted from every column.)

Intervals in Equal Temperament are out of resonance by exactly the opposite of what this shifted Ptolemaic table shows above. If the Ptolemaic/Just table says that G needs to be tuned 2 cents sharp, compared to Equal Temperament, then in Equal Temperament 5-ths are 2 cents flat from resonance. And so on for every number in the table: major 3-rds are 14 cents sharp, minor 3-rds are 16 cents flat, etc., except for the minor 7-th, whose ideal ratio is debatable.
 

Just by a Hair

The problem of notes being off by 80:81 in the Ptolemaic system was first explained by a greek musician and mathematician named Didymus. Because the ratio 80:81 is "just a hair", and "hair" in Greek is "kome" = comma, it's called "The Comma of Didymus." It's also called a "syntonic comma," and in modern units it's 21.5 cents. The sound of which is noticeably grating. An Equal Tempered half step, as from B to C or E to F, defines 100 cents (an Equal Tempered whole step, as from A to B, is 200 cents) so the comma of Didymus is about 1/4 sharp.

In fact, in music theory the Comma of Didymus is practically the definition of a nasty sound. It may be "just a hair," but whenever two tones differ by a comma, the combination makes an awful mix, and that's part of why there are so many different tuning systems. All tuning systems try and fail to do three things at once: come close to purely resonant tones, hide the comma of Didymus, and make the steps from note to note equal. Mathematically, it's only possible to do one of those things, so every tuning system is a different kind of compromise between the three.

Chords where a big part of a comma is exposed are often called "wolf tones," because of the beat frequencies seem to make a howling sound.

The comma is "hidden" in Meantone systems by distributing it in small pieces, a little bit into each note on the scale, so that each note is a little bit off resonance with the tonic, but no chord that you're going to play is really horribly off.

There are other, slightly different commas, although all things called a "comma" are close to 1/4 sharp (25 cents), and all of them sound bad. A comma that's almost as famous Didymus' is the "Pythagorean Comma" of 23.5 cents. That's the amount that Ab is different from G# in Pythagorean Intonation.

The Well Temperaments are all based on breaking up the Pythagorean comma into several small pieces, and subtracting those pieces, one each, from what would have been a Pythagorean interval (pure 5-th) between some of the notes on the circle of 5-ths. The other intervals are left Pythagorean. Usually. With the comma subtracted away, Ab comes out the same as G# (and every other enharmonic pair come out to be exactly the same as well). Which notes you pick to subtract between, and how many pieces  you break the comma up into is what makes Well Temperaments different. 12-Tone Equal Temperament is the ultimate Well Temperament, with the comma equally divided up into 12 pieces of about 2 cents each, that are subtracted from every one of the 5-ths between the 12 notes on the scale.

Pythagorean Intonation
 

Pythagorean Intonation
	C	D	E	F	G	A	B
#	+10	+14	+18	+8	+12	+16	+20
|–|	–4	0	+4	–6	–2	+2	+6
b	–18	–14	–10	–20	–16	–12	–8

This is the system was known to the ancient Greeks, and seems to have been the most widely used during medieval times. It's big advantage is that it has an easy rule for tuning by ear. Pythagorean tuning is pretty close to modern Equal Temperament, but its chords are about twice as far from pure resonance. (Except for chords with only the tonic, 5-th, 9-th, 11-th, and their octaves, which do have all pure resonances. An open 5-th chord is an example.)

Please notice the pattern of numbers in the natural (|–_|) row is like the fake phone number "202 3113", multiplied by –2. It shows up in all of the systems here, except Ptolemaic/Just. There is more about that pattern, below.

This system is based on the discovery of the circle of 5-ths – putatively by the mystical greek mathematician Pythagoras. The tuning process for Pythagorean Intonation is the way many piano tuners tuned by ear, before the common use of the electronic tuning meter in the late 1900's, and the procedure goes like this:

Listen for a pure resonance of a 5-th between F and C, and C and G, adjusting both F and G to match the tone of C. A "pure" resonance has been found when the beat frequencies go away – when there's no more "wow-wow-wow" sound when the two notes are played together. After G is tuned, proceed to tune D to the G, then similarly go from D to A, then from A to E, then from E to B. If you're tuning a harp, you quit there. If you're tuning a piano, you go up the circle of 5-ths (and down) to get the sharps (and flats). That's still the way that some folk musicians with well trained ears tune their harps.

Of course, after that, anyone with any sense goes back and "tempers" the tuning to make the chords sound better. That's an important practical consideration that I'm sure performing musicians in classical Greek times used, but theory-obsessed Pythagoras chose to ignore. In fact, because this tuning is so close to Equal Temperament, you could get Equal Temperament from the "tune everything by 5-ths" procedure, if you just tuned all the 5-ths a tiny bit flat (–2 cents). It seems natural that musicians tuning by ear would learn by trial and error to fix the bad chords by tuning each 5-th just a little flatter than resonance. It would get them Equal Temperament if they tuned to a gentle, 2 heartbeat vibrato in the middle of the scale. If they overshot that a little, with a faster vibrato, it would produce some moderate form of a Meantone Temperament, like 1/6 Comma Meantone. Trying to temper Pythagorean tuning naturally leads one into the more normal Meantone systems.

Without tempering, this scheme of tuning by 5-ths gets all of the 5-ths (and 4-ths, and 9-ths) perfectly, beautifully resonant (or as good as the tuner's ear). But that forces the major 3-rds to all be 22 cents sharp; likewise the major 6-ths and 7-ths. The minor 3-rds, 6-ths, and 7-ths are all 22 cents flat. Which is all perfectly hideous: it's a full comma off, which sounds quite bad. Aficionados of medieval music prefer to demure and say that their sharp 3-rds are "lively." Humbug. During medieval times, when untempered Pythagorean Intonation was used slavishly (because educated musicians read music theory books by Pythagoreas, one of the revered ancient Greeks), both major and minor 3-rds were considered dissonant chords, for good reason. So in Pythagorean tuning, one generally needs to stick with open 5-th chords, spiced up with 9-ths.

Except for 12-Tone Equal Temperament, the Pythagorean enharmonic gap is the smallest of the Meantone systems mentioned here. The gap is a little more than 23 cents, but the note order is weird: A# is higher than Bb, and all the other enharmonic sharps and flats similarly cross over. So although the normal order of rising tones in a conventional Meantone system is "A A# Bb B," etc., in Pythagorean their order is "A Bb A# B," etc. The scheme for sharps and flats given at the top still works, but you have to be ready for the negative multiplier and the wrong-seeming crossover of the flats and sharps.
 

Lucy Tuning
 
 

The Lucy Tuning system is a Meantone tuning beyond 1/4 Comma Meantone. It is the most recently invented of all the systems here, and the most extreme departure from 12-Tone Equal Temperament. It was discovered by John Harrsion, the Scotish clock inventor of "Longitude" fame.

Don't be fooled by just one row each of sharps and flats filled in in this bigger table: the complete table goes up forever into ## and ###, etc., and down forever into bb and bbb, etc. This table is also typical of Meantone systems in that everything is moderated: all of the sharps are flattemed, and all of the flats are sharpened, all of the big steps (wholetones) are a little short (9 cents flat), and all of the little steps (semitones) are a little long (23 cents sharp).

This is probably the best table to study for the symmetry patterns between sharps and flats. For example: D# is equal and opposite to Db. And Bb is equal and opposite to F#. Which of the notes will be opposite depends on how the table was shifted; in this case the table is shifted to make the highs the same as the lows which makes D come out as 0. Tables published elsewhere are typically shifted to force C = 0 for easy comparison of notes in chords to the tonic, and they would have the similar symmetries but in different places in the table.

In Lucy Tuning the major 3-rd is flat of 4:5 resonance by 4.3 cents, the perfect 5-th is flat of 2:3 by 6.5 cents, and the minor 3-rd is flat from 5:6 by 2.1 cents. The major 9-th is off by 13 cents. The augmented 6-th is flat of the 4:7 resonance (normally a minor 7-th in Equal Temperament) by 13 cents. Although it gives a distinctive sound, Lucy Tuning does not appear to optimise any resonance, at least for simple harmonics and timbres.

Lucy tuning still has a small but enthusiastic following, and you can find other pages on the web that discuss it passionately:

 http://lehua.ilhawaii.net/~lucy/

Please, however, be careful with what you read. I've found the web page very good with Lucy Tuning details, but I've noticed a few facts about other tuning systems that I think are mistaken.

Doubtless, of course, I've made mistakes of my own here. You can tell me all about them by e-mailing me at this e-mail address. (You have to remove the letters "NOSPAM" from "NOSPAMtom_lougheed@pacbell.net" that are there just to stop junk mail.) Please write gently, as my feelings can be hurt.

1/4 Comma Meantone Temperament
 

1/4 Comma Meantone Temperament
	C	D	E	F	G	A	B
#	–17	–24	–31	–14	–21	–27	–34
|–|	+7	0	–7	+10	+3	–3	–10
b	+31	+24	+17	+34	+27	+21	+14

This is the "new" tuning system used during the Renaissance, when harmonies started to include 3-rds in a big way, although many continued to use Pythagorean Intonation, and still others were experimenting with different systems. It so overshadows other Meantone systems in the modern mind that when someone just says "Meantone" this is usually the particular Meantone that they meant.

The middle row of this system fits the 202 3113 pattern when the pattern is multiplied by 3.4 cents.

Don't be fooled by the single extra sharps and flats already filled in: the table goes up forever into ## and ###, etc., and down forever into bb and bbb, etc. Notice that unlike the wierd Pythagorean, this is typical of Meantone systems: everything is moderated. All of the sharps are a little flat, and all of the flats are a little sharp, all of the big steps (wholetones) are a little short (7 cents flat), and all of the little steps (semitones) are a little long (17 cents sharp).

1/4 Comma Meantone Temperament gets major 3-rds exactly right (4:5 resonance with the tonic) all the way up the diatonic scale. Every 5-th and every minor 3-rd are off by 5.4 cents – about  1/4 of the famous Comma of Didymus. The minor 7-th and the major 9-th are 11 cents off. The augmented 6-th is flat from "the lost chord" of 4:7 by 3 cents.

This is close to the tuning used by J. S. Bach and people living around his time. But what their systems of " Well Temperament" was an advanced variation off this system that inserted sharps and flats in a different way. Unfortunately, the Well Temperamanet most closely related to this one has a few hideous chords in it.

The enharmonic gap for 1/4 Comma Meantone is 41 cents, which at about 2 commas is the largest gap for these systems. Sharps and flats sound very different in 1/4 Comma Meantone, and it isn't really possible to find a good compromise note in between a sharp/flat pair like A# and Bb, because any note in-between them will be off by about a comma – too ugly to endure.

1/6 Comma Meantone
 

1/6 Comma Meantone Temperament
	C	D	E	F	G	A	B
#	–22.5	–17.5	–12.5	–10	–14	–20	–25
|–|	+5	0	–5	+7.5	+2.5	–2.5	–7.5
b	+12.5	+17.5	+22.5	+25	+20	+14	+10

This system is a compromise tuning between 1/4 Comma Meantone and 12-Tone Equal temperament. major 3-rds are sharp by approximately "1/6 Comma" in this tuning.

1/6 Comma Meantone Temperament is optimal in the sense of making the worst off-resonance in a 9-th chord (using augmented 6-th instead of minor 7-th, like G:B:D:E#:A) as small as possible at 8 cents, which winds up being a little more than half the error of 12-Tone Equal Temperment.

Don't be fooled by the sharps and flats that are filled in: the table goes up forever into ## and ###, etc., and down forever into bb and bbb, etc. And everything is moderated: the sharps are all a little flat, and of the flats are all a little sharp, all of the long steps (wholetones) are a little short (5 cents flat), and all of the short steps (semitones) are a little long (13 cents sharp).