Aaron and Meantone

Pietro and the Ambiguities

Attempting to recreate a Meantone tuning of 1529 for the modern piano

In the study of historical temperaments there are numerous references to Pietro Aaron (or Aron) and Meantone. Most say little more than he was probably describing quarter comma meantone and discuss theoretical concerns rather than his actual tuning procedures. The bibliographies give the source as Toscanello in Musica, an edition published by Aaron in Venice in 1523, or the revised one of 1529. Murray Barbour in his 1953 book Tuning and Temperament, an Historical Survey quotes a small part of Aaron’s method. A few words are given to it in Jorgensen's "Big Red" ie Tuning, An Historical Survey" and in New Groves. It is in a series of books from Colorado Music Press (1970) that the complete text of Toscanello is translated. From this we will attempt to devise a tuning scheme from Aaron's actual words and hopefully render a temperament as he did in the early 16th century.

While there are some doubts among modern scholars, which will be discussed, the tuning constructed from Pietro Aaron's instructions easily results in what today we call "quarter comma meantone". Meantone is a system of tuning that produces pure 3rds by tuning a series of narrow tempered 5ths. When 12 pure 5ths are tuned as in the circle of 5ths the result is an octave that is sharp from pure. This sharpness or difference is the age old Pythagorean comma. Now if four of those 5ths are tuned pure (C--G--D--A--E) the E (a 3rd from the starting note) is sharp from a pure third by an amount called the syntonic comma or the comma of Didymus.

Reason suggests that if twelve 5ths are tuned slightly flat the comma of Pythagorus will disappear, and this the foundation Equal Temperament. If four 5ths are tuned even flatter then the syntonic comma is eliminated and a pure 3rd results which results in Meantone. To achieve this, each 5th is narrowed by one-fourth of the syntonic comma, hence the name "quarter comma meantone".

But there is a major restriction because 3 contiguous pure 3rds do not make an octave. Thus C--E and E--G# can be pure but G#--C is so out of tune it cannot be called musical. We notice that G#--C is really not a 3rd but more properly a diminished 4th. It is Ab--C that is needed but since G# has already been tuned we either have to add another note to the keyboard or be content with a "howling" "3rd".

As if this weren't enough it must be remembered that since we are tuning 5ths narrower than ET, a circle of twelve such 5ths will come out short of the octave. This last 5th is so out of tune that it is called the "wolf" because its beating is so fast as to howl. In spite of the wolf 5th and four "wolf 3rds" meantone was apparently the predominant tuning in the 14th, 15th and 16th centuries perhaps because the way it sounded on the organ.

Knowledge of the syntonic or comma of Didymus does not tell tuners exactly much to flatten the 5th to achieve pure 3rds. One can experiment by tuning four pure 5ths, noting how sharp the 3rd is and then narrowing or tempering the same four 5ths to achieve a pure 3rd. Today with computer based spreadsheets, beat tables can be calculated to help establish a system with checks and possible short cuts to achieve the actual temperament. It is interesting though to study how our predecessors did it and we find in Pietro Aaron's method an intriguing and apparently forgotten way to tune Meantone.

Presented below are his instructions word for word followed by comments on various parts. For clarity, the notes being tuned are given in bold--- the notes already tuned are in Italics. Capital letters are from the translation and assumed to be the original. Items in {} are mine, the ones in [ ] are the translator’s.

Chapter XLI

About the Temperament and Manner of Tuning the Instrument

The following sets out briefly, as easily as I can, how much is necessary for the performer to know about the temperament and union of the notes, because there are many who proceed in such matters with little or no system and not much practice. Therefore, note that we make our accord and temperament in three parts. If you want to tune and temper your instrument, you must first consider the note or position called C fa ut, using whatever intonation you please. When you have decided this, take the octave above C fa ut and make it always just. Then the major third above, E la mi, wants to be sonorous and just, that is, as pure as possible. When this is done, take the fifth in the middle, G sol re ut, and make it a little flat. Then another fifth above follows, D la sol re, tuned in the same way as G sol re ut. Then tune D sol re as the octave to d la sol re. Following that, take the fifth above D sol re, a la mi re. It should lack as much from E la mi as from D sol re, that is, it should be equally distant from each. All the fifths fall short of perfection, the note above being flat, so that the fifths above C fa ut, D sol re, and E la mi, which are G sol re ut, a la mi re, and b fa _ mi, always fall short and lack their perfection.

For the second series, tune the fifth below c sol fa ut, which note is perfect and just, tune F fa ut, which has to be the opposite of the others mentioned above; that is, it is tuned a little high, passing a bit beyond perfection, from which results a just and good temperament. The thirds and sixths are diminished by this temperament. Tune the semitone of B fa mi, {Bb} below F fa ut, and that of E la mi [E-Flat], below b fa_ mi, which is a fifth, in the same manner as that between F fa ut and c sol fa ut.

In the third and last series, tune major semitones between their thirds, such as the semitone of C fa ut [C-sharp] above A re, which you should tune together with the fifth, e la mi, so that it is a major third with A re and a minor third with E la mi. The same is to be done with the third between D sol re and a la mi re, the semitone of F fa ut [F-sharp], just as the previous note. Proceeding thus to the end of your instrument, tune every octave justly, and this arrangement will bring about the true temperament of the notes. Finis.

Now we shall examine the treatise in detail from the point of view of a modern piano tuner. As we shall see, different comprehension of meaning and vague points of reference lead to ambiguities but never the less offer an interesting challenge for study and tuning experience.

You must first consider the note or position called C fa ut, using whatever intonation you please.

We may call C fa ut either C3 or C4 for the purpose of "laying the bearings" the tuner's term for setting the temperament octave. It is up to the individual which octave he feels comfortable in. C4 is used by piano tuners to denote middle C. The "4" signifies it is the fourth C from the bottom on the modern piano. The C above Middle C is called C5. So the notes between C4 and C5 would be D4, E4, F4 and so on. Likewise the notes below C4 would be B3, Bb3, A3, G3 etc.

The phrase "using whatever intonation you please" I believe means pitch source. In those days it could have been the C from an organ, a recorder or perhaps a pipe specifically designed as a pitch reference. Tuning forks were not invented until the early 1700s.

When you have decided this, take the octave above C fa ut and make it always just.

We will take "just" to mean beatless as in a pure interval. To avoid ambiguity ourselves perhaps "just" should be capitalized as in Just Intonation to signify it from just another "just". Notice here he does not give the name of the note of the octave above C fa ut, however if we started with C4 we are now tuning C5.

Then the major third above, E la mi, wants to be sonorous and just, that is, as pure as possible.

There is no doubt Aron is indicating a pure third--one without beats. He calls it E la mi, and we can assume it is the 3rd above C fa ut or our E4. However one could also interpret that he means the E of the octave above C fa ut since C5 was the last note tuned. So is E la mi E4 or E5? Actually for practical purposes it is a good idea to tune both of these E's at this time. This note once established as a pure 3rd from either C will not be changed and will serve as an anchor so to speak for the rest of the temperament. Other notes though may need re-adjusting as the tuning proceeds.

…take the fifth in the middle, G sol re ut, and make it a little flat. Then another fifth above follows, d la sol re, tuned in the same way as G sol re ut. Then tune D sol re as the octave to d la sol re.

First we ask "In the middle of what"? Between C fa ut and E la mi? There is no 5th in the middle of C4 and E4. If though, between C4 and E5 then G4 would be "in the middle". But we have also tuned the C octave so perhaps he means in the middle of that. At any rate there is no question that since G sol re ut is to be a "little flat" it must be tuned to C fa ut.

So far we have tuned the octave C4--C5, and the 3rd C4--E4. Also E5 for study purposes. Now we are to tune G4 "a little flat" (to C4) and then the fifth above, D la sol re (D5), tuned a little flat, and then its octave down (D sol re) or D4.

…take the fifth above D sol re, a la mi re. It should lack as much from E la mi as from D sol re, that is it should be equally distant from each.

This is taken to mean, "tune A4 to D4". Again E la mi comes into question. If it is E4 then he is saying the 5th D4-A4 is "equally distant" from the 4th E4-A4. However if you are following this with a tuning hammer it makes great sense to consider that the A we are tuning is between D4 and E5. He is always comparing 5ths, 4ths are never mentioned. It is more practical to get A4 "equally distant" from D4 and E5, since tuning one note affects two 5ths at the same time. But it also works to check A4 with E4 because if this beats too much then A or D must be changed. (Remember that E4 having been established as pure will not be changed.) But Aaron does not mention checking or tuning to 4ths and does not mention beats at all. Experimenting with the tuning hammer though, will show that tuning A4 to get the beats of E4--A4 (4th) as close as possible to D4--A4 (5th), and if necessary re-adjusting D4 and/or G4, helps greatly in rendering these first four notes in good tune following the idea that four sufficiently (and "equally") narrowed 5ths will produce a pure 3rd.

This is a good example of interpreting history by recreating the experience. When at the instrument with the tuning hammer, temperament strip and mutes our experiences are much the same as Pietro Aaron's over 450 years ago. All he was hearing was the sound of the 5ths and changing (tempering) those sounds to make G, D and A "equal" sounding as 5ths without changing C or E. In listening to these 5ths during tuning I think the meaning of "lack as much from", and "equally distant from" become empirically clear. By themselves these 5ths will sound "out of tune" since they are twice as flat as today’s Equal Temperament 5ths. The goal is to make them sound out of tune the same as possible. The magic of Meantone is experienced when these 5ths are played in triads---the out of tune sound disappears.

All the fifths fall short of perfection, the note above being flat, so that the fifths above C fa ut, D sol re, and E la mi, which are G sol re ut, a la mi re, and b fa _ mi, always fall short and lack their perfection.

Here we are to tune B4 to E4, and here E la mi is obviously E4. This completes the first series (accord?) which encompasses all of the white notes of the keyboard except F, or all the notes, which can be tuned as 5ths up from C. Since five 5ths are tuned, there is a second 3rd formed, that of G4--B4. Aaron does not mention that this 3rd should be pure which is perplexing since it would be an easy check for accuracy. One of the little ambiguities.

For the second series, tune the fifth below c sol fa ut, which note is perfect and just, tune F fa ut, which has to be the opposite of the others mentioned above, that is, it is tuned a little high, passing a bit beyond perfection, from which results a just and good temperament.

Notice "c sol fa ut". This note until now has not been mentioned by name. He says it is "perfect and just" and that can only mean as an octave up from C fa ut. This then is C5. So we are to tune the 5th below C5, or F4, which must be tuned, "a little high" or sharp. Of course sharpening the bottom note of a 5th produces a narrow 5th which is the requisite of this temperament. The words "passing a bit beyond perfection" are almost the exact words I heard when I was learning to tune. "Bring it to perfect, then draw it through".

The thirds and sixths are diminished by this temperament.

Diminished from what? Certainly not from pure. Is this an indication there were other temperaments in which 3rds and 6ths were sharp? And if this is 1/4 comma Meantone the F just tuned should form a pure 3rd with A and a nearly pure 6th with D. So why does he say "diminished" instead of "pure" or "nearly pure"? This could be a reference to the fact that when tuning a series of pure fifths the 3rds and 6ths come out sharp due to the Pythagorean comma. Perhaps Pythagorean tuning was common in his time and he was comparing to that.

Tune the semitone of b fa _ mi below F fa ut, and {the semitone} [E-flat] of E la mi, below b fa _ mi, which is a fifth, in the same manner as that between F fa ut and c sol fa ut.

Here we tune Bb3 to F4, and then Eb below that, or Eb3. Also notice the ubiquitous E la mi now seems to be E3.

In the third and last series, tune major semitones between their thirds, such as the semitone of C fa ut [C-sharp] above A re, which you should tune together with the fifth, e la mi, so that it is a major third with A re and a minor third with E la mi.

Poor E la mi. Unless it is an error of translation or original error, it appears that e la mi is the same as E la mi, that the capitalization doesn't matter---that E la mi may have a place in 3 successive octaves. It is not immediately clear what is meant by "major semitones between their thirds" unless he means the semitone that forms two 3rds (major/minor) within the 5th which seems to be what he explains in the next sentence. At any rate we can say that C#4 is tuned to A3 or A la. But A la has not yet been tuned, however its octave a la mi re or A4 has. Perhaps tuning the octaves is supposed to be understood. He mentions this C#4 should be a major third to A3 and a minor third to the 5th of A, or E4. Which is interesting because if A-C# is pure then C#-E will be narrower than pure because the 5th A-E is narrow. However with the tuning hammer the sound of a major 3rd and resulting minor 3rd within a flattened 5th is immediately heard. The tendency is to get the two 3rds sounding the "best" and that happens when A--C# is pure or nearly so.

It is interesting to note in his book "Tuning and Temperament" Barbour states, "Finally in the third stage, C#, and F# are tuned as pure thirds to A and D respectively." However in the Colorado Music Press translation we have seen that Aaron does not state they are to be tuned pure. Is there a discrepancy or mistake of translation? Scholars would have to examine the original Italian. This may be what Mark Lindley the author of the articles on temperament in New Groves bases his statement, "Many scholars attribute 1/4 comma mean-tone to Aaron in his tuning instructions of 1523, but an equally legitimate interpretation of Aaron's text suggests that while he would not have faulted regular 1/4-comma mean-tone neither did he specify it." [New Groves, "Temperaments", p 662] Now if Aaron actually said that C# and F# were tuned as pure 3rds, that would indeed be 1/4 comma Meantone. Even so, using Aaron's instructions the tendency is to tune these 3rds pure (as possible) which yields a very close 1/4 comma mean-tone. And we wonder what is the "equally legitimate interpretation" Lindley mentions.

The same is to be done with the third between D sol re and a la me re, [or F-sharp]. Proceeding thus to the end of your instrument, tune every octave justly.

All of the black notes have been tuned except G#. We must assume this is the next and final step---tune G# to E la mi as a "major third" as was C# and F#. Again, unless there is an error of translation, Aaron does not say these 3rds should be tuned as "pure as possible" as he said for the C--E third. In fact C--E is the only 3rd he specifies as pure. When he mentions, "the thirds and sixths are diminished by this temperament," again he does not say they should be "sonorous and just". (The sixths actually have a slow rate, a vibrato almost that might lend a sonorous quality)

There is a lot about this temperament Aaron does not mention, namely the one exceptional 5th so out of tune it "howls" or the so-called "wolf tone", G#--Eb (technically a dim 6th). There are also four very sharp "thirds" (G#--C, F#--Bb, C#--F, B--Eb actually dim 4ths in music theory). These "wolf 3rds" occur because it is impossible to get three pure 3rds to form a just octave. It appears Aaron leaves many issues to the tuner's intuition and skill or expects them to become obvious in practice. Whether this is evidence that something other than 1/ 4 comma meantone is what Aaron intended, can be debated. If all 5ths are to be "equally distant from each" the result is meantone. But Aaron really says this about only one 5th, just as he specifically mentioned only one pure third. He does say, "All the fifths fall short of perfection". We must infer that he means all fifths fall equally short. If all 5ths are tuned in the same manor as C--G, G--A, A--E, pure 3rds and a wolf 5th result in a temperament best described by 1/4 comma Meantone.

How much to actually temper these 5ths in practice was a guess since Aaron could give no beat rates as the exact nature and determination of such rates would remain a mystery for over three hundred years. Even today rather than counting the beats of tempered 5ths it is more expedient to judge them the by tonal quality. Modern tuning theory reveals that Meantone 5ths are tempered nearly twice as much as Equal Temperament 5ths. Even so, on a modern piano instead of counting these faster rates the task instead focuses on getting the 5ths to sound "alike" while making them "come out" and agree with the E previously tuned. Perhaps one key 5th might be counted against a metronome as a check if a beat table were consulted.

Beat tables constructed according to theory reveal numerous checks. At A440 A3--E4 beats 2 times a second. From the same root, the major 6th beats slightly slower than the minor 3rd. From middle C the 6th beats 4 times a second.

Because there are so many pure 3rds, short cuts are looked for such as tuning the first two thirds, pure ie; C--E, E--G#, then tuning G4 flat to C and then the G4--B4 3rd pure, and see if the B4 makes a "good" 5th with E4. Then tune F3 sharp, (narrow) to C4, then tune A3 pure to F3 and see if A3 makes a "meantone fifth" with E4. Such schemes go on and on with endless variations giving many ways to "figure it out".

When all is said and done as far as the theory of Meantone, and the different schemes based on modern knowledge, it is interesting and refreshing to return to Pietro Aaron's instructions of 1523 and appreciate how simple they are and yet how close they can render quarter comma Meantone to its theoretical norm.

Aaron's Tuning Instructions Adapted For the Piano

The notes in bold are the notes being tuned.

First Series

Source---C4 "using what ever intonation you please"

C4---E4 "wants to sonorous and just, that is, as pure as possible"

C4---G4 "make it a little flat"

G4---D5 "tuned the same way"

D5---D4 Octave "pure"

D4---A4 "should be equally distant from D4, (alsoD5) and E4"

E4---B4 "all fifths...fall short and lack their perfection"

Second Series

C4---C5 Octave, "perfect and just"

C5---F4 "tuned a little high, passing a bit beyond perfection"

F4---Bb3 "same manner as F4

Bb3---Eb3 "same"

Third Series

(A4---A3) octave (not specified in original instructions)

A3---C#3 forms a "major third" with A and a "minor third" with E4

D4---F#4 forms a major 3rd with D and minor 3rd with A

E4---G#4 major 3rd with E and (minor 3rd with B)

(B4--B3) (not specified in original)

"The thirds and sixths are diminished by this temperament"

"Proceeding thus to the end of your instrument, tune every octave justly, and this will bring about the true temperament of the notes." Pietro Aaron ca 1523

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Notes.

For the ETD (electronic tuning device) from pitch center of C

A4 will end up 437.4 if machine is set at 440 from the beginning

For A to be 440 set the machine to 442.62

For A to be 442, set the machine to 444.63

From ET

0.00. (cents) C

-23.95..........C#

-6.84...........D

10.26...........Eb

-13.68..........E

3.42 .........F

-20.53..........F#

-3.42..........G

-27.36..........G#

-10.26..........A

6.84.........Bb

-17.10..........B

Sources

Toscanello in Music. Translated by Peter Bergquist. Colorado Springs [Colorado College Music Press] 1970.

Tuning and Temperament, a Historical Survey. East Lansing: Michigan State College Press, 1953

New Groves, "Temperaments", (Mark Lindley).