First of all in twelve equal then E# = F and B# = C. So then the keys will just duplicate the existing keys, which isn’t very interesting. But what about tunings which make these so called “enharmonic pitches” actually different in pitch? Well then the answer is, Yes!.

HISTORICAL BACKGROUND

First, a bit of history. The idea of using split keys goes back to the sixteenth century and late fifteeenth century. The 12 note approach was still quite new at that time, and developed gradually by adding extra keys to the earlier “music recta” gamut of Bb-F-C-G-D-A-E-B. to get to the full twelve tone keyboard (e.g. Eb-G#). by the early fourteenth century.

Since the tuning could start at different points in that cycle of fifths, and since the pythagorean twelve tone tuning doesn’t close out after twelve fifths, it was natural to continue the circle of fifths to add extra notes, and organs with some split keys were constructed in the late 15th century - the 1468 Cesena Cathedral (with three extra keys) and the 1480 Lucca (2 extra notes), the earliest in the list on page 162 and following of Italian split-keyboard instruments with fewer than nineteen divisions to the octaeve - Denzil Wraight.

The 1480 Lucca organ for instance might have been tuned from Ab - D# with alternatives of Eb/D# and G#/Ab)

Also theoretically, in the early fifteenth century, Prosdicimo of Beldemando (1412 or 1413), proposes a 17-note Pythagorean tuning of Gb-A# (comment by Margo Schulter in The Xenharmonic Alliance II where she gives more details).

But let’s leave Pythagorean tunings for now as the way the sharps and flats work for them is rather unintuitive, will come back to that later in the section on 53 equal.

LETS START WITH NINETEEN TONE

Let’s start with Gioseffo Zarlino (1517–1590) with his nineteen tone keyboard. Has enharmonic versions of all the “black keys” plus extra keys between B and C and between E and F for a B#=Cb and E#=Fb.

Zarlino probably tuned this to a slightly irregular tuning with some of the intervals larger than others. He was particularly keen on a system called 2/7 comma meantone (see Zarlino's 2/7-comma meantone memperament) which has none of the intervals exactly pure with their partials locked in, and instead has them all more or less equally resonant. See An historical survey of meantone temperaments

However other sixteenth century writers, Costely (1570) and Salinas (1577) as well as Praetorius in 1619 describe keyboards of similar design tuned with an equal spacing between the keys.

Here is a realization of one of Costeley’s pieces, Seigneur Dieu ta pitié in Finale by Roger Wibberley

More exactly, they tuned to 1/3 comma meantone, which is almost exactly nineteen equal. It gives pure minor thirds and their inversion, the major sixths (quarter comma meantone gives pure major thirds and minor sixths). See 1-3 Syntonic Comma Meantone.

It works because if you keep stacking minor thirds (6/5), then after going up by a minor third 19 times and dropping down by an octave five times, you get close to an octave above the original note, and in the process you go to every key on the keyboard.

You can see this because, since you multiply the frequency by 6/5 for a pure minor third (the interval between the fifth and the sixth harmonics) and divide by 2 to go down an octave, that process takes you to: (6/5)^19/2^4 = 1.9967, very close to 2/1 for the octave.

In this system the sharps and flats are ordered as

C, C♯, D♭, D and E♯ = F♭ and B♯ = C♭.

The enharmonic equivalences are:

E♯ = F♭ . B♯ = C♭. Also C## = Db, Dbb = C# etc

This of course also means it has nineteen keys instead of the usual twelve.

So, what did it sound like? Well here at least is a recording of Edward Parmentier playing a 17th-Century French Harpsichord Music in 1/3 comma meantone ( free on Spotify). It’s played on a Keith Hill copy of a 1640 harpsichord by Joannes Couchet (information from: Music: A Mathematical Offering)

For a modern piece in this tuning, see for instance Easely Blackwood’s Fanfare in 19-EDO

“This tuning contains diatonic scales in which the major second spans three chromatic degrees, and the minor second two. Triads are smooth, but the scale sounds slightly out of tune because the leading tone seems low with respect to the tonic. Diatonic behavior is virtually iden­tical to that of 12-note tuning, but chromatic behavior is very different. For example, a perfect fourth is divisible into two equal parts, while an augmented sixth and a diminished seventh sound identical. The Erude is in a sonata form where the first theme is diatonic and the second is chromatic. The development modulates entirely around the circle of nineteen fifths. An extended coda employs both diatonic and chro­matic elements.”

Blackwood: Microtonal Compositions

ANOTHER EARLY KEYBOARD - THIRTY ONE TONE

Then there’s Nicola Vicentino who constructed his Archicembalo. It could be tuned in two different ways, but one of them was an extended quarter comma meantone system, almost exactly 31 equal - at least in theory. Here quarter comma meantone is a system with the major thirds almost exactly pure. Mathematically, if you stack 31 pure major thirds (multiply frequency by 5/4 thirty one times) and then drop down nine octaves (divide by two nine times), you get (5/4)^31/2^9 = 1.972 which is close to 2/1, the octave.

So this time, it’s not enough to have split keys of 19 equal. Instead he needed to have two keyboards, the top one with 17 keys (no accidentals between B and C or between E and F) and the bottom with 19 for a total of 36 notes to an octave. When he tuned it to extended quarter comma meantone he had some duplicate keys.

In this system, 31 equal temperament then you have double sharps and double flats with the pitches arranged like this, from, say, C to D:

B Cb B# C, D♭♭, C♯, D♭, C♯♯, D

So, now you have separate keys for E# and Fb, and separate keys for B# and Cb

Note that the B# is sharper than the Cb in this system.

The enharmonic equivalences now are:

B## = Dbb, Cbb = A## etc

This shows the notes consecutively with sharps and flats notation, screenshot from beta for next upload of my program Bounce Metronome Pro - this is its onscreen keyboard which you can use to play in the tuning using pc keyboard, and mouse.

Though it’s more often notated using half sharps and half flats like this

(screenshot from the Scala program)

Or in Ascii in various ways such as using + for half sharp and - for half flat.

C C+ C# Eb E- E

So now we have 31 different keys to play our music in.

However he might not have tuned it to an exact 31 equal. His description is a bit strange, he says that "in the same rows [in] which one plays the perfect fifths, there will one find also the major thirds more perfectly tuned than those which we use"

In other words that if you play say the C on the front keyboard with the sharper E on the back keyboard the result is more pure than usual. But that doesn’t make sense if it is in 31 equal or extended meantone because then C to E on the same keyboard should be pure already.

Karol Berger suggested that perhaps instead he used a somewhat uneven systemwith some of the tones up to 0.2 comma flat, ranging up to 1/3 comma sharp. See Theories of chromatic and Enharmonic Music in Late 16th Century Italiy - Chromatic systems (or non-systems) from Vicentino to Monteverdi. and summarized by Mark Lindley in . An historical survey of meantone temperaments to 1620

HUYGENS, EXTENDED MEANTONE, AND THIRTY ONE EQUAL

Huygens recognized that extended quarter comma meantone was almost exactly 31 equal in 1661. But whether earlier theorists realized that as well I don’t know. Huygens thought that Salinas said that Vicentino’s Archicembalo was tuned in 31 equal.

Huygens himself built a "mobile keyboard" consisting of 12 keys per octave that could be moved around to play different subsets of a set of 31 strings or pipes tuned to 31-et, so permitting play in all 31 keys of 31 equal. Details of how it worked not very clear but is clear it was an actual physical instrument. Quantifying Music. So this may be the first instrument actually tuned to 31 equal - rather than to extended meantone and then only approximating 31 equal because it’s meantone

WHAT ABOUT VICENTINO’S COMPOSITIONS?

So, what does extended meantone sound like? Here is a performance of one of Vicentino’s own compositions played on a 24 tone harpsichord tuned using an extended meantone system - presumably this is a piece that didn’t need all the pitches on his keyboard:

You can also hear a recording of a live performance of Vicentino played on a reconstructed Archicembalo, with many exotic transitions here Vicentino's enharmonic madrigals

DETAILS OF HIS COMPOSITIONAL TECHNIQUE WITH SOUND SAMPLES OF HIS COMPOSITIONAL EXAMPLES

This page goes into details of his compositional technique. It’s rather techy but if you scroll down the page, it also has many example fragments of his music to listen to Wild, Vicentino’s 31-tone Compositional Theory

FOKKER’S 31 EQUAL ORGAN IN THE NETHERLANDS

Adriaan Fokker' organ which still survives in the Netherlands is in 31 equal.

- he was composing for 31 equal back in the 1940s?

This is what it is like to play :)

You can also try your hand at playing an onscreen version of his keyboard tuned to thirty one equal here: fokker keyboard online though it’s only one note at a time with a bit of sustain so if you play two notes quickly you get some overlap.

FIFTY THREE EQUAL

Ellis’s 1885 translation of Helmholtz's "On the Sensations of Tone" published in 1885 has a long appendix describing many instruments of his day to explore many tones to the octave with modulation. He was particularly interested in 53 equal which is a kind of extended pythagorean - if you use pure fifths instead of pure major thirds and keep going you get a nearly exact 53 tone system. (3/2)^53/2^30 = 2.004, very close to an octave 2/1.

This system not only gives pure fifths. It also has close to pure major and minor thirds. and does decent approximations of some intervals with higher harmonics, such as 7/4 and 13/10. and 13/8, for list of some of them, see 53 equal temperament

This is Ellis’s description of Paul Whites harmonium designed to play in 53 equal in his appendix to his translation of Helmholtz’s On the Sensations of Tone.

See also: Full text of "On the sensations of tone as a physiological basis for the theory of music"

TRICKY TO NOTATE SHARPS AND FLATS IN TUNINGS LIKD 53–ET WITH THE FIFTH WIDER THAN TWELVE EQUAL

So now we have 53 different keys. It’s quite tricky to notate, as it is based on the pythagorean system.

When you have pure fifths, or indeed any fifths sharper than the ones of twelve equal, then you get the rather curious situation that the sharps and flats go like this:

C, Db C#, D

i.e. the flat of the note above is flatter than the sharp of the note below.

Why does that happen?

In the circle of fifthS:

Fb, Cb, Gb, Db, Ab, Eb, Bb. F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B# C##

then in twelve equal, as we know, Db is the same as C# and so on.

If the fifth you use is narrower than for twelve equal, as it is in 31 equal and 19 equal, then the notes in the cycle of fifth get closer together (before octave reduction), so the C# will be a bit flatter than for twelve equal and the Db will be a bit sharper so you get the order that seems most natural at first C, C#, Db,D

But if the fifth you use is wider than in twelve equal, it goes the other way around the notes get spread out further. So, the C# is a bit higher in pitch than in twelve equal and the Db is a bit lower in pitch so then it has to go C Db C# D. In 53 equal then the Db and C# are very close together too, like this:

C [90] Db [24] C# [90] D

For more about this see Pythagorean Tuning - further down the page she has a diagram showing how the flats and sharps interrelate.

53 EQUAL IS EVEN MORE CONFUSING WITH DOUBLE SHARPS AND DOUBLE FLATS

I fully expect your head to spin at this point, do feel free to skip to the next section if it does! Almost nobody finds this intuitive or easy. And you can get by fine without knowing the positions of the double and triple flats and sharps in these tunings.

You can manage fine just using # and b and then some symbol for one step in your tuning, e.g. / or \ to go up one step or down one step.

So anyway, it gets even more confusing when you go on to the double sharps and double flats. B#, of course, with the same circle of fifths reasoning,is sharper than C (and Cb is flatter than B). And the Ebb of course has to be flatter than the D (because it’s more than half way there already on its way down from the E, or you can figure that out from the same chain of fifths reasoning too). So, going as far as the double sharps and double flats, the order of the notes is:

C B# Db C# Ebb D

But we have many more notes to fit in here, another four indeed.

Calculation indented so easy to skip:

The perfect fifth is approximated as almost exactly 31 steps of 53 equal. So you can work it out from there

C G D = 62 % 53 = 9 (here 62 % 53 means the remainder on dividing 62 by 53)

So we have 9 notes in total to fit in there.

One chromatic semitone C G D A E B F C# is (7 *31) % 53 = 5. So C# is 5 steps higher than C.

The diatonic semitone is C G D A E B

(5 *31) % 53 = 49 so C at 53 is 4 steps higher than B.

One cycle of fifths is (12 *31) % 53 = 1. So B# is one step higher than C in (C, G, D, A, E, B, C#, G#, D#, A#, E#, B#) as expected from our intuitive reasoning above.

So, to summarise,
diatonic whole tone, e.g. C to D is 9 steps
diatonic semitone e.g B to C is 4 steps.
chromatic semitone, e.g. C to C# is 5 steps
a chain of twelve fifths reduced to the octave, e.g. C to B# is 1 step.

Similarly Ebb is one step lower than D

(Dbb, Abb, Ebb, Bbb, Fb Cb, Gb, Db, Ab, Eb, Bb. F, C, G, D )

So filling in the section between B and C, since the chromatic semitone e.g. Db to Dbb etc is five steps, we can see that the Dbb is one step below C, the A## is one step above B, so it’s

Cb, B, A##, ?, Dbb, C, B#, ?, ?, Db, C#, ?, ?, Ebb, D

Continuing to fill in more chromatic semitones, remembering that you go up or down by 5 steps in 53-et for a chromatic semitone:

Cb, B, A##, ?, Dbb, C, B#, ?, Ebbb, Db, C#, B##, ?, Ebb, D

Continuing in the same way, and also doing the section from D to E in a similar way to the section C to D we get:

Cb, B, A##, Ebbb=G###, Dbb, C, B#, A###, Ebbb, Db, C#, B##, Fbbb, Eb, D, C##, B###, Fbb, Eb, D#, C###, Gbbb Fb, E, D##, Abbb=C###, Gbb, F, E#

You can’t even use half sharps and half flats to simplify it as C# is five steps above C. You can use x for double sharp, which may be useful if you are used to this symbol, to get

Cb, B, Ax, Ebbb=Gx#, Dbb, C, B#, Ax#, Ebbb, Db, C#, Bx, Fbbb, Eb, D, Cx, Bx#, Fbb, Eb, D#, Cx#, Gbbb Fb, E, Dx, Abbb=Cx#, Gbb, F, E#

If you want to test your understanding of this, try it again, start with . B . .. . C . . . . Db C# . . . .D . . . . Eb D# . . . . E , and then try filling in the dots with similar reasoning

(on a piece of paper or whatever, without looking back at the reasoning above, just based on the size of the chromatic semitone of 5 steps and the diatonic semitone at 4 steps and twelve fifths reduced to the octave as 1 step).

Anyway - as you can see, though you can notate everything using sharps and flats right up to 53-et, it gets very complicated.

UPSHOT, YOU NOW HAVE FOUR NEW NOTES BETWEEN B AND C, AND B# SHARPER THAN C, AND Cb FLATTER THAN B!

But anyway now between B and C you have four new notes, but the B# is now sharper than C and the Cb is now flatter than B:

Cb, B, A##, Ebbb=G###, Dbb, C. B#

or using x for double sharp:

Cb, B, Ax, Ebbb=Gx#, Dbb, C. B#

The enharmonic equivalences now are

Ebbb=G###, Abbb=C###, and numerous relations involving four flats or sharps such as B#### = Gbbb,

or using x for double sharp:

Ebbb=Gx#, Abbb=Cx#, and numerous relations involving four flats or sharps such as Bxx = Gbbb,

Musicians tend to use notations with fractions of a sharp or a flat at this point - things like half or a third or a sixth sharp or flat etc, and to have different systems of notation for each tuning system.

This is the result

ORTHOTONOPHONIUM - 53 OR 72 EQUAL

The Orthotonophonium dating back to 1914 by Arthur von Oettingen could play in 53 equal or 72 equal.

72 equal remains a popular tuning amongst some microtonalists. It’s not so good at approximating pure intervals exactly as 53 equal. But it does have twelve equal as part of it. Indeed you get it by dividing each of the twelve equal semitones into six exactly equal parts. With all the other tunings so far, then you can get to every note in the tuning using chains of fifths.

But 72 equal doesn’t work like that. It’s basically just six copies of twelve equal, shifted by a sixth of a semitone each (twelfth tones). And the “circle of fifths” takes you around between the notes of only one of those copies.

ENHARMONICS IN 72 EQUAL

So the enharmonics in 72 equal are the same as for twelve equal:

B# = C, E# = F, C# = Db etc

But now you have these offset tunings, which you can label, using some notation or other for the steps, e.g. / for one step sharp and \ for one step flat you get

B#/ = C/, E#/ = F/, C#/ = Db/ etc

B#// = C//, E#// = F//, C#// = Db// etc

B#/// = C///, E#/// = F///, C#/// = Db/// etc

and similarly

B#\ = C\, E#\ = F\, C#\ = Db\ etc

OTHER NOTATION SYSTEMS

This can be simplified by using notations such as Maneri / Sims

+1 = ^. +2=>, +3=]

-1 = v. -2=<, -3=[

I.e.

Maneri / Sims: ^>] to raise, v<[ to lower.

So then it’s

B#^ = C^, E#^ = F^, C#^ = Db^ etc

B#> = C>, E#> = F>, C#> = Db> etc

B#] = C], E#] = F], C#] = Db] etc

and similarly

B#v = Cv, E#v = Fv, C#v = Dbv etc

You could in principle play 72 equal using six pianists and six ordinary keyboards, each tuned to twelve equal and each piano sharper than the previous one by a sixth of a semitone.

This is an example, Wyschnegradsky - Arc en ciel

for six microtonally tuned twelve equal pianos

and

Georg Friedrich Haas - limited approximations for 6 micro-tonally tuned pianos and orchestra (2010)

Other notations you can use here are

HEWM +>^ to raise a note, and -<v to lower

Scala notation A144: /}| to raise and \{d to lower

Sagital 72 tone: /f^ and \tv (the t is visually similar to an upside down f)

Or we can just use multiple /s or \s

Then we can notate 53 equal with a similar system.

B\=Cb, B, B/ B// = C\\, C\, C, C/=B# , C//, C///, Db=C////, C#=D\\\\, D\\\, D\\, Eb=D\, D

where of course we could use e.g. Maneri Sims to replace / by ^, // by >, /// by ].

Or we can use the flats and sharps combined with / though that gets a bit confusing perhaps:

B\=Cb, B, B/ B// = C\\, C\, C, C/=B# , C//, C///, Db=C////, C#=D\\\\, D\\\, D\\, Eb=D\, D

Actually quite a lot of effort has gone into how best to notate such tunings, with one of the most thorough the Sagittal microtonal notation, with the aim of a unifying approach, for many different tuning systems. It’s based on notating everything relative to the pythogorean twelve tone system together with notation symbols for various commas. E.g. you get to E as 5/4 from the pythagorean E flattened by a syntonic comma, so if you have a symbol for the syntonic comma it turns out it’s easy to notate 5/4, 15/8, 8/5, etc. Similarly with other commas you can notate any of the small number ratios like 7/6, 11/8 etc. So it’s a very logical notation system.

They also use it for equal temperaments using notations for tempered commas (in cases where the comma doesn’t vanish).

It looks like this:

Anyway there are many more microtonally interesting equal temperament systems than this. Indeed probably almost any number of equal notes to the octave up to about 100 has it’s advocates for one reason than another. And there are people who go well beyond 100, even can get enthusiasts fretting guitars with more than 100 frets to an octave.

A few of these ETs have almost no decent approximations to a fifth, or a major third or a minor third, but that makes them even more intriguing and appealing to some composers.

But these tuning systems 19 equal, 31 equal, 53 equal and 72 equal are amongst the historically most important of the equal temperament systems. Well apart from twelve equal of course.

ASIDE ON TWELVE EQUAL

Twelve equal is interesting too. Some of you probably think it wasn’t developed until really late - like the early twentieth century. Few pianos were tuned to twelve equal for sure until then, Chopin even didn’t tune his piano to twelve equal, but instead used subtle late victorian tunings.

Some of you on the other hand may think that Bach’s “Well tempered Clavier”was written for twelve equal, as that’s a widespread “urban myth” in this topic area.

But no, actually it was written for well temperaments - tunings that are approximately equal but the different keys intentionally different in tuning. Keys likc C, G etc, related keys have purer sounding notes, often some of them pure fifths for instance while remoter keys have less pure pitch intervals.

But -back in Bach’s time and earlier they did have twelve equal as the tuning for the lute or guitar. It’s because the frets are much more straightforward. But they wouldn’t have thought of it as the “right tuning” and wouldn’t have liked the sound of it. Instead, they would probably try to bend the pitches to the desired tuning as they played. But the mistuning (to their ears) would be less noticeable perhaps on a plucked instrument.

HOW DID THEY TUNE TWELVE EQUAL BACK THEN?

How though did they manage to tune to twelve equal without logarithms to calculate the fret positions.

Well, in principle, they could have used twelfth roots as we would nowadays to make the twelve equal tuning. It was within their mathematical capabilities more or less. This is how they could have done it (but didn’t) - maths indented:

The ratio of the frequencies of two notes an exact semitone apart is sqrt(sqrt(cube root(2))) = 1.05946309436 so 1.05946309436^12 = 2. Square roots are easy to do geometrically, by constructing a square and drawing its http://diagonal.As for the cube, you can do that using the exact solutions to the problem of Duplicating the cube#History, which was solved back in the time of Plato. So you could also solve that one in a conceptually simple way by first duplicating the cube, which gives you the cube root of 2 by a complicated geometrical construction using for instance marked rulers that you slide into position against other lines you have drawn. Then you use another geometrical construction for the square root, and do that twice. You now have two lines with their lengths at a ratio of 1 : twelfth root of 2, and using that you could go on to construct your fretting pattern.

And the theoreticians did explore this idea. Zarlino suggested you could tune to twelve equal using an ancient Greek instrument design called the mesolabium, see page 55 of Tuning and temperament : a historical survey : Barbour, J. Murray (James Murray), which also describes other possible geometrical approaches.

However, in practice, one favoured way to do it was to use the ratio 18/17 for the twelve equal semitone - which is just a smidgen flat, but as it turns out is actually, or can be, a better way to tune a lute to twelve equal.

“..18:17 , which happens to make a very good prescription for placing the frets down the neck of a lute for equal temperament. It puts the octave shy of the string's midpoint by some 1/3 of 1% of the total length (comparable on a tenor lute to the width of the fret itself). This might be considered a defect from a certain theoretical point of view, but in reality 18:17 works better than twelfth root of 2 as the latter makes no allowance for the string's greater tension when it is pressed down to the fret. On a good instrument (that is, with a low action) the 18:17 rule renders the string just about long enough to compensate”

Chromatic systems (or non-systems) from Vicentino to Monteverdi

This idea or using 18:17 for the semitone for twelve equal goes back to Vincenzo Galilei (c. 1520 – 2 July 1591) , See Tuning and Temperament (you need to click on the Page xx link on that page in google books snippet view to see the quote). Incidentally, he was the father of the famous Galileo Galilei who made many of the first astronomical observations through a telescope, such as the craters on the Moon and the moons of Jupiter, and champion of the Copernican view that the Earth orbits the sun.

For more about the 18:17 approach see page 57 and following of Tuning and temperament. Kepler pointed out that it would not lead to the exact octave (theoretically) and there are various ways this could be fixed to give a better, though still not exact, solution, without need for twelfth roots etc.

SEE ALSO

I’ve just done lots more of those cycle of fifths notation screenshots from the next beta of Bounce here: Bounce Metronome Pro - Photos