Early developments of keyboards with more than 12 notes per octave
From Microtonal
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 BbFCGDAEB. to get to the full twelve tone keyboard (e.g. EbG#). 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 Denzil Wraight's "Italian splitkeyboard instruments with fewer than nineteen divisions to the octave"^{[1]}.
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 17note Pythagorean tuning of GbA# (comment by Margo Schulter in The Xenharmonic Alliance II (^{[2]} 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.
Contents
 1 Ninteen tone
 2 Another early keyboard  thirty one tone
 3 Huygens, extended meantone, and thirty one equal
 4 Vicentino's compositions in extended meantone
 5 Details of his compositional technique with sound samples of his compositional examples
 6 Fokker’s 31 equal organ in the Netherlands
 7 Fifty three equal
 8 Tricky to notate sharps and flats in tunings like 53et with the fifth wider than twelve equal
 9 53 equal is even more confusing with double sharps and double flats
 10 Upshot, you now have four new notes between b and c, and b# sharper than c, and cb flatter than b!
 11 Orthotonophonium  53 or 72 equal
 12 Enharmonics in 72 equal
 13 Other notation systems
 14 Georg Friedrich Haas  limited approximations for 6 microtonally tuned pianos and orchestra (2010)
 15 Aside on twelve equal
 16 How they tuned twelve equal without logarithms
 17 See also
 18 External links
 19 References
Ninteen tone[edit  edit source]
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.
 Sketch of Zarlino's keyboard with split keys from the sixteenth century used to play music tuned with nineteen notes to an octave. It was probably tuned to 2/7 comma meantone, which has them slightly unevenly spaced.
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. In quarter comma meantone the major thirds are pure, but minor thirds are smaller than pure by a quarter of a comma. In third comma meantone the minor thirds are pure and the major thirds smaller by a third of a comma. In 2/7 comma meantone, both are smaller than pure by 2/7 of a comma^{[3]}. So, it has none of the thirds exactly pure with their partials locked in, and instead has them all more or less equally resonant. It does have a pure chromatic semitone at 25:24.
For more about the background see: An historical survey of meantone temperaments^{[4]}s)
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 (https://www.youtube.com/channel/UCjyrj9HDoKXO5Yr2nPP6nBg)
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 13 Syntonic Comma Meantone ^{[5]}.
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 (^{[6]}, 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 17thCentury French Harpsichord Music in 1/3 comma meantone (^{[7]} ( free on Spotify). It’s played on a Keith Hill copy of a 1640 harpsichord by Joannes Couchet (information from: Music: A Mathematical Offering (^{[8]})
For a modern piece in this tuning, see for instance Easely Blackwood’ s Fanfare in 19EDO
 “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 12note 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.”^{[9]}
Another early keyboard  thirty one tone[edit  edit source]
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 (^{[10]}= 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 Bounce Metronome  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 nonsystems) from Vicentino to Monteverdi [http://www.academia.edu/291960/Chromatic_systems_or_nonsystems_from_Vicentino_to_Monteverdi). and summarized by Mark Lindley in "An historical survey of meantone temperaments to 1620" ^{[11]}
Huygens, extended meantone, and thirty one equal[edit  edit source]
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 31et, 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. ^{[12]}. 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.
Vicentino's compositions in extended meantone[edit  edit source]
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 (^{[13]}
Details of his compositional technique with sound samples of his compositional examples[edit  edit source]
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 31tone Compositional Theory (external link)
Fokker’s 31 equal organ in the Netherlands[edit  edit source]
Adriaan_Fokker's 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 fokker keyboard online</ref>) 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[edit  edit source]
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 (^{[14]}), 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#Comparison to other scales
This is Ellis’s description of Paul White’s harmonium designed to play in 53 equal (^{[15]} 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" (^{[16]})
Tricky to notate sharps and flats in tunings like 53et with the fifth wider than twelve equal[edit  edit source]
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 [91 cents] Db [23 cents] C# [91 cents] D
(the intervals there are more exactly 22.6415 cents and 90.566 cents)
For more about this see Margo Schulter's site on Pythagorean Tuning ^{[17]}  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[edit  edit source]
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 collapsed, so easy to skip:
Extended content 


So, to summarise,
diatonic whole tone is 9 steps, e.g. C to D
diatonic semitoneis 4 steps e.g B to C .
chromatic semitone is 5 steps, e.g. C to C#
a chain of twelve fifths reduced to the octave is 1 step, e.g. C to B#.
Similarly Ebb is one step lower than D (from the chain of twelve fifths: Ebb, Bbb, Fb Cb, Gb, Db, Ab, Eb, Bb. F, C, G, D )
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 53et 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 53et, 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![edit  edit source]
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, rather than to use these multiple sharps and flats.
Anyway, this is the result
Orthotonophonium  53 or 72 equal[edit  edit source]
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[edit  edit source]
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[edit  edit source]
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 microtonally tuned pianos and orchestra (2010)[edit  edit source]
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. One of the most thorough is the Sagittal notation^{[20]}, developed by George Secor and David Keenan with the aim of providing a unifying approach to notating microtonal music, 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:
For more details see Sagittal notation.
Anyway there are many microtonally interesting equal temperament systems. Indeed almost any number of equal notes to the octave up to about 100 has it's advocates for one reason than another. Also, there are people who go well beyond 100, and you even get enthusiasts fretting guitars with more than 100 frets to an octave.
A few of these ETs with few numbers of notes to an octave 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[edit  edit source]
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 they tuned twelve equal without logarithms[edit  edit source]
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 and collapsed:
Extended content 


The theoreticians did explore this idea. Zarlino suggested you could tune to twelve equal using an ancient Greek geometrical instrument design called the mesolabium, see page 55 of Tuning and temperament : a historical survey : Barbour, J. Murray (James Murray), (^{[21]} 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 nonsystems) from Vicentino to Monteverdi [http://www.academia.edu/291960/Chromatic_systems_or_nonsystems_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 (^{[22]}). 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 (^{[23]}. 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[edit  edit source]
 Why twelve notes as one attractive arrangement
 How did Western music settle on a 12 note scale
 What is Microtonal music?
External links[edit  edit source]
 Many more screenshots of those cycle of fifths notations here: Bounce Metronome Pro screenshots (Facebook)
This article is based on my Quora answer to a question here: To be totally unconventional, should a piano with keys of E# and B# be produced? Would it be more versatile and more interesting?
References[edit  edit source]
 ↑ Italian splitkeyboard instruments with fewer than nineteen divisions to the octave Denzil Wraight
 ↑ comment by Margo Schulter in The Xenharmonic Alliance II with early history of keyboards with more than twelve notes to an octave
 ↑ Zarlino's 2/7comma meantone memperament
 ↑ An historical survey of meantone temperaments
 ↑ "1/3 syntonic comma meantone", Xenharmonic wiki.]
 ↑ 6/5)^19/2^4 (google calc)
 ↑ Edward Parmentier playing a 17thCentury French Harpsichord Music in 1/3 comma meantone
 ↑ Music: A Mathematical Offering
 ↑ Blackwood: Microtonal Compositions
 ↑ (5/4)^31/2^9
 ↑ An historical survey of meantone temperaments to 1620, Mark Lindley
 ↑ Quantifying Music
 ↑ [1]
 ↑ 3/2)^53/2^30
 ↑ [2]
 ↑ Full text of "On the sensations of tone as a physiological basis for the theory of music"
 ↑ Pythagorean Tuning Mrgo Schulter
 ↑ (7*31) % 53
 ↑ (5*31) % 53>
 ↑ Sagittal microtonal notation
 ↑ Tuning and temperament : a historical surve
 ↑ Tuning and Temperament: A Historical Survey, J. Murray Barbour (you need to click on the Page xx link on that page in google books snippet view to see the quote)
 ↑ Tuning and Temperament at archive.org