How to make a Kite Guitar
General design considerations
Converting an existing guitar vs. building a new one
Converting is far more affordable than building, and will be the focus of this section. If building, the only difference is the location of the frets and the string gauges. The process for intonating the saddle points also differs.
Even-frets vs. odd-frets
There are two types of Kite Guitar fretboards, even-frets and odd-frets. In the former, all or almost all of the frets are an even number of 41-equal steps from the nut. In the latter, it's an odd number. The even-frets layout is primarily for isomorphic ("same-shape") tunings and the odd-frets layout is primarily for open tunings. Most of the research and development to date has focused on the even-frets layout.
An odd-frets guitar can be converted to an even-frets one simply by capoing. An even-frets one can be converted to odd-frets similarly if there is an additional fret (or fret slot that accepts a temporary fret) near the nut. There are also advantages musically to this extra fret even if not using open tunings. The extra fret is named the "a-fret" if it's between the nut and the 1st fret, "b-fret" if it's between the 1st and 2nd frets, etc. A b-fret or b-fret-slot might be better for acoustics, which tend to have higher nuts. An empty fret slot does not interfere at all with normal play. Thus there is absolutely no downside to having an a-slot and/or a b-slot, and it's highly recommended.
6 strings vs. 7 strings vs. 8 strings
Assuming one is not using an open tuning, a Kite guitar with 6 strings is a little limiting. 7 strings or even 8 is better. Arguably a slightly longer scale, say 26.5-27", is nice because it makes the frets less cramped. Fortunately 7- and 8-string electric guitars often have longer scales anyway. But on the other hand, the Kite guitar's frets are not much tighter than a normal mandolin's, and some feel a longer scale isn't necessary.
While 7- and 8-string electric guitars are plentiful, acoustic ones are rarer. (See Extended range guitar.) One way to get one is to convert a 12-string guitar. The neck will be sufficiently strong and there will be enough tuners. There's fewer strings but more courses, so the string spacing can be very tight. To avoid this, the new fretboard can be wider than the old one. The fretboard overhang can be filled with bondo to create a nice-feeling neck.
Another possibility is to convert a 6-string classical nylon-string to 7 or 8 strings. The fingerboard is wide enough that it may suffice as is. If not, again the new fretboard can be slightly wider. The tension is low enough that an extra string or two won't break the guitar. The 3 holes on each side of the headstock that the tuner pegs go through can be filled and 4 new holes drilled. Or a steel-string-style tuner or two can be added at the top of the headstock.
There will need to be new holes in the tie block. They can be drilled at an angle, entering from near the upper edge of the back of the tie block. To help get the precise angle and spacing, one can make a guide block out of a hard wood like maple. This block will have holes drilled into it that line up with where the new holes will be.
Because both the nut and the tie block holes will be replaced, and because the new fretboard can be a little wider or narrower than the old one, one has total control over the string spacing.
Nut width
This is a crucial decision when there are 7 or 8 strings, as too wide a nut is hard on the wrist, but too narrow a nut makes fretting cleanly difficult. Nylon strings need a wider nut not only because the strings themselves are wider, but also because they vibrate more widely than steel strings and need more room. See Extended range guitar#Nut and saddle widths. See also guitarplayerhq.com/7-string-guitar-nut-width.
Number of frets
In any given key, the Kite guitar has multiple "rainbow zones" on the neck. Assuming the tonic falls in the "sweet spot" between the 4th and 11th fret, it takes about 28 frets to provide 2 zones in every key, but it takes the full 41 frets to provide 3 zones. This 3rd zone increases the range the lead guitarist has to solo in by a 5th or so. The highest frets are very tight, but still playable melodically. Chording is very difficult. Having a 41st fret makes intonating the guitar easier, see below. In general, if you can fit in 41 frets, do so.
There is a 'home zone" around the 14th fret that is the rainbow zone when the low open string is the tonic. There is a 2nd home zone around the 28th fret. To get a complete 2nd home zone, one needs about 32 frets. This should be the minimum number of frets even on an acoustic or classical without a cutaway. Fortunately this translates to almost 19 conventional frets, which almost all guitars have.
The fret spacing is 1.7 times tighter than a 12-equal guitar. This chart compares it to the standard fret spacing. The spacing between the nut and the first fret is about the same as the space between the 12-equal 9th and 10th frets. Increasing the overall scale length will widen the spacing.
Replacing each fret vs. replacing the entire fretboard
When converting a guitar, it's best to replace the entire fretboard, rather than removing the frets and putting new frets in the old fretboard. The 41-equal 5th is 702.5¢, so two frets will be only 2.5¢ away from the old ones, two will be 5¢ away, etc. So the old and new fret slots overlap, making conversion difficult. The following table shows the distance from the old fret to the new fret for close pairs. One could just use the old slots (or even the old frets) and accept a few cents error. But in certain keys a 5¢ error will make the downmajor 6th or 10th that's already 6¢ flat a full 11¢ flat.
| old fret | 7 | 10 | 14 | 17 | 20 | 21 |
|---|---|---|---|---|---|---|
| new fret | 12 | 17 | 24 | 29 | 34 | 36 |
| cents | 2.5¢ | -5¢ | 5¢ | -2.5¢ | -10¢ | 7.5¢ |
| distance | 0.024" | -0.040" | 0.032" | -0.013" | -0.045" | 0.032" |
Removing the entire fretboard also has the advantage that you can get a pre-slotted computer-cut fretboard fairly cheaply that has extremely accurate slot placement (see below).
When building a guitar, the bridge is positioned relative to the fretboard. When converting a guitar, it's crucial to place the fretboard accurately relative to the bridge. One method: first put the frets on the fretboard. Then clamp it to the neck using narrow wooden blocks that won't interfere with the strings. Then string it up, test the intonation, and adjust the fretboard placement as needed (see Saddle Compensation below). Finally, mark the correct position, remove the strings, and glue down the fretboard. These pictures illustrate the clamping on a standard 12-equal guitar:
Fret Placement
On a standard guitar, the nth fret is L * (1 - 2^(-n/12)) from the nut, where L is the scale length. On a Kite guitar, for an even-fret layout, it's L * (1 - 4^(-n/41)). In other words, simply replace the 12th root of 2 with the 41st root of 4. For the a-fret, use n = 0.5. Or use this spreadsheet:
http://tallkite.com/misc_files/KiteGuitarFretPlacementCalculator.ods
As an alternative to doing the work yourself, various suppliers can make pre-slotted fingerboards complete with radius, taper and inlays. All you need to do is glue it on and put in the frets.
- Precision Pearl (Texas)
- Starrett Guitars (Colorado)
- Tonedevil Guitars (Idaho)
- JLJ Instruments (Maryland)
Fret Markers
On an even-frets layout, dots (fretboard markers) are placed every 4 frets in a cycle of single-double-triple. So, the 4th fret has a single dot, the 8th fret has double dots, the 12th fret has triple dots, and then the 16th fret is back to single, and so on. Thus, a 36-fret guitar (pictured) has 18 dots on 9 frets, and a 41-fret guitar has 19 dots on 10 frets.
The small dots on the side of the neck follow the same single/double/triple pattern. The double and triple dots are oriented like the usual 12-equal double dots. Further up the neck, the triple dots are too wide to fit between the frets, but this is not a problem.
Because the frets get closer as one goes up the neck, the double dots are closer to the triple dots than the single dots. As a result, if the distance between the double dots is the same as the distance between any two of the triple dots, the side of each "kite" formed by the dots is a concave line. To make a nice straight line, the distance between the double dots needs to be increased by a factor of 2/(1+2^(-8/41)) = about 1.0675, i.e. 6.75% greater.
String Gauges
A 6-string Kite guitar tuned in 3rds can be strung with a standard set of strings, but it's not ideal. The high strings will be somewhat slack, and the low strings will be somewhat tight. To find the appropriate gauges, use the D'Addario method: calculate each string's tension from its unit weight, length and pitch (frequency) by the formula T = (UW x (2 x L x F)2) / 386.4. For open strings, the length is the guitar's scale. The frequency in hertz of the Nth string of 8 strings tuned in the standard downmajor 3rds with a low string of vD is 440 * (2 ^ (-7/12 + (21 - 13*N) / 41)). For a 6-string guitar in mid-6 tuning, N ranges from 2 to 7. Or use the frequency table below. The unit weight is pounds per inch, and is a function of string gauge and string type (plain vs. wound, etc.). D'Addario has published their unit weights, thus the individual tensions can be calculated for a given set of strings. One can work backwards from this and select string gauges/types that give uniform tensions using this spreadsheet: TallKite.com/misc_files/StringTensionCalculator.ods The desired tension depends on the instrument, and of course personal taste. A steel-string acoustic guitar might have 25-30 lbs. tension for each string. A 12-equal 25.5" electric guitar strung with a standard 10-46 set has 15-20 lbs. With a 9-42 set it has 13-16 lbs.
- A longer scale means a higher tension or a smaller gauge or a lower pitch (frequency)
- A higher tension means a longer scale or a bigger gauge or a higher pitch
- A bigger gauge means a shorter scale or a higher tension or a lower pitch
- A higher pitch means a shorter scale or a higher tension or a smaller gauge
Microtonalist and luthier Tom WInspear can provide custom string sets at his website www.winspearinstrumental.com. His approach is to extrapolate from familiar string sets. He says this about string gauges: "Gauges can be scaled at the same ratios as frequency. A 41-equal downmajor 3rd is 2^(13/41) = 1.2458, thus from string to string the gauge changes by 24.58%. But you can't do that across the plain to wound transition. To tune to different keys, increase the gauges by 5.95% for each 12-equal semitone of transposition, or 1.705% for each 41-equal step. All this assumes a 25.5" scale. For a scale of S inches, multiply each gauge by 25.5/S and round off. For scales longer than 25.5", err on the side of heavier and round up, as longer scales do feel more flexible loaded with the same tension. Likewise, for scales less than 25.5", err on the side of lighter and round down. However, the plain strings should always be rounded slightly down, and should utilize .0005" increment plain strings where available."
JustStrings.com sells custom gauges singly or in bulk. Recommended (somewhat light) gauges for a 27" acoustic guitar: 11.5 15 18 24 30 36 46 56 (3 plain, 5 wound). For a 25.5" or 26.5" electric: 10 13 16 22 26 32 42 52, the wound 4th string could instead be a 19 plain.
Saddle and Nut Compensation
Since the Kite guitar is so much more in tune than the 12-equal guitar, extra care should be taken with saddle compensation.
Method #1: To find the saddle compensation on a standard guitar, one compares the harmonic at the 12th fret with the fretted note at the 12th fret. For the Kite guitar, by a weird coincidence, one does the same! But the 12th fret now makes the 3rd harmonic, not the 2nd. Thus the two notes should be an octave apart, not a unison. If using a tuner, this is not a problem. But if using your ear, a unison is easier to hear than an octave. To get a unison, when you fret the string, play the 2nd harmonic with your other hand. With your forefinger or middle finger, touch the string midway between the 32nd and 33rd frets. Then stretch your hand and pluck with your thumb as close as you can get to the midpoint between your finger and the bridge. If this isn't feasible (e.g. with a bass guitar), you can capo the string at the 12th fret and use both hands to play the harmonic. (And to be extremely precise, the fretted note should be 0.48¢ sharper than the harmonic. The 3rd harmonic is 701.96¢ and the 41-equal interval is 702.44¢.)
On a standard guitar, there's a formula for saddle compensation. Move the saddle point back by about 0.015" for every cent that the 12th fret note is sharp of the open string's 2nd harmonic. The 0.015" figure is more precisely the scale length times ln(2)/1200, which is approximately scaleLength/1731. Saddle compensation flattens the 12th fret note twice as much as the open string note. So if the 12th fret note is 3¢ sharp, flattening the open string note by 3¢ (about 0.045") flattens the 12th fret note by 6¢, and the interval between them is flattened by 3¢ to an exact octave.
On a Kite guitar, the scaleLength/1731 formula still holds. But saddle compensation affects the 12th fret note only one and a half times as much as the open string note. (Because an octave has frequency ratio 2/1 = twice as much, and a fifth has 3/2 = one and a half as much.) Hence for each cent of sharpness, one must flatten by two cents.
For example, suppose the 12th fret note is 2¢ sharp of the 3rd harmonic. It's supposed to be 0.48¢ sharp, so the actual sharpness is only 1.5¢. (In practice, if one's tuner isn't this accurate, one might simply round down a bit.) Move the saddle point back by twice this, 3¢ or 0.045". This will flatten the open string by 3¢ and the 12th fret note by 4.5¢, narrowing the interval by 1.5¢ to an exact 41-equal 5th. On the saddle, mark a point 0.045" behind the exit point, and file up to the mark.
Alternative method #1: If the guitar has a 41st fret, compensation can be done more easily and accurately by comparing the harmonic at the 41st fret (the 4th harmonic) with the fretted note at the 41st fret. They should be an exact unison, so no need to subtract a half cent, and no need to play the harmonic of the fretted note. The 4th harmonic is a double octave, with frequency ratio 4/1, so saddle compensation affects the 41st fret note four times as much as the open string note. Hence for each cent of sharpness, one must flatten by one-third cent.
In the previous example, the 12th fret harmonic was 2¢ sharper than the fretted note. This would make the 41st fret note 9¢ sharp of the 4th harmonic. Move the saddle point back by 1/3 this, 3¢ or 0.045". This will flatten the open string by 3¢ and the 41st fret note by 12¢, narrowing the interval by 9¢ to an exact double octave. This method is more accurate because tuners aren't perfect, and an error affects the compensation distance only one-sixth as much. This method also works for 12-equal guitars on the 24th fret.
Other alternatives: The 8th harmonic is at fret 4 and the 10th one is at fret 3. The 9th harmonic is midway between them. Play the 8th, 9th and 10th harmonics to get a do-re-mi melody. Now play those same harmonics just a few inches from the bridge. Practice until you can cleanly play the 9th harmonic with one hand. Next play that harmonic while fretting at the 24th fret (major 9th = 9/4 ratio). The fretted harmonic should be 1 cent sharper. For every cent of sharpness above that, flatten at the saddle by four-fifths of a cent. For example, if the fretted harmonic is 6¢ sharp, that's 5 extra cents of sharpness. Move the saddle point back by four-fifths of this, 4¢. This will flatten the open string by 4¢ and the 24th fret note by 9¢, narrowing the interval by 5¢ to an exact 41-equal major 9th.
The 7th harmonic is between the 4th and 5th frets. Find that same harmonic about 3-4" from the bridge. Play it one-handed both open and while fretting at the 37th fret (minor 7th plus an 8ve = 7/2 ratio). The fretted harmonic should be 3 cents flatter. For every cent of sharpness above that, flatten at the saddle by two-fifths of a cent. For example, if the fretted harmonic is 2¢ sharp, that's 5 extra cents of sharpness. Move the saddle point back by two-fifths of this, 2¢. This will flatten the open string by 2¢ and the 37th fret note by 7¢, narrowing the interval by 5¢ to an exact 41-equal downminor 7th.
| fret | interval | ratio | harmonic | fretted note should be | for each cent of sharpness | location to play shared harmonic |
|---|---|---|---|---|---|---|
| 12 | 5th | 3/2 | 3rd | 0.5¢ sharp | flatten by two cents | 2/3 of string: between frets 32 and 33 |
| 24 | maj 9th | 9/4 | 9th | 1¢ sharp | flatten by four-fifths of a cent | 8/9 of string: past the fretboard up by the bridge |
| 37 | vmin 7th | 7/2 | 7th | 3¢ flat | flatten by two-fifths of a cent | 6/7 of string: past the fretboard up by the bridge |
| 41 | dbl 8ve | 4/1 | 4th | the same | flatten by one-third of a cent | 1/4 of string: at the same fret 41 |
Method #2: The first method serves as a rough check of the saddle points. But it's much safer to check multiple frets. The cents table below (printable pdf here) has the pitch of every single note on the fretboard. The 2nd page of the pdf omits some redundant information to make room to pencil in discrepancies in cents. But the open strings aren't reliable, because the nut is not yet compensated (nut compensation must be done after saddle compensation). Use a capo to remove the nut issue. Capo the string at the 1st fret (or 2nd or 3rd, if the capo doesn't fit your 8-string very well). Tune the capo'ed string to the table, then compare the other frets to the table. Important: do not remove the capo during this process, as that will change the tension, and thus the pitch. It's usually sufficient to check every 4th fret, i.e. every dot. Look for the general trend. If the saddle point is too far back, the higher frets will be increasingly flat. Too far forward, and they will trend sharp. If there's an outlier that breaks the pattern, check its neighboring frets. No guitar is perfect. If some frets are sharp and some equally flat, that's the best you can get. Once you find the trend, estimate how much cents error would be expected at the 5th dot, which is almost an octave. That's roughly how many cents to compensate by. (To be super-precise, you could increase the cents by about 3%, so that 6¢ becomes 6.2¢.) Compensate as in method #1 with the scaleLength/1731 formula.
Nut compensation can be done similarly to a standard guitar, by comparing the open string to the fretted notes. But extra care might be taken here too. One can shorten the fingerboard by around 0.030" (more if the nut action is high) to slightly overcompensate, then de-compensate empirically by filing the front of the nut to move the exit points back. One can determine the exact amount to file by finding the sharpness in cents with a tuner, then using the scaleLength/1731 formula. The front of the nut can be filed lengthwise to move all the exit points at once, or up and down to move individual exit points.
Final notes:
- String gauges affect compensation, so try to choose the correct gauges first.
- One can avoid nut compensation by using a zero fret or by having a very low action.
- Adjustable saddle and nut for acoustic guitars (similar to electric guitars): https://www.portlandguitar.com/collections/bridges
For more on saddle and nut compensation, see
- https://www.doolinguitars.com/intonation/intonation4.html (Mike Doolin)
- https://www.portlandguitar.com/pages/guitar-intonation (Jay Dickinson)
- https://www.portlandguitar.com/pages/perfect-intonation-long-version (Jay Dickinson)
- http://schrammguitars.com/intonation.html (John and William Gilbert)
- https://www.proguitar.com/academy/guitar/intonation/byers-classical (Greg Byers)
String Spacing
With 7 or 8 strings, it's important to avoid both a too-wide neck and a too-tight string spacing. Every millimeter counts. One can reduce neck width by minimizing the distance from the outer string to the edge of the fretboard. To do this, minimize the amount of rounding of the edge of the fretboard. And preserve as much usable fret length as possible by beveling the ends of the frets at a steeper angle and not over-rounding the corner of the fret where the top and the end meet.
The spacing can be slightly improved further as follows:
Conventional wisdom holds that there are two ways to space the strings: center-to-center (C2C) and edge-to-edge (E2E). For the right hand, E2E is better than C2C, because otherwise it's harder to fit one's finger between the thicker strings. E2E spacing ensures that the gap between strings is uniform, and each string is equally easy to pluck.
On the left hand, if the spacing is too tight, when one frets a string and plays the neighboring string either open or fretted further back, the finger can dampen the neighboring string. Thus the important gap is the gap between every other string. That is, when fretting the 2nd string, the important gap is between the inner edges of the 1st and 3rd strings. When fretting the 3rd string, it's between the 2nd and 4th string. (When fretting the 1st string, the gap is between the 1st and 2nd string, but if the 2nd string is more or less in the center of the 1st-to-3rd gap, the 1st-to-2nd gap will be sufficiently large.)
This spacing is called edge-to-next-edge (E2NE). It is different from the other two spacings. C2C spacing results in the thicker strings being more crowded and harder to fret cleanly. E2E spacing results in the thinner strings being more crowded.
But specifying that these gaps be uniform doesn't completely specify E2NE spacing, because one could shift every other string sideways without changing these gaps. So we need an additional requirement. Ideally each string should be midway between the nearest edges of the two neighboring strings, i.e. perfectly centered in its gap. The center-to-edge spacing would be constant for each string. But this is impossible. For example, the distance from the center of the 2nd string to the nearest edge of the 3rd string must be less than the distance from the center of the 3rd string to the nearest edge of the 2nd string, because the 2nd string is thinner.
It is not yet known how to maximize centeredness. Consider the center of the 2nd string, and the center of the gap between the 1st and 3rd strings' edges. Let x be the distance between the two, measured so that positive x corresponds to being closer to the 1st string. Each string except the 1st and last will have a similar distance from the center of the gap it is in, measured in the same direction, called its off-centeredness. This distance is different from x but utterly dependent on it. For a given set of string gauges, how can one find the x that minimizes all the off-centerednesses? For gauges 11.5 15 18 24 30 36 46 56, the best x is zero.
In the next table, R1, R2, etc. is the radius of each string, and D is a constant roughly equal to 1/7th of the nut width. The value of D is not consistent from column to column. In E2E spacing, all off-centerednesses are zero. In C2C spacing, each string is off-center towards its thicker neighbor.
| C2C | E2E | E2NE | E2NE off-centeredness | |
|---|---|---|---|---|
| 2nd string | D | D + R1 + R2 | D - x + R1 | x |
| 3rd string | 2D | 2D + R1 + 2R2 + R3 | 2D + R1 + R3 | -x - (R3-R2) |
| 4th string | 3D | 3D + R1 + 2R2 + 2R3 + R4 | 3D - x + R1 + R2 + R4 | x + (R3-R2) - (R4-R3) |
| 5th string | 4D | 4D + R1 + 2R2 + 2R3 + 2R4 + R5 | 4D + R1 + 2R3 + R5 | -x - (R3-R2) + (R4-R3) - (R5-R4) |
| 6th string | 5D | 5D + R1 + 2R2 + 2R3 + 2R4 + 2R5 + R6 | 5D - x + R1 + R2 + 2R4 + R6 | x + (R3-R2) - (R4-R3) + (R5-R4) - (R6-R5) |
| 7th string | 6D | 6D + R1 + 2R2 + 2R3 + 2R4 + 2R5 + 2R6 + R7 | 6D + R1 + 2R3 + 2R5 + R7 | -x - (R3-R2) + (R4-R3) - (R5-R4) + (R6-R5) - (R7-R6) |
| 8th string | 7D | 7D + R1 + 2R2 + 2R3 + 2R4 + 2R5 + 2R6 + 2R7 + R8 | 7D - x + R1 + R2 + 2R4 + 2R6 + R8 | (N/A) |
Note that the nut is slotted edge-to-next-edge but the bridge is edge-to-edge, so as one plays further up the neck, the spacing deviates from the ideal. Furthermore each string's sideways movement increases as you get away from the nut. But the spacing widens further up the neck, making fretting cleanly easier, so this is not a problem.
Resources
Graphtech.com carries a wide variety of tusq saddles (synthetic ivory) as well as other guitar parts.
Thomastik-infeld.com sells kf110 string sets, which are low-tension steel strings that supposedly can be used on a classical guitar.
Tables
Cents
Every note on the Kite Guitar fretboard. The outer columns show the dots on the fretboard. The low note is vD and the tuning is in downmajor 3rds. The note names in the table are 12-equal, not 41-equal. The low vD is written "D -29.3", meaning 12-equal D minus 29.3¢. The 7 natural notes in 41-equal are bolded and underlined italic. The full set of 41-equal names are here: File:The Kite Tuning 5.png
| 8th string | 7th string | 6th string | 5th string | 4th string | 3rd string | 2nd string | 1st string | |||||||||||
| 0 | D | -29.3 | F# | -48.8 | A | +31.7 | C# | +12.2 | F | -7.3 | A | -26.8 | C# | -46.3 | E | +34.1 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | D | +29.3 | F# | +9.8 | Bb | -9.8 | D | -29.3 | F# | -48.8 | A | +31.7 | C# | +12.2 | F | -7.3 | ||
| 2 | Eb | -12.2 | G | -31.7 | Bb | +48.8 | D | +29.3 | F# | +9.8 | Bb | -9.8 | D | -29.3 | F# | -48.8 | ||
| 3 | Eb | +46.3 | G | +26.8 | B | +7.3 | Eb | -12.2 | G | -31.7 | Bb | +48.8 | D | +29.3 | F# | +9.8 | ||
| * | 4 | E | +4.9 | G# | -14.6 | C | -34.1 | Eb | +46.3 | G | +26.8 | B | +7.3 | Eb | -12.2 | G | -31.7 | * |
| 5 | F | -36.6 | G# | +43.9 | C | +24.4 | E | +4.9 | G# | -14.6 | C | -34.1 | Eb | +46.3 | G | +26.8 | ||
| 6 | F | +22.0 | A | +2.4 | C# | -17.1 | F | -36.6 | G# | +43.9 | C | +24.4 | E | +4.9 | G# | -14.6 | ||
| 7 | F# | -19.5 | Bb | -39.0 | C# | +41.5 | F | +22.0 | A | +2.4 | C# | -17.1 | F | -36.6 | G# | +43.9 | ||
| * * | 8 | F# | +39.0 | Bb | +19.5 | D | +0.0 | F# | -19.5 | Bb | -39.0 | C# | +41.5 | F | +22.0 | A | +2.4 | * * |
| 9 | G | -2.4 | B | -22.0 | Eb | -41.5 | F# | +39.0 | Bb | +19.5 | D | +0.0 | F# | -19.5 | Bb | -39.0 | ||
| 10 | G# | -43.9 | B | +36.6 | Eb | +17.1 | G | -2.4 | B | -22.0 | Eb | -41.5 | F# | +39.0 | Bb | +19.5 | ||
| 11 | G# | +14.6 | C | -4.9 | E | -24.4 | G# | -43.9 | B | +36.6 | Eb | +17.1 | G | -2.4 | B | -22.0 | ||
| * * * | 12 | A | -26.8 | C# | -46.3 | E | +34.1 | G# | +14.6 | C | -4.9 | E | -24.4 | G# | -43.9 | B | +36.6 | * * * |
| 13 | A | +31.7 | C# | +12.2 | F | -7.3 | A | -26.8 | C# | -46.3 | E | +34.1 | G# | +14.6 | C | -4.9 | ||
| 14 | Bb | -9.8 | D | -29.3 | F# | -48.8 | A | +31.7 | C# | +12.2 | F | -7.3 | A | -26.8 | C# | -46.3 | ||
| 15 | Bb | +48.8 | D | +29.3 | F# | +9.8 | Bb | -9.8 | D | -29.3 | F# | -48.8 | A | +31.7 | C# | +12.2 | ||
| * | 16 | B | +7.3 | Eb | -12.2 | G | -31.7 | Bb | +48.8 | D | +29.3 | F# | +9.8 | Bb | -9.8 | D | -29.3 | * |
| 17 | C | -34.1 | Eb | +46.3 | G | +26.8 | B | +7.3 | Eb | -12.2 | G | -31.7 | Bb | +48.8 | D | +29.3 | ||
| 18 | C | +24.4 | E | +4.9 | G# | -14.6 | C | -34.1 | Eb | +46.3 | G | +26.8 | B | +7.3 | Eb | -12.2 | ||
| 19 | C# | -17.1 | F | -36.6 | G# | +43.9 | C | +24.4 | E | +4.9 | G# | -14.6 | C | -34.1 | Eb | +46.3 | ||
| * * | 20 | C# | +41.5 | F | +22.0 | A | +2.4 | C# | -17.1 | F | -36.6 | G# | +43.9 | C | +24.4 | E | +4.9 | * * |
| 21 | D | +0.0 | F# | -19.5 | Bb | -39.0 | C# | +41.5 | F | +22.0 | A | +2.4 | C# | -17.1 | F | -36.6 | ||
| 22 | Eb | -41.5 | F# | +39.0 | Bb | +19.5 | D | +0.0 | F# | -19.5 | Bb | -39.0 | C# | +41.5 | F | +22.0 | ||
| 23 | Eb | +17.1 | G | -2.4 | B | -22.0 | Eb | -41.5 | F# | +39.0 | Bb | +19.5 | D | +0.0 | F# | -19.5 | ||
| * * * | 24 | E | -24.4 | G# | -43.9 | B | +36.6 | Eb | +17.1 | G | -2.4 | B | -22.0 | Eb | -41.5 | F# | +39.0 | * * * |
| 25 | E | +34.1 | G# | +14.6 | C | -4.9 | E | -24.4 | G# | -43.9 | B | +36.6 | Eb | +17.1 | G | -2.4 | ||
| 26 | F | -7.3 | A | -26.8 | C# | -46.3 | E | +34.1 | G# | +14.6 | C | -4.9 | E | -24.4 | G# | -43.9 | ||
| 27 | F# | -48.8 | A | +31.7 | C# | +12.2 | F | -7.3 | A | -26.8 | C# | -46.3 | E | +34.1 | G# | +14.6 | ||
| * | 28 | F# | +9.8 | Bb | -9.8 | D | -29.3 | F# | -48.8 | A | +31.7 | C# | +12.2 | F | -7.3 | A | -26.8 | * |
| 29 | G | -31.7 | Bb | +48.8 | D | +29.3 | F# | +9.8 | Bb | -9.8 | D | -29.3 | F# | -48.8 | A | +31.7 | ||
| 30 | G | +26.8 | B | +7.3 | Eb | -12.2 | G | -31.7 | Bb | +48.8 | D | +29.3 | F# | +9.8 | Bb | -9.8 | ||
| 31 | G# | -14.6 | C | -34.1 | Eb | +46.3 | G | +26.8 | B | +7.3 | Eb | -12.2 | G | -31.7 | Bb | +48.8 | ||
| * * | 32 | G# | +43.9 | C | +24.4 | E | +4.9 | G# | -14.6 | C | -34.1 | Eb | +46.3 | G | +26.8 | B | +7.3 | * * |
| 33 | A | +2.4 | C# | -17.1 | F | -36.6 | G# | +43.9 | C | +24.4 | E | +4.9 | G# | -14.6 | C | -34.1 | ||
| 34 | Bb | -39.0 | C# | +41.5 | F | +22.0 | A | +2.4 | C# | -17.1 | F | -36.6 | G# | +43.9 | C | +24.4 | ||
| 35 | Bb | +19.5 | D | +0.0 | F# | -19.5 | Bb | -39.0 | C# | +41.5 | F | +22.0 | A | +2.4 | C# | -17.1 | ||
| * * * | 36 | B | -22.0 | Eb | -41.5 | F# | +39.0 | Bb | +19.5 | D | +0.0 | F# | -19.5 | Bb | -39.0 | C# | +41.5 | * * * |
| 37 | B | +36.6 | Eb | +17.1 | G | -2.4 | B | -22.0 | Eb | -41.5 | F# | +39.0 | Bb | +19.5 | D | +0.0 | ||
| 38 | C | -4.9 | E | -24.4 | G# | -43.9 | B | +36.6 | Eb | +17.1 | G | -2.4 | B | -22.0 | Eb | -41.5 | ||
| 39 | C# | -46.3 | E | +34.1 | G# | +14.6 | C | -4.9 | E | -24.4 | G# | -43.9 | B | +36.6 | Eb | +17.1 | ||
| * | 40 | C# | +12.2 | F | -7.3 | A | -26.8 | C# | -46.3 | E | +34.1 | G# | +14.6 | C | -4.9 | E | -24.4 | * |
| 41 | D | -29.3 | F# | -48.8 | A | +31.7 | C# | +12.2 | F | -7.3 | A | -26.8 | C# | -46.3 | E | +34.1 | ||
Frequencies
41-equal frequencies in Hertz. D is tuned to standard A-440 pitch. vA is roughly 432hz, and vvB is roughly the ubiquitous 60hz mains hum.
| 0th octave | 1st octave | 2nd octave | 3rd octave | middle-C | 5th octave | 6th octave | 7th octave | 8th octave | |
| C | 16.305589 | 32.611178 | 65.222357 | 130.44471 | 260.88943 | 521.77886 | 1043.5577 | 2087.1154 | 4174.2308 |
| ^C | 16.583595 | 33.167191 | 66.334382 | 132.66876 | 265.33753 | 530.67505 | 1061.3501 | 2122.7002 | 4245.4004 |
| ^^C / vDb | 16.866341 | 33.732683 | 67.465366 | 134.93073 | 269.86146 | 539.72293 | 1079.4459 | 2158.8917 | 4317.7834 |
| vC# / Db | 17.153908 | 34.307817 | 68.615633 | 137.23127 | 274.46253 | 548.92506 | 1097.8501 | 2195.7003 | 4391.4005 |
| C# / ^Db | 17.446378 | 34.892756 | 69.785512 | 139.57102 | 279.14205 | 558.28410 | 1116.5682 | 2233.1364 | 4466.2728 |
| ^C# / vvD | 17.743834 | 35.487669 | 70.975337 | 141.95067 | 283.90135 | 567.80270 | 1135.6054 | 2271.2108 | 4542.4216 |
| vD | 18.046362 | 36.092724 | 72.185449 | 144.37090 | 288.74179 | 577.48359 | 1154.9672 | 2309.9344 | 4619.8687 |
| D | 18.354048 | 36.708096 | 73.416192 | 146.83238 | 293.66477 | 587.32954 | 1174.6591 | 2349.3181 | 4698.6363 |
| ^D | 18.666980 | 37.333960 | 74.667919 | 149.33584 | 298.67168 | 597.34335 | 1194.6867 | 2389.3734 | 4778.7468 |
| ^^D / vEb | 18.985247 | 37.970494 | 75.940988 | 151.88198 | 303.76395 | 607.52791 | 1215.0558 | 2430.1116 | 4860.2232 |
| vD# / Eb | 19.308941 | 38.617881 | 77.235763 | 154.47153 | 308.94305 | 617.88610 | 1235.7722 | 2471.5444 | 4943.0888 |
| D# / ^Eb | 19.638153 | 39.276306 | 78.552613 | 157.10523 | 314.21045 | 628.42090 | 1256.8418 | 2513.6836 | 5027.3672 |
| ^D# / vvE | 19.972979 | 39.945957 | 79.891915 | 159.78383 | 319.56766 | 639.13532 | 1278.2706 | 2556.5413 | 5113.0825 |
| vE | 20.313513 | 40.627026 | 81.254051 | 162.50810 | 325.01621 | 650.03241 | 1300.0648 | 2600.1296 | 5200.2593 |
| E | 20.659853 | 41.319706 | 82.639412 | 165.27882 | 330.55765 | 661.11530 | 1322.2306 | 2644.4612 | 5288.9224 |
| ^E | 21.012098 | 42.024197 | 84.048393 | 168.09679 | 336.19357 | 672.38714 | 1344.7743 | 2689.5486 | 5379.0972 |
| vF | 21.370349 | 42.740698 | 85.481397 | 170.96279 | 341.92559 | 683.85117 | 1367.7023 | 2735.4047 | 5470.8094 |
| F | 21.734708 | 43.469416 | 86.938833 | 173.87767 | 347.75533 | 695.51066 | 1391.0213 | 2782.0426 | 5564.0853 |
| ^F | 22.105279 | 44.210559 | 88.421118 | 176.84224 | 353.68447 | 707.36894 | 1414.7379 | 2829.4758 | 5658.9515 |
| ^^F / vGb | 22.482169 | 44.964337 | 89.928675 | 179.85735 | 359.71470 | 719.42940 | 1438.8588 | 2877.7176 | 5755.4352 |
| vF# / Gb | 22.865484 | 45.730968 | 91.461936 | 182.92387 | 365.84774 | 731.69549 | 1463.3910 | 2926.7819 | 5853.5639 |
| F# / ^Gb | 23.255335 | 46.510669 | 93.021339 | 186.04268 | 372.08535 | 744.17071 | 1488.3414 | 2976.6828 | 5953.3657 |
| ^F# / vvG | 23.651832 | 47.303664 | 94.607329 | 189.21466 | 378.42931 | 756.85863 | 1513.7173 | 3027.4345 | 6054.8690 |
| vG | 24.055090 | 48.110180 | 96.220359 | 192.44072 | 384.88144 | 769.76287 | 1539.5257 | 3079.0515 | 6158.1030 |
| G | 24.465223 | 48.930446 | 97.860892 | 195.72178 | 391.44357 | 782.88714 | 1565.7743 | 3131.5485 | 6263.0971 |
| ^G | 24.882349 | 49.764698 | 99.529395 | 199.05879 | 398.11758 | 796.23516 | 1592.4703 | 3184.9406 | 6369.8813 |
| ^^G / vAb | 25.306586 | 50.613173 | 101.226346 | 202.45269 | 404.90538 | 809.81077 | 1619.6215 | 3239.2431 | 6478.4861 |
| vG# / Ab | 25.738057 | 51.476115 | 102.952229 | 205.90446 | 411.80892 | 823.61783 | 1647.2357 | 3294.4713 | 6588.9427 |
| G# / ^Ab | 26.176885 | 52.353769 | 104.707538 | 209.41508 | 418.83015 | 837.66031 | 1675.3206 | 3350.6412 | 6701.2825 |
| ^G# / vvA | 26.623194 | 53.246388 | 106.492775 | 212.98555 | 425.97110 | 851.94220 | 1703.8844 | 3407.7688 | 6815.5376 |
| vA | 27.077112 | 54.154225 | 108.308450 | 216.61690 | 433.23380 | 866.46760 | 1732.9352 | 3465.8704 | 6931.7408 |
| A | 27.538770 | 55.077541 | 110.155081 | 220.31016 | 440.62032 | 881.24065 | 1762.4813 | 3524.9626 | 7049.9252 |
| ^A | 28.008299 | 56.016599 | 112.033197 | 224.06639 | 448.13279 | 896.26558 | 1792.5312 | 3585.0623 | 7170.1246 |
| ^^A / vBb | 28.485834 | 56.971667 | 113.943335 | 227.88667 | 455.77334 | 911.54668 | 1823.0934 | 3646.1867 | 7292.3734 |
| vA# / Bb | 28.971510 | 57.943020 | 115.886039 | 231.77208 | 463.54416 | 927.08831 | 1854.1766 | 3708.3533 | 7416.7065 |
| A# / ^Bb | 29.465467 | 58.930933 | 117.861867 | 235.72373 | 471.44747 | 942.89493 | 1885.7899 | 3771.5797 | 7543.1595 |
| ^A# / vvB | 29.967845 | 59.935691 | 119.871381 | 239.74276 | 479.48553 | 958.97105 | 1917.9421 | 3835.8842 | 7671.7684 |
| vB | 30.478789 | 60.957579 | 121.915158 | 243.83032 | 487.66063 | 975.32126 | 1950.6425 | 3901.2850 | 7802.5701 |
| B | 30.998445 | 61.996890 | 123.993780 | 247.98756 | 495.97512 | 991.95024 | 1983.9005 | 3967.8010 | 7935.6019 |
| ^B | 31.526961 | 63.053921 | 126.107842 | 252.21568 | 504.43137 | 1008.86274 | 2017.7255 | 4035.4510 | 8070.9019 |
| vC | 32.064487 | 64.128974 | 128.257949 | 256.51590 | 513.03179 | 1026.06359 | 2052.1272 | 4104.2544 | 8208.5087 |
| C | 32.611178 | 65.222357 | 130.444714 | 260.88943 | 521.77886 | 1043.55771 | 2087.1154 | 4174.2308 | 8348.4617 |