Which Frets Do You Need on a Guitar? - II
The Diatonic Fretting Map and what do those dots mean on your meantone guitar?
This is the second part of a series on which frets you need on a guitar.
I’ll just start off by sharing my newly designed Diatonic Fretting Map. The previous article already showed something similar: a Pure Data patch with Ts values and Lilypond style note names tied to fictive frets.
And some readers might recognize both images from an earlier, simpler version in plain text, I shared when writing about n-tone equal temperaments. Just have a look at the map down here if you like, then I'll explain a bit more afterwards…
![Diatonic Fretting Map by Luthier Alfons Diatonic Fretting Map by Luthier Alfons](https://substackcdn.com/image/fetch/w_1456,c_limit,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F3e8b77c9-1e9a-4185-b709-c7c4d885e9a0_2000x1911.jpeg)
The map above is a short version of the Diatonic Fretting Map on my website. It shows the first perfect fourth of a 6-string guitar tuned to the regular EADGBE.1
Each line shows:
A Ts distance from the reference note within the fret distance boundaries of this fourth2
The corresponding note names for that Ts distance (as can be generated with my Pure Data patch)
The corresponding fret number for each of the listed equal temperaments, sorted by their semitone-to-whole-tone ratios in ascending order (1/2, 4/7, 3/5, 2/3)
Some lines feature a colored dot to indicate sharps (blue) versus flats (yellow), as used on my meantone guitar3
As I might have mentioned in earlier posts, the theoretical options for note names within these boundaries are infinite, so I limited them to the most common, least obscure.
31-TET
T = 5 parts, S = 3 parts
The spacing of the lines is based on 31-TET. Values that share a fret in 31-TET I fitted into this spacing. I actually only added them, to show which notes would be situated on these frets if you would actually place them on a 31-TET guitar. They’re all shown in dark red, which means, they’re too obscure to be used in common diatonic music.
The coloring shows the preference. No color (actually light beige) means that I would use this fret, while light red means, I tend to omit it (which I did at my first meantone fretting design implemented). The light green means a clear yes, the preferred option over the enharmonically equivalent (i.e. same fret in 12-TET, remember?). The dark red, as mentioned, a clear no.
Since frets on a guitar get closer together as they approach the bridge, the spaces can get too small to be able to comfortably play. For this reason, on my meantone guitars — which are almost equally tempered to 31-TET, hence their similar semitone-to-whole-tone ratio — I omitted even more frets on the higher octave, choosing only one of the enharmonically equivalent options. More on this later…
43-TET
T = 7 parts, S = 4 parts
Note that the dark red shared fret options all fill out notes for two different frets in 43-TET, which gives us values for almost all frets in 43-TET. In the first fourth shown here (18 frets), only the 9th and the 16th fret are still missing. On the map that shows the whole octave, also the 34th is omitted from the list. These note names are even more obscure than the dark red ones, and would overlap with more common notes in the other temperaments shown.
This means, on a 43-TET guitar, I wouldn’t put any other frets than on a 31-TET guitar. Indeed, I perhaps wouldn’t even use this temperament on a guitar. The reason is that the semitone-to-whole-tone (S/T) ratio 4/7 gets closer to 12-TET, so the difference gets less noticeable. Another effect is that enharmonically equivalent notes — always one fret apart, except for 12-TET — get closer together as the number of frets in an octave increases, so playability suffers. This makes it less worth the effort. However, when omitting enharmonically equivalent frets in the higher octave, a S/T ratio closer to 1/2 spaces the frets more evenly, and thus widens the smallest gaps.
19-TET
T = 3 parts, S = 2 parts
The highest S/T ratio listed here is 19-TET. That’s great news for the enharmonically equivalent frets in the lower octave. It gives a little extra space to your fingers between them. Nevertheless, on the higher octave, it defeats the purpose of omitting them. Since an F♯ and a G♭ are one fret apart, but the F and the F♯ as well, omitting all flats for example would still leave us with small distances of one fret.
The resulting pattern of parts between actual frets would be something like 1-2-1-2-2-1, compared to 2-3-2-3-3-2 in 31-TET, or 3-4-3-4-4-3 in 43-TET. Since we can’t increase the space of the smallest gap, the price for omitting frets is just too high, I guess, while the price for leaving them in place (i.e. inserting them) is lower. I will find out whether this is the right approach when I will soon be building my first 19-TET guitar.
The price for 19-TET compared to meantone or 31-TET is a little bit less perfection for both the perfect fifth and the major third. In return, the minor third is almost tuned to perfection (6:5). But, a really cool feature of 19-TET is that no frets have to be omitted, that they all make sense. If you have a look at my selection for the meantone guitar’s lower octave (light green and beige), you’ll notice that it includes all frets in 19-TET. This is also true for all frets up to the octave that are not listed here (but in the extended version of the fretting map they are).
The especially good part is, that some frets I omitted for my meantone guitar are included implicitely, since they overlap with others. It’s the same advantage the well-known 12-TET has: the less equally tempered frets, the more overlap there is, the less need to omit frets for playability. Checkout the light red lines. Ts(0,2) is now included, since in 19-TET it overlaps with Ts(2,-1). The same applies to Ts(3,-1) and Ts(1,2).
Which frets to omit?
Now, the choice for frets you’ll need depends on the repertoire you want to be able to play. In which keys is your repertoire written? Being a guitarist for over 30 years, from what I know from pop and rock music, is that common keys — ordered from flat to sharp — are F, Dm, C, Am, G, Em, D, Bm, A, F♯m, E, C♯m, and B. The flattest two only feature a B♭, while the Dm already introduces the first sharp (C♯) as a leading note to D. The rest of the keys don’t feature any flats, but do contain quite some sharps: F♯, C♯, G♯, D♯, A♯, and E♯ and B♯ as leading notes for the sharpest minor keys listed. This circumstance made me prefer sharps over flats, unless a fret is getting “too sharp” compared to its neighbors.
Jazz standards, however, due to the common use of brass sections, are often written in flatter keys, and usually contain chords like B♭7, E♭7, and A♭7. Jazz guitarists might prefer other frets, but may just as well transpose the piece they’re playing from E♭ to E, for example, and eventually tune down their guitar a bit. Furthermore, Jazz can be a bit tricky to play in other temperaments due to typical 12-TET tricks, like the tritone replacement, where a dominant seventh chord resolves into a completely new context, e.g. D♭7 → Cmaj7, instead of D♭7 → G♭maj7, which would be the expected and correct context4.
The line of fifths and the sharpness score
The line of fifths is an infinite collection of notes sorted by their flatness/sharpness. Consider the seven notes from our diatonic scale without modulation. If we order them in this line (one fifth apart from each other) we’ll get the following sequence:
F → C → G → D → A → E → B
When we add notes to this line on either end, we’ll notice we obtain flats and sharps in the order of the diatonic modulation:
… → A♭ → E♭ → B♭ → […] → F♯ → C♯ → G♯ → …
If we want to truly give our frets a score based on its actual sharpness, we can give each note a score, and then take the sum of those numbers for a fret score. The central note in our diatonic scale is the D. If we assign it a zero, we can then add or subtract one for each fifth away from the D. I’ll now add the scores to the central notes written above:
F=-3 → C=-2 → G=-1 → D=0 → A=+1 → E=+2 → B=+3
… → A♭=-6 → E♭=-5 → B♭=-4 → […] → F♯=+4 → C♯=+5 → G♯=+6 → …
By the way, enharmonically equivalent notes are always twelve fifths apart, as you can see comparing the A♭ with the G♯. Since they overlap in 12-TET, many people refer to the line of fifths as the circle of fifths. Others might argue that it’s actually a spiral of fifths, since it’s an infinite line which has overlaps only in pitch, but not in meaning.
The sharpness score for frets
Now we get the idea, we can calculate fret scores. Guitars are generally more sharp than flat, if we consider the score at the nut. EADGBE gives us:
If we retune the G (-1) to an F♯ (+4), as is common for lutes, the score even increases substantially.
Such a score can be calculated for all Ts distances. Of course, one could create an ultra-long line of fifths and look up all the individual notes. But it’s quicker to just apply the same formula to all notes on the fret by looking at the Ts distance for that fret.
For example, a whole tone increase, Ts(1,0), is equal to two fifths up (e.g. C → G → D). If we add this score for all six strings, it increases the fret score by 12. The semitone has a reverse effect. It’s located five fifths down (B → E → A → D → G → C), so each increase of one semitone in the Ts value for the fret, that is Ts(0,1), equals a fret sharpness score decrease of 30 in total for the six strings.
To summarize this, we could say that all Ts distances have a sharpness factor that can be applied to all the notes on the fret at once, so multiplied by the number of strings. This should equal the sum of the sharpness scores for all individual notes. If you’d like, test it with the map, to start with the most straight forward examples. Remember that our score at Ts(0,0) was 7.
Ts(1,0)
F♯=+4 → B=+3 → E=+2 → A=+1 → C♯=+5 → F♯=+4
7 + (6 * 2) = 19
Ts(0,1)
F=-3 → B♭=-4 → E♭=-5 → A♭=-6 → C=-2 → F=-3
7 + (6 * -5) = -23
The dots
![Xelo (meantone version), a versatile and classic style electric guitar by Luthier Alons, the present writer Xelo (meantone version), a versatile and classic style electric guitar by Luthier Alons, the present writer](https://substackcdn.com/image/fetch/w_1456,c_limit,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Fddbc435b-d218-46f0-ade8-507511c15b44_3234x3234.jpeg)
The blue and yellow colored dots in the center of the fingerboard I already explained a bit. In short, I chose blue dots to identify frets with a highly positive sharpness score, where I used the yellow for the highly negative ones. Of course, the colors can be changed. Where you put the limits is also up to the designer, or the player of the instrument. The 1/4 comma meantone guitar on the picture above was created as indicated by the map.
Then, on both edges of the fingerboard you’ll notice dots in a neutral color. For this particular guitar I matched the mint green from the scratch plate. On the left (or top in playing position) I put dots to indicate the odd fret numbers in 12-TET, since guitarists know well to navigate on a 12-TET guitar. These dots indicate to which fret they relate. So, if two dotted or two dotless frets are placed next to each other it means that these two frets represent each other’s enharmonic equivalents.
On the right (bottom) side you’ll just find the dots found on almost every guitar on frets 3, 5, 7, 9, 12 (doubled), 15, 17, 19, and 21. Two neighboring dots also indicate an enharmonically equivalent note.
Overlapping notes and their Ts distance from each other
On my meantone guitars, all these frets only have one function. They don’t overlap with other notes. In 31-TET they theoretically would overlap with all notes at Ts distances of Ts(3,-5) or Ts(-3,5) and their multitudes away, but these notes are even less relevant than others I already decided to omit.
The case of 19-TET is different. Overlapping notes are only Ts(2,-3) or Ts(-2,3) away from each other. Consider the beige line at Ts(2,-1). I omitted the light red line above at Ts(0,2), due to my preference for the higher sharpness score. However, if you consider their actual sharpness scores…
…you’ll notice that the omitted flat fret is even closer to zero (more neutral) than my sharper preference. And since they’re both located at the 4th fret in 19-TET, they’re both worth to be referenced. I already mentioned this as an advantage of 19-TET. The only problem is my dotting strategy.
As you can see on the picture of the guitar, the fourth fret got a blue dot for being sharp and mint green dots on both sides to indicate its equality to the third fret in 12-TET. In 19-TET this is only half the truth, so we might change the colors where this occurs. My idea would be to use a color less contrasting with the fingerboard wood, a color tone somewhere in between the wood and the dots. A flat dot (yellow in my case) can just be added next to the sharp (blue) one, to indicate that this particular fret has a double function.
Over the next few months, I will be building my first 19-TET guitar. I will show you the result when the guitar (or just the fingerboard) is ready.
As stated above, sharing my work is really appreciated. And it’s completely free, even without ads or sponsored content. Although I don’t earn anything having more people read my free articles, it’s still helping me, since more people get to know me. And some of them might at some point start to appreciate my work.
And of course I’m really hoping that those who appreciate what I’m doing, when considering to invest in a music and tuning theory book (I’m starting to write soon), new pickups for their guitar, a tuner, or even a new guitar (in regular or special temperament), at least consider spending their money with me.
The exact tuning of the EADGBE depends on the temperament. One of my articles in the near future will be about how to perfectly tune a meantone or 19-TET guitar with a specialized tuner and custom settings that go beyond what a regular chromatic tuner can do.
That is, based on their position on the fingerboard and thus their frequencies for all of the listed temperaments (12-, 43-, 31, and 19-TET). On the second line for example, Ts(-1,2) would have a lower pitch than the lowest boundary at Ts(0,0) for all semitone-to-whole-tone ratios below 1/2. In the same way, Ts(3,-1) would be higher in pitch than Ts(2,1).
The dots in parentheses were omitted in the lower octave, but added to the higher octave.
The tensioning diminished fifth within the D♭7 chord (F-C♭) is not resolved into the major third of the G♭maj7 chord (G♭-B♭), but resolved as if it were an augmented fourth (F-B) into the major third (or minor sixth) of the Cmaj7 chord (E-C). This results in a sudden very noticeable context change, a shortcut to extreme modulation as a side effect of 12-TET, in which enharmonically equivalent notes (C♭ and B) have the same pitch. The pitch equivalence functions like a wormhole to another place far away.