Just Intonation on a Normal Guitar by Restringing and Retuning

Discussion in 'Music Theory, Lessons & Techniques' started by ixlramp, Apr 10, 2019.

  1. ixlramp

    ixlramp SS.org Regular

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    Just Intonation
    -------------------

    What 'Just Intonation' (JI) is, is best explained in detail elsewhere, but briefly:

    It is 'perfectly tuned harmony'.

    The modern dominant worldwide tonal system is called '12 Tone Equal Temperament' (12TET). It has 12 equally-spaced pitches within an octave. However, these pitches are actually only approximations to perfectly tuned intervals.
    The octave is exact.
    The 'fourth' and 'fifth' are very close approximations, with an error of only 2 cents.
    The 'second' has a non-problematic error of 4 cents.
    'Third's, 'sixth's and 'seventh's have significant errors of 12-18 cents.
    ('cent' = 1/100th of a semitone.)

    If you have had the experience of thirds sounding out of tune, but when you retune them by ear they are then out of tune with 12TET, this is why.

    The JI major third is 3.86 semitones, 14 cents flat of the 12TET major third.
    The JI minor third is 3.16 semitones, 16 cents sharp of the 12TET minor third.

    Perfectly tuned intervals are not equally spaced within an octave, however it was generally decided that the advantages of equally spaced pitches (and regularly spaced frets on a guitar) outweigh the disadvantages of imperfect harmony.

    ///////

    The system
    ---------------

    My system of restringing and retuning allows various JI scales to be played on a normal guitar with normal 12TET frets. The alternative is to use a customised guitar with JI fretting, for example:

    14-note octave just intonation guitar.jpg

    ^ Owned by Matthew Grasso

    My system is a thirds tuning created by 2 interlaced 12TET fifths tunings.
    For example on a 7 string guitar: Strings 1, 3, 5, 7 are one 12TET fifths tuning, strings 2, 4, 6 are the other.
    These 2 12TET fifths tunings have a non-12TET pitch offset between them, which can be altered to switch between various JI scales.

    JI major scale
    ------------------

    For the first JI scale detailed in this thread, the JI major scale, this pitch offset is 3.86 semitones, the JI major third.
    The result is a thirds tuning composed of 2 alternating thirds of sizes 3.86 semitones (JI major third) and 3.14 semitones (very close to the JI minor third).
    Moving across by 2 strings always results in a 12TET fifth (7.00 semitones).

    My system closely approximates the JI major scale to within 4 cents.

    JImajor7st_new.png

    ^ JI major scale.
    Numbers on the left are the intervals between the open strings in cents.

    The 2 interlaced fifths tunings are considered to be 2 'groups' of strings.
    One group contains 12TET pitches, the other group contains pitches that are 14 cents flat of 12TET.
    On the '12TET group' of strings the following scale degrees of the JI major scale are played:
    Tonic, second, fourth, fifth, octave.
    On the 'flattened group' of strings the following scale degrees of the JI major scale are played:
    Third, sixth, seventh.

    These scale degrees are shown on the diagram above. The red circles with no number are the tonic or octave, the other scale degrees are shown as numbers 2-7.

    Playing 3 strings straight across a fret plays the JI major triad chord, 4 strings straight across a fret plays the JI major seventh chord.

    ///////

    String gauges
    ------------------

    Obviously a custom set of gauges is needed. I have used D'Addario tension data to create suitable sequences of gauges to choose a subsequence from according to the pitch range you desire.

    PL0085 or PL009
    PL0105 or PL011
    PL013
    PL016
    NW022
    NW028
    NW036
    NW046
    NW056
    NW070

    For bass guitar:

    PSG022w
    PSB028w
    PSB035
    PSB045
    PSB055
    PSB070
    PSB085
    PSB105
    PSB130

    Here's a suggestion for a 6 string guitar tuned to FACEGB. The tension is similar to the common 9-42 and 10-46 sets. It shares 3 strings with standard tuning.

    B PL013 standard B
    G PL016 standard G
    E NW022
    C NW028
    A NW036 standard A
    F NW046 standard E + 1 semitone

    ///////

    Tuning
    ---------

    As a starting point, tune the strings to a 12TET tuning of alternating major and minor 12TET thirds, for example FACEGB.

    The next step is to slightly detune all the major third intervals, by 14 cents, to make them JI major third intervals.
    For the FACEGB example, this means detuning the A E B strings (which themselves are a 12TET fifths tuning, and are the 'flattened group' of strings described above).
    Once 1 string has been detuned, the other strings in the 'flattened group' can be easily tuned in 12TET fifths to that string.

    So a method of tuning 2 strings to a JI major third interval of 3.86 semitones is required. Most guitar tuners cannot do this.

    Tuning a JI major third using harmonics
    --------------------------------------------------

    First, practice playing open string harmonics to acheive clear harmonics with good sustain.
    The specific harmonics to practice are the 4th harmonic, located precisely at 'fret 5', and the 5th harmonic, located at roughly 'fret 3.9' (not at 'fret 4' as many sources suggest).

    * Start with 2 adjacent strings tuned to a 12TET major third.
    * Play the 5th harmonic of the lower string and the 4th harmonic of the higher string, such that the 2 harmonics sustain together.
    * They will be similar pitches but slightly out of tune.
    * Repeatedly play the harmonics, very slowly and carefully detune the higher string until the 2 harmonics are identical in pitch. The higher string only needs detuning by 14 cents to do this.
    * The 2 open strings will then be tuned to a JI major third.

    ///////

    JI minor scale
    ------------------

    Once your guitar is tuned it can also play the JI minor scale, by simply placing the tonic note on an adjacent string, moving the whole pattern across by 1 string:

    JIminor7st_new.png

    ^ JI minor.
    'b' means 'flattened' or 'minor'.

    Effectively, the 2 groups of strings now have the JI minor third as the pitch offest between them.

    Or, retune to swap the positions of the JI major and JI minor thirds.

    Playing 3 strings straight across a fret plays the JI minor triad chord, 4 strings straight across a fret plays the JI minor seventh chord.

    The string gauges recommended are chosen to work for either of major-minor or minor-major alternating JI thirds.
     
    Last edited: Apr 10, 2019
    StevenC, bostjan, 777timesgod and 2 others like this.
  2. Humbuck

    Humbuck Can't stop, won't stop

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    Can you repeat that?
     
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  3. Hollowway

    Hollowway Extended Ranger

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    I wish to hell I understood microtonality so I could understand and try that.
     
  4. Winspear

    Winspear Tom Winspear

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    If you're keen, just dive in. I read about it for months but nothing about micro clicked with me until I started doing it, and doublechecking the math etc myself. It all became very simple then
     
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  5. ixlramp

    ixlramp SS.org Regular

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    LOL, and i thought i had made it clear =)
    No need to understand all of the above, or understand microtonality, all that's required is restringing with the special gauges, retuning as explained, and then play the patterns shown.
    Any specific questions?

    This method is similar to that of http://sevenstring.org/threads/retune-to-play-quartertone-scales-microtonal-beginners-guide.161530/
    The same set of gauges and same basic thirds tuning is used for both, so once set up you can play quartertones too.
    For further explanation of Just Intonation (JI), and what 'harmony' actually is, see my posts from http://sevenstring.org/threads/school-me-on-tuning-temperaments.331425/page-2#post-4920114
    onwards.

    Most microtonal metal music uses EDOs (equal divisions of the octave): The octave is split into an equal number of pitches, like 24EDO (quartertones), 22EDO, 19EDO or 17EDO. However, just like the standard 12 Tone Equal Temperament (12TET), these EDOs are only approximations of perfectly tuned intervals.
    JI metal is rare though, but the perfectly tuned intervals are ideal for distortion, as distortion really highlights the imperfect tuning of the standard 12TET.

    I have got the impression that this is part of the reason behind the popularity of the 'power chord': It starts as the major triad but distortion highlights the out of tune major third so it is dropped from the chord. In JI the major triad is more usable with distortion.

    I wonder how many know what the major triad actually is, why it is that way?
    It started as a JI chord where the 3 notes have frequencies in the ratio 4:5:6, such as 400, 500, 600 Hz. This triple-simple-ratio is the reason for it's special harmony.
    If you place the JI major triad on the tonic, the fourth and the fifth, the resulting collection of pitches are the JI major scale, this is where the major scale originates.
    The minor scale originates in a similar way.
    JI explains what harmony is too. Yet all this is not taught in music education, 12TET is the unexplained starting point for everything, despite being full of the traces of JI.
     
  6. ixlramp

    ixlramp SS.org Regular

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    Here are the complete 7-string diagrams showing all notes within one octave of frets.

    JI major scale
    ------------------

    JImaj_7str_full.png

    JI minor scale
    ------------------

    JImin_7str_full.png

    • Horizontal lines are strings, vertical lines are frets.
    • Small black squares below the grid represent standard fretboard / side dots.
    • Red squares are the tonic notes, black circles are the other scale degrees.
    • Scale degrees that are open notes are shown on the left just outside the grid.
    • The numbers on the left are the intervals between the open strings in semitones.
    3.86 + 3.14 = 7.00 semitones, the 12TET fifth.
    3.86 semitones is the JI major third.
    3.14 semitones is very close to the JI minor third (3.16 semitones).

    The pattern of scale degrees is identical in both diagrams, and plays either JI major or JI minor scales depending on the tuning of the open notes.
    Later i will show a more unusual JI scale that again has the same pattern of scale degrees, and again is acheived by just altering the open tuning.

    ///////

    I like that pattern, it's actually repetitions of a vertical line of 7 scale degrees.

    If anyone wants to understand and/or try this and is having problems, let me know and i will try to explain and help.
     
  7. ElRay

    ElRay Mostly Harmless

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    Man, just when I thought I decided what I was going to do with my 92528. I was definitely going to go all M3, and potentially keep the open string octaves (3,6,9) and 3rds (2,5,8) JI, and then tune the 5ths on 1,4,7 to by Just to the other strings. The idea was to get some key color back because a chord would be slightly different if the root was on octave, 3rd it 5th string.

    Now I’m tempted to get away from the regular tuning. The big question is how well this plays with other people in vanilla 12 EDO?
     
  8. ixlramp

    ixlramp SS.org Regular

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    My system, and your idea, can easily be retuned to 12TET if you ever need to do that.

    Mixing JI intervals with the 12TET versions of those intervals might sound ok if you are playing chords which have their root notes being 12TET pitches. The chord will be 'rooted' in 12TET and have a very harmonius sound, which may be ok even if another musician is playing the 12TET version of a JI interval you are playing.
     
  9. ixlramp

    ixlramp SS.org Regular

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    Full neck diagram for JI major and JI minor scales
    ----------------------------------------------------------------

    JImajmin_7str_full_2oct.png

    Above is the full neck diagram for playing JI major and JI minor scales. Which scale results depends on how the open strings are tuned, as explained in previous posts and diagrams.

    Playing up through the scale degrees
    ------------------------------------------------

    JImajmin_7str_full_arrows.png

    The diagram above shows how to play up through the scale degrees, playing across the fretboard.

    JImajmin_7str_full_2oct_arrows.png

    The diagram above shows how to play up through the scale degrees, playing up the fretboard.
    Because the degrees of the scale are split across pairs of strings, playing up the fretboard requires playing on 2 strings.

    These diagrams will also apply to the 'JI septimal minor scale', which is a 3rd scale (a significantly microtonal one) that can be played by just retuning the open strings. Details later.
     
  10. ixlramp

    ixlramp SS.org Regular

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    ElRay,
    My system can work for an (approximately) all major 3rds tuning. The open string intervals would be tuned (low to high):
    JI major: 3.86 4.14 3.86 4.14 ... JI minor: 4.14 3.86 4.14 3.86 ...
    Then just shift parts of the patterns accordingly.

    My system works with various open intervals. Strings just need to alternate between '12TET' and '12TET plus a microtonal offset'. Smaller intervals result in better playability though.
     
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  11. ixlramp

    ixlramp SS.org Regular

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    Seeing as at least 2 people like to use all major 3rds tuning i'll cook up some diagrams for that.
     
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  12. ixlramp

    ixlramp SS.org Regular

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    JI major scale for major thirds tuning
    ----------------------------------------------

    JImaj_8st_M3tun.png

    • One octave of frets is shown so the pattern can be repeated for a full fretboard diagram.
    • Numbers on the left are open string intervals in semitones.
    • Tune the 3.86 semitone interval (the JI major third) using the harmonics method explained earlier.
    • 3.86 + 4.14 = 8.00, the 12TET minor sixth.
    • 8.00 * 3 = 24.00 = 2 octaves, the pattern repeats every 6 strings.
    • Scale degree symbols are shown for one octave.
    • Unlike the 3.86 / 3.14 system, this pattern cannot be used for JI minor.
    When playing up through the scale degrees, i suggest going from the 7th to the tonic that is 5 frets up and 1 string down. This results in playing diagonally across the fretboard.

    This major thirds version of my system requires a little more shifting, chords do not line up across 1 fret as much, but it helps by not reducing the range of the guitar so much.
    It is possible to retune to/from a 12TET major thirds tuning with no noticeable change in string tension.
     
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  13. ixlramp

    ixlramp SS.org Regular

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    Now, to be clear, we are returning to the initial system described. That is, 2 strings across forming an interval of 7 semitones, and string gauges as recommended.

    JI septimal subminor scale
    ----------------------------------

    JIsepsubmin_7st.png

    This is the first significantly 'microtonal' scale i am presenting.
    This is essentially a minor scale with the minor intervals, the 3rd, 6th and 7th, flattened further, and is therefore called 'subminor'. The scale has a darker character than a minor scale.

    'septimal' is used in the name because it is derived from harmonics up to the 7th harmonic. The 'septimal subminor 7th' is actually the 7th harmonic, at 9.69 semitones.

    The subminor intervals in this scale are often used in blues and jazz through the use of string bending. The 'septimal subminor 3rd' and septimal subminor 7th are often called 'blue notes'.

    The pattern of notes and method of playing up through the scale degrees are identical to JI major and JI minor scales, as we are again retuning the strings to alter the pitch of the 3rd, 6th and 7th scale degrees.

    The exact interval sizes for this JI scale are shown below. My system very closely approximates all these to within 0.04 semitones (4 'cents').

    Scale degree. Semitones above tonic. Name. Frequency. My system's approximation

    1 0 'tonic' 1/1 0
    2 2.04 'major 2nd' 9/8 2.00
    3 2.67 'septimal subminor 3rd' 7/6 2.67
    4 4.98 '4th' 4/3 5.00
    5 7.02 '5th' 3/2 7.00
    6 7.65 'septimal subminor 6th' 14/9 7.67
    7 9.69 'septimal subminor 7th' 7/4 9.67

    Tuning
    ---------

    As a starting point, tune the strings to a 12TET tuning of alternating minor and major 12TET 3rds, starting with a minor 3rd as the lowest interval.

    The next step is to slightly detune all the minor 3rd intervals by 33 cents, to make them JI septimal subminor 3rd intervals. Counting strings from low to high, strings 2, 4, 6 are detuned.

    Tuning a JI septimal subminor 3rd using harmonics
    -----------------------------------------------------------------

    First, practice playing open string harmonics to acheive clear harmonics with good sustain.
    The specific harmonics to practice are the 6th harmonic, located at 'fret 3.2', and the 7th harmonic, located at 'fret 2.7'.

    * Start with 2 adjacent strings tuned to a 12TET minor 3rd interval (3 semitones).
    * Play the 7th harmonic of the lower string and the 6th harmonic of the higher string, such that the 2 harmonics sustain together.
    * They will be similar pitches but out of tune.
    * Repeatedly play the harmonics, very slowly and carefully detune the higher string until the 2 harmonics are identical in pitch. The higher string needs detuning by 33 cents to do this.
    * The 2 open strings will then be tuned to a JI septimal subminor 3rd.

    Other JI intervals playable with this open tuning
    ------------------------------------------------------------

    Semitones above tonic. Name. Frequency. My system's approximation.
    Notes.

    4.71 'septimal 4th' 21/16 4.67
    1 string up and 2 frets up, from tonic
     
  14. ixlramp

    ixlramp SS.org Regular

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    The core idea
    ------------------

    On a normally tuned guitar, all the scale degrees are played on every string.

    My system has 2 types of string, which alternate with each other across the fretboard:

    * One type of string is called a 'tonic string', the tonic of the scale is played on these strings, plus some of the other scale degrees.
    All the tonic strings are tuned to 12TET.

    * The other type of string is called an 'offset string'. They are not tuned to 12TET but are offset from it by a non-12TET interval, for example by 0.86, 0.14 or 0.67 semitones.
    The remaining scale degrees are played on these strings.
    The pitches of these strings can be finely adjusted to allow various JI scales to be played.

    This can be thought of as string pairs , each pair containing a tonic string and an offset string.
    It is as if each string of a normal guitar splits into a pair, with the tonic and some scale degrees played on one string, and the other scale degrees played on the other.
    (However, an instrument with an even number of strings is not required or even preferable.)

    To play non-12TET scales, instead of moving frets, my system leaves the frets unchanged and makes use of retuning strings.
    Movable frets are difficult to engineer and are time-consuming to move and tune, however, retunable strings are already simply engineered, and retuning is quick and easy.

    Particular intervals between the open strings are not actually a part of the core idea, they can vary.
    So, having 12TET 5th intervals between the tonic strings is not essential.
    For JI major, the interval between a tonic string and the offset string above it does not have to be 3.86, it could be 0.86, 1.86, 2.86, 4.86, 5.86, 6.86 etc.

    I have found that the tuning being alternating major and minor 3rds works quite well, but according to your need, tunings that are all-2nds, all-major 3rds, all-4ths, or smaller intervals alternating with larger ones, could be used.
    These have consequences for instrument range and the patterns for chords and melodic lines.

    Modulation to other scales
    ----------------------------------

    A pure JI instrument, for example a guitar with frets placed in the positions that correspond to JI, cannot freely modulate to other scales. For example from JI C major to JI E major. This is one of the primary reasons that 12TET has been widely adopted.

    However, since my system uses a 12TET fretting, it is actually possible to modulate.
    For example, if the system is set up for JI major:
    JI major scales can be played with the tonic note on any of the 12TET pitches on the tonic string.
    But also, since moving the tonic note up or down by one string results in JI minor being playable, it is possible to modulate to 12 JI minor scales too.

    Playing with 12TET instruments
    ----------------------------------------

    If you do not play the offset strings, the tuning is a normal 12TET 5ths tuning.

    The offset strings can, of course, be retuned to 12TET pitches. Such a tuning could be alternating 12TET major and minor 3rds or alternating 12TET 4ths and major 2nds.

    Now for example, if the system is set up for JI major:
    The system is 12TET plus 12 extra pitches, each being a JI major 3rd above a 12TET pitch.
    When playing together with 12TET instruments and playing a chord, it is played with the root note of the chord being a 12TET pitch, this keeps the chord sounding reasonably in tune with the other instruments, but you are able to use the JI 3rd, 6th or 7th in that chord to make it more harmonious.

    Stealth
    ---------

    Because the guitar has normal frets, it will not be visually obvious that you are playing anything other than a normal guitar. This could be helpful to focus attention on the music instead of the distractions of 'those weird frets'.
     
  15. ixlramp

    ixlramp SS.org Regular

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    Why it works
    ----------------

    Many JI scales have some scale degrees that are close to 12TET, plus some other scale degrees that are all offset from 12TET by roughly equal amounts. For example:

    JI major scale
    Degree. Exact semitones. Name. Frequency

    1 0.00 'tonic' 1/1
    2 2.04 'major 2nd' 9/8
    3 3.86 'major 3rd' 5/4
    4 4.98 '4th' 4/3
    5 7.02 '5th' 3/2
    6 8.84 'major 6th' 5/3
    7 10.88 'major 7th' 15/8

    The scale degrees below are within 0.04 semitones of 12TET, so are closely approximated by the 'tonic strings' of my system.

    Tonic strings.
    Degree. Exact semitones. Name. Frequency. Approximation semitones

    1 0.00 'tonic' 1/1 0
    2 2.04 'major 2nd' 9/8 2.00
    4 4.98 '4th' 4/3 5.00
    5 7.02 '5th' 3/2 7.00

    The scale degrees below are all offset from 12TET by roughly +0.86 semitones.
    With the 'offset strings' of my system set to 12TET + 0.86 semitones, these are approximated to within 0.02 semitones.

    Offset strings (+0.86 semitones).
    Degree. Exact semitones. Name. Frequency. Approximation semitones

    3 3.86 'major 3rd' 5/4 3.86
    6 8.84 'major 6th' 5/3 8.86
    7 10.88 'major 7th' 15/8 10.86
     
  16. ixlramp

    ixlramp SS.org Regular

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    Tuning a JI major third using ear and tuner
    ------------------------------------------------------

    The first post in this thread explains a way to tune a JI major third by playing 2 harmonics simultaneously.
    It is perhaps a better idea (and easier) to tune the interval using a combination of tuner and ear.
    It helps to use a tuner that has some kind of rough indication of how many cents sharp or flat a pitch is.

    * Start wth the interval tuned to a 12TET major third, 4.00 semitones.
    * Play the higher note. Watching the tuner, detune this note slightly so that it appears to be roughly 14 cents flat, this will get you close.
    * Play both notes. Using your ears, adjust the higher pitch precisely such that the interval sounds as 'smooth' and harmonious as possible. You may need to use distortion to make the amount of harmony more obvious.
    * After some practice, you may be able to tune this interval by ear alone.

    Advantages of the major thirds form
    ---------------------------------------------

    Above in post 12 i detailed the 'major thirds' form of this system.
    I'm beginning to think this may be a superior system to the 'alternating major/minor thirds' one:

    * Instrument range is not decreased as much.
    * To switch between major and minor tonalities, the 'offset' strings (relative to the tonic strings) are retuned between 3.86 and 4.16 semitones, a smaller change than between 3.16 and 3.86 semitones, so the string tension change (and the effect on setup) is smaller.
    * The original system results in patterns that tend to line up across frets. However, barre-chording 4 strings is hard work for 1 finger. Splitting a chord across multiple frets uses more fingers. Playing arpeggios through such a chord is easier.
     
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  17. ixlramp

    ixlramp SS.org Regular

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    I just restrung my 35" scale 6 string bass into all-major-thirds, to experiment with this form of my system.
    I calculated a set of gauges which i share below.

    For the first time i am experimenting with using roundwound guitar strings (D'Addario Prosteels) on a bass due to their lower cost, greater flexibility and larger choice of more finely stepped gauges.
    I use low tension strings and don't need a big stiff core capable of withstanding 60 pounds of tension, i never use more than half of that.
    These guitar strings are long enough to cross the nut. To reach the more distant tuner posts i am joining them to sections of old strings using the 'figure 8' fencing knot.
    To anchor in a bass bridge i thread an old bass ball-end onto the string.

    My bass is set up primarily for 2 hand tapping so i need flexibility. I have been thinking about how most tapping instrument companies provide custom optimised strings, which are often taperwound / exposed-core and i suspect are made with thinner cores for more flexibility (not needing high strength).

    For the plain steel strings i am using D'Addario loop-end plain steel strings, which are extremely long, and adding improvised ball-ends (M6 x 6mm 'set screws' which screw into the loop).

    //////////////////////////////////

    I use steel roundwounds because nickel allergy can be caused through exposure, and i suspect i am someone more prone to allergies.
    So the restriction on my choice of strings is using D'Addario guitar ProSteels.
    These come in a more limited range of gauges, the 2 largest gauges are .070 and .056, which nicely have a similar tension when tuned 4 semitones apart, so the suggested gauges are based on those 2 gauges as the lowest.

    The 'tension profile' is optimised for tapping, so is rather heavy-bottom-light-top:
    Tension falls steadily across the roundwounds from low to high.
    Tension drops significantly from roundwounds to plain steels.
    All plain steels are equal tension, mostly to preserve the volume of the highest plain steels.

    This all worked out well, resulting in the critical wound -> plain transition being .024w .016p like a common guitar string set, avoiding a thick and stiff plain steel but also avoiding a thin roundwound with a thin and fragile core.

    //////////////////////////////////

    Tensions from D'Addario stringtensionpro.com
    Guitar XL ProSteels and plain steel strings
    25.5" scale
    Tensions stated in 'pound-force'.

    F#4 PL008 13.1 standard E +2
    D4 PL010 12.9 standard E -2
    A#3 PL013 13.7 standard B -1
    F#3 PL016 13.1 standard G -1
    D3 PSG024 14.9 standard D
    A#2 PSG032 16.4 standard A +1
    F#2 PSG042 17.0 standard E +2
    D2 PSG056 17.4 standard E -2
    A#1 PSG070 18.1 standard B -1

    Tensions for D'Addario XL Nickel Wound strings are similar.

    /////////////////////////////////

    To be most relevant to forum members i present these with tensions for a 25.5" guitar in what seems a reasonable tuning based around an unchanged standard D. But of course you should tune all the strings up or down to your preferred tension.

    Many of these gauges turn out to be similar to the common 9-42 standard tuning set, which makes sense as it is an all-major-thirds tuning with heavy-bottom-light-top tension.
     
    Last edited: Aug 18, 2019
    Winspear likes this.
  18. ixlramp

    ixlramp SS.org Regular

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    UK
    To be clear, this post is about the 'all major thirds' version of my system.

    So far in this thread i have simplified things by only visualising the tones of particular scales for each open string tuning. Actually, each open string tuning makes more than 7 Just Intonation (JI) intervals playable, so more scales are possible.

    The number of playable JI intervals depends on how closely you want to approximate them. If approximating closely to within 4 cents, each open tuning makes 10 JI intervals possible. If approximating to within 6 cents, each open tuning makes 14 JI intervals possible.
    In this post i will deal with approximating to within 4 cents.

    Each image below is for a different open string tuning. Each shows a table of the playable JI intervals and an 8 string fretboard diagram showing the positions of each interval.

    The fretboard diagrams are in standard orientation: open notes on the left, lowest string at the bottom.
    The 'tonic' strings are marked with 'T', the tonic of the scale must be placed on these strings.
    The 'offset' strings are marked with 'O' and are offset from 12 Tone Equal Temperament (12TET) by the number of cents shown after the symbol, either '-14' or '+14'.
    The base 12TET open string tuning (before offsetting) is all the intervals between the open strings being 4 semitones.

    ///////

    The first image is for the open string tuning used for playing the JI Major scale.
    The additional JI intervals are:
    * An alternative, slightly flattened, major second, called the 'small major second', of size 182 cents.
    * The 'small tritone' of size 590 cents.
    * The 'pythagorean minor seventh' of size 996 cents.

    Compared to the JI Major scale:
    * Using the minor seventh instead of the major seventh results in a JI Mixolydian scale.
    * Using the tritone instead of the fourth results in a JI Lydian scale.
    * Using both the alternatives above results in a JI Lydian Dominant scale.
    Many more scales are possible, the above examples are subject to the condition of being 7 tone scales composed only of wholetone and semitone steps.

    O-14image.png

    The second image is for the open string tuning used for playing the JI Minor scale.
    The additional JI intervals are:
    * The 'minor second' of size 112 cents.
    * The 'large tritone' of size 610 cents.
    * An alternative, slightly flattened, minor seventh, calld the 'pythagorean minor seventh', of size 996 cents.

    Compared to the JI Minor scale:
    * Using the minor second instead of the major second results in a JI Phrygian scale.
    * Using the tritone instead of the fifth results in a JI Locrian #2 scale.
    * Using both the alternatives above results in a JI Locrian scale.
    Many more scales are possible, the above examples are subject to the condition of being 7 tone scales composed only of wholetone and semitone steps.

    O+14image.png
     
  19. ElRay

    ElRay Mostly Harmless

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    Thank you! Thank you! Thank you!

    I need to get that nut cut and the 92528 set-up.
     
    Winspear likes this.

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