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Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

  • 1.  Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 12-06-2018 11:29
    Edited by Roshan Kakiya 10 days ago
      |   view attached

    Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma


    Roshan Kakiya's Well Temperament (created by correcting the Pythagorean comma):

    http://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=db956071-2c21-439d-8ae5-be761bfbf01e&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf



    Just Intonation Dilemma (partially solved by correcting the Syntonic comma):

    http://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=3ce711a8-1939-4ab8-9f88-c85435a71885&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf


    Steps Required to Completely Solve the Just Intonation Dilemma

    1. Correct the Syntonic comma:

    The Syntonic comma can be corrected by keeping C-G (Fifth) pure, narrowing C-D (Major Second) by 1/3 Syntonic comma, narrowing F-G (Major Second) by 1/3 Syntonic comma and widening D-F (Minor Third) by 2/3 Syntonic comma.


    2. Establish the Fifth-Fourth relationship and the Fourth-Fifth relationship:

    Fifth-Fourth relationship = Pure Fifth-Tempered Fourth = C-G-D-A-E-B-F#

    Pure Fifth = 701.96 cents
    Tempered Fourth = 505.21 cents

    Fourth-Fifth relationship = Pure Fourth-Tempered Fifth = F#-C#-G#-D#-A#-F-C

    Pure Fourth = 498.04 cents
    Tempered Fifth = 694.79 cents


    3. Use the relationships established in Step 2 to create a temperament:

    C-G (Pure Fifth)
    G-D (Tempered Fourth)
    D-A (Pure Fifth)
    A-E (Tempered Fourth)
    E-B (Pure Fifth)
    B-F# (Tempered Fourth)
    F#-C# (Pure Fourth)
    C#-G# (Tempered Fifth)
    G#-D# (Pure Fourth)
    D#-A# (Tempered Fifth)
    A#-F (Pure Fourth)
    F-C (Tempered Fifth)

    C 0.00
    C# 92.18
    D 196.74
    D# 288.92
    E 393.48
    F 485.66
    F# 590.22
    G 701.96
    G# 786.96
    A 898.70
    A# 983.71
    B 1095.44
    C 1180.45

    The Octave should have a value of 1200 cents. However, the correction of the Syntonic comma has led to the creation of another comma called the Diaschisma which has a value of 19.55 cents (1200 cents - 1180.45 cents).


    4. Correcting the Diaschisma:

    Pure Fifth = 701.96 cents
    Tempered Fifth = 694.79 cents

    There are 6 Pure Fifths and 6 Tempered Fifths. The Circle of Fifths is not complete because of the Diaschisma. The 6 Pure Fifths can remain intact. Each of the 6 Tempered Fifths must be widened in order to correct the Diaschisma. The Diaschisma can be corrected by widening each Tempered Fifth by 1/6 Diaschisma.

    Revised Tempered Fifth = 698.04 cents


    5. Producing the Final Temperament:

    Here is a list of all the intervals that are needed to produce the Final Temperament:

    Pure Fifth = 701.96 cents
    Revised Tempered Fifth = 698.04 cents
    Pure Fourth = 498.04 cents
    Revised Tempered Fourth = 501.96 cents

    C-G (Pure Fifth)
    G-D (Revised Tempered Fourth)
    D-A (Pure Fifth)
    A-E (Revised Tempered Fourth)
    E-B (Pure Fifth)
    B-F# (Revised Tempered Fourth)
    F#-C# (Pure Fourth)
    C#-G# (Revised Tempered Fifth)
    G#-D# (Pure Fourth)
    D#-A# (Revised Tempered Fifth)
    A#-F (Pure Fourth)
    F-C (Revised Tempered Fifth)

    C 0.00
    C# 101.96
    D 200.00
    D# 301.96
    E 400.00
    F 501.96
    F# 600.00
    G 701.96
    G# 800.00
    A 901.96
    A# 1000.00
    B 1101.96
    C 1200.00


    The Final Temperament is Roshan Kakiya's Well Temperament!

    Conclusion: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma.

    ------------------------------
    Roshan Kakiya
    ------------------------------


  • 2.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 29 days ago
    Edited by Roshan Kakiya 29 days ago
      |   view attached
    I have included all the files that are related to my well temperament in a ZIP file.

    The ZIP file contains:

    • 2 Excel files.
    • 2 Images.
    • 1 Scala file.


    2 Excel files:

    1. First method: Correction of the Pythagorean comma (23.46 cents).
    2. Second method: Correction of the Syntonic comma (21.51 cents) and then the Diaschisma (19.55 cents).


    2 Images:

    1. A diagram of the Circle of Fifths.
    2. A table and a graph of the theoretical beat rates (available at http://rollingball.com/TemperamentsFrames.htm under the Modern Well section).


    1 Scala file:

    • For retuning MIDI files with the use of Scala.

    ------------------------------
    Roshan Kakiya
    ------------------------------

    Attachment(s)



  • 3.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 26 days ago
      |   view attached
    Whilst plugging in the figures for this sort of temperament into the spreadsheet that I did recently looking at shimmer of piano sound I hadn't wanted to comment until trying it out on something and recently had the opportunity to synthesise it with Pianoteq.

    With Pianoteq I can't hear a difference. But it's not quite the same as tuning a real instrument and to be honest one's not going to be able to take a view without tuning it on a very standard piano such as a Steinway B or C.

    However, one should take into account that 2 cents is the very minimum that it's said that we can distinguish in pitch - so this scale makes the very barest difference - and taking the difference between 440 and 415, around 440 there's around 4 cents to each beat - so in making a 2 cent change we're looking at only a 1 beat in 2 seconds difference.

    In practice, having experienced trying a Bill Bremmer equal beating style of tuning where actually I found the beats distracting a temperament as far away from equal as Vallotti I don't really detect enough key colour to be able to identify it as anything other than equal. In my opinion one only starts to depart from the equal temperament experience when notes depart at least 4 to 6 cents away from the equal.

    I'm not at all pouring cold water on your work other than to take issue with you calling 1.96 cents not 2 cents - as a matter of false accuracy - but simply saying that to make a real judgment upon it one's got to tune a real instrument to it.

    Were some notes to be +2 away from ET and others -2 then I think that perhaps you might start to hear it.

    On a real instrument the spreadsheet analysis attached suggests that it may well remove quite a lot of close beating harmonics from the resonances of the instrument and be as effective as Kellner. In practice in tuning it's then necessary to decide on the stretching curve to be used, as to whether to stretch normally, or to tune the central three octaves straight without stretch.

    Proportion of same frequencies
    Equal
    38%
    Kellner
    43%
    Roshan
    42%

    Around 1 beat
    Equal
    15%
    Kellner
    9%
    Roshan
    8%

    1 to 5 beats
    Equal
    33%
    Kellner
    28%
    Roshan
    28%

    2 to 5 beats
    Equal
    16%
    Kellner
    17%
    Roshan
    18%

    Best wishes

    David P

    ------------------------------
    David Pinnegar BSc ARCS
    Curator and House Tuner - Hammerwood Park, East Grinstead, Sussex UK
    antespam@gmail.com

    Seminar 6th May 2019 - http://hammerwood.mistral.co.uk/tuning-seminar.pdf "The Importance of Tuning for Better Performance"
    ------------------------------



  • 4.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 25 days ago
    Edited by Roshan Kakiya 25 days ago
    "On a real instrument the spreadsheet analysis attached suggests that it may well remove quite a lot of close beating harmonics from the resonances of the instrument and be as effective as Kellner."


    This seems to be the most intriguing aspect of my well temperament because it can be as effective as Kellner whilst deviating only slightly from equal temperament.

    I seem to have "killed two birds with one stone", as the saying goes.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 5.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 11 days ago
    Edited by Roshan Kakiya 10 days ago
    Roshan Kakiya's Well Temperament is the best temperament that I have created. I recommend using this temperament rather than any of the other temperaments that I have created.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 6.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Can you publish instructions to tune this aurally?

    Pwg

    ------------------------------
    Peter Grey
    Stratham NH
    603-686-2395
    pianodoctor57@gmail.com
    ------------------------------



  • 7.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Edited by Roshan Kakiya 10 days ago
    6 Pure Fifths and Fourths (Pure Fifth = 701.96 cents and Pure Fourth = 498.04 cents):

    C-G, D-A, E-B, Gb-Db, Ab-Eb and Bb-F.


    6 Tempered Fifths and Fourths (Tempered Fifth = 698.04 cents and Tempered Fourth = 501.96 cents):

    G-D, A-E, B-Gb, Db-Ab, Eb-Bb and F-C.


    All Octaves should be pure (Pure Octave = 1200 cents).


    Young II and Vallotti both contain 6 Pure Fifths and 6 Tempered Fifths (reduced by 1/6 of the Pythagorean comma). Therefore, use their instructions to aurally tune the Pure Fifths, Pure Fourths, Tempered Fifths, Tempered Fourths and Octaves outlined above for my well temperament.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 8.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Roshan said: All Octaves should be pure (Pure Octave = 1200 cents).

    Richard replies: I've not been following your posts recently.  But, this statement flies in the face of any piano tuning I know. In the real world of pianos, there are several octave widths possible, and 1200 cents is the narrowest and least used by most piano tuners. And that fact makes me wonder about your solution. The natural world does not allow for totally just intonation in pianos. Only selected intervals can be brought close to just in order to make selected intervals more "harmonious" and "calm." And each piano will have a different harmonic structure making universal solutions to the just intonation dilemma impossible. That's not necessarily a bad thing. Most piano tuners choose ET insofar as it is achievable in practical terms. But your suggestion is possible, just as any option other than ET is possible. 

    How do your tempered fifths sound? How many beats per second? How do the major thirds sound? How many beats per second? If you tune the 2/1 octave pure, the 4/2 and 6/3 octaves will have annoying beats. That will be exaggerated as the tuning expands outside the middle of the piano.

    As I see it, your solution applies to early keyboards. The high tensions in pianos thwart simple mathematical descriptions. 

    Richard West








  • 9.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    The mathematics that I have shown is based on the assumption that there is zero inharmonicity.

    If inharmonicity is present, the octaves should be stretched as much as is needed to make them as beatless as possible.

    The fifths and the fourths can be tuned in the middle octaves of the piano. Unisons can be tuned thereafter to completely tune the piano.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 10.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Rohan said:  If inharmonicity is present, the octaves should be stretched as much as is needed to make them as beatless as possible. The fifths and the fourths can be tuned in the middle octaves of the piano.

    Richard replies: Okay, then. It seems like we're back to square one. Inharmonicity is present and to differing degrees from piano to piano and actually from note to note within a particular instrument.  How do we define having an octave "as beatless as possible?" Totally beatless at every coincidental partial of an octave is impossible. So how do I decide how wide to make my octaves, and what are the consequences in the tuning of the intervals of the middle of the piano? How do those initial decisions get expanded to the top and bottom notes?

    Richard





  • 11.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Edited by Roshan Kakiya 10 days ago
    The same instructions for tuning Young II and Vallotti can be applied to my well temperament. All of these temperaments contain 6 pure fifths and 6 fifths reduced by 1/6 of the Pythagorean comma.

    The only difference between these three temperaments is the placement of the 6 pure fifths and the 6 fifths reduced by 1/6 of the Pythagorean comma.

    There are practical methods for tuning Young II and Vallotti. The same practical methods can be used for my well temperament.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 12.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Let's say I need 1205 cents in my 2/1 octave. What numbers would I enter into my ETD to adjust for the stretch, i.e., how do the numbers work out for the rest of the notes in the octave? If I wanted to aurally verify the results, what would some of the intervals that have beats sound like, especially major thirds?

    Richard








  • 13.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Edited by Roshan Kakiya 10 days ago
    The stretch can be accounted for by using a stretch ratio.

    Stretch ratio = 1205 / 1200.

    Multiply all 12 intervals by this stretch ratio.


    Approximate examples:

    C-C# = Stretched Minor Second = 101.96 cents × (1205 / 1200) = 102.38 cents.

    C-E = Stretched Major Third = 400 cents × (1205 / 1200) = 401.67 cents.

    C-G = Stretched Fifth = 701.96 cents × (1205 / 1200) = 704.88 cents.


    The spreadsheet that I have attached to my original post can be used to calculate the exact figures.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 14.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    So, from what I gather here it is basically:

    Down a pure 4th
    Up a tempered 5th
    Down a pure 4th
    Up a tempered 5th
    Etc.

    Now, can any of the math whizz's among us tell me what the approx beat speed of (at the very least) that first G-D tempered 5th would be?  I can doubtlessly figure out the rest from there.

    1 bps? 1.25 bps? .95 bps?...  what would it be from the data presented?

    On the octaves, my initial suspicion is that I would simply listen for a "sweet spot" in the midrange and then expand as needed/wanted from there. Theoretically, after establishing the temperament region, try my best to replicate it up and down with an adjustment for inharmonicity.

    It's worth a try. I hate to praise or condemn until I have at least given it a fair trial.  (But I do need an initial bps target).

    Pwg

    ------------------------------
    Peter Grey
    Stratham NH
    603-686-2395
    pianodoctor57@gmail.com
    ------------------------------



  • 15.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Edited by Roshan Kakiya 10 days ago
    Jason Kanter has produced the following charts and table for the beat rates:

    http://rollingball.com/images/KakiyaWell.gif

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 16.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    I have difficulty figuring out the table but my guess is the the tempered 5ths are going to beat at roughly 2 bps...maybe slightly less. (At the 3:2 partial level...twice that at the 6:4).

    Is this a reasonable estimate?

    Pwg

    ------------------------------
    Peter Grey
    Stratham NH
    603-686-2395
    pianodoctor57@gmail.com
    ------------------------------



  • 17.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    "These are the same sixth-comma fifths as Vallotti, and beating twice as fast as the same interval in Equal Temperament".

    Source: https://www.hpschd.nu/index.html?nav/nav-4.html&t/welcome.html&https://www.hpschd.nu/tech/tmp/young.html


    My well temperament's tempered fifths are also sixth-comma fifths so they are the same as the tempered fifths of Young II and Vallotti.

    Therefore, my well temperament's tempered fifths will beat twice as fast as equal temperament's tempered fifths.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 18.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Here's a bearing plan for Roshan's well temperament.
    From A4 tune A3.
    Starting with A3 we will tune the temperament from C3 to C4.
    From A3 tune D3 a pure beatless fifth.
    From D3 tune G3 a wide fourth beating 1.3 bps. (No one can distinguish beats to the tenth of a second, but we can try.)
    From G3 tune C3 a pure beatless fifth.
    From C3 tune C4.
    ...Test G3-C4 a pure beatless fourth.
    From C4 tune F3 a narrow fifth beating 1.2 bps.
    ...Test C3-F3 a wide fourth also beating 1.2 bps.
    Now we have FCGDA and can test the resulting M3.
    ...Test M3 F3-A3 to beat 6.9 bps.
    From A3 tune E3 a wide fourth, to beat 1.5 bps.
    ...Test M3 C3-E3 to beat 5.2 bps.
    From E3 tune B3 a pure beatless fifth.
    ...Test M3 G3-B3 to beat 7.8 bps.
    From B3 tune F#3 a wide fourth, to beat 1.7 bps.
    ...Test M3 D3-F#3 to beat 5.8 bps.
    From F#3 tune C#3 a pure beatless fourth.
    ...Test M3 C#3-F3 to beat 5.5 bps. Compare to D3-F#3 at 5.8 bps.
    From C#3 tune G#3 a narrow fifth, beating 0.9 bps.
    ...Test M3 E3-G#3 to beat 6.5 bps.
    From G#3 tune D#3 a pure beatless fourth.
    ...Test M3 D#3-G3 to beat 6.2 bps.
    From D#3 tune A#3 a narrow fifth beating 1.1 bps.
    ...Test M3 F#3-G#3 to beat 7.3 bps.

    ------------------------------
    Jason Kanter
    Lynnwood WA
    425-830-1561
    ------------------------------



  • 19.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 10 days ago
    Edited by Scott Kerns 9 days ago
    For those of us who use an ETA, what would the offsets from equal temperament be to input into our devices?

    ------------------------------
    "That Tuning Guy"
    Scott Kerns
    www.thattuningguy.com
    Tunic OnlyPure, TuneLab & Smart Piano Tuner user
    ------------------------------



  • 20.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 9 days ago
    Edited by Roshan Kakiya 9 days ago
    Offsets in Cents from Equal Temperament for Electronic Tuning Devices

    C 0.00
    C# +1.96
    D 0.00
    D# +1.96
    E 0.00
    F +1.96
    F# 0.00
    G +1.96
    G# 0.00
    A +1.96
    A# 0.00
    B +1.96
    C 0.00

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 21.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 9 days ago
    Thanks! I'll give it a whirl! 👍

    ------------------------------
    "That Tuning Guy"
    Scott Kerns
    www.thattuningguy.com
    Tunic OnlyPure, TuneLab & Smart Piano Tuner user
    ------------------------------



  • 22.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 9 days ago
    Edited by Kent Swafford 9 days ago
    This is not well-temperament. Implicit in the concept of well-temperament, besides being usable in all keys is the concept of key coloration, of which this temperament has none, to the extent that all the M3rds are the same amount of tempering as in pure octave ET.

    This scheme divides the 12 notes into the 2 whole-tone scales, both in pure octave ET, and offsets/sharpens one of the whole-tone scales by 2 cents. This leaves the M3rds the same temper as ET, but alternates the fourths and fifths between pure and 4 cents tempered.

    So in addition to the uniformly fast-beating M3rds, the temperament produces the uneven fourths and fifths of unequal temperament but without the “benefit" of slow-beating M3rds in the close to home keys.

    It isn’t any kind of compromise, but rather, combines the worst of both of the equal and unequal worlds.


    C 0.00

    D 0.00

    E 0.00

    F# 0.00

    G# 0.00

    A# 0.00


    C# +1.96

    D# +1.96

    F +1.96

    G +1.96

    A +1.96

    B +1.96




  • 23.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 9 days ago
    Edited by Roshan Kakiya 8 days ago
    Kent,

    My temperament deviates only slightly from pure octave equal temperament and, according to David's analysis, can be as effective as Kellner.

    Are there any other solutions that can also achieve these two things?

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 24.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 8 days ago
    Well actually I have said that from analysis of the frequencies and their interaction with harmonics in the spreadsheet I've circulated in another thread it _may_ be as effective as Kellner for a specific purpose - knocking scale notes and harmonics apart so that there is less shimmer to the piano sound, so that better stillness can be achieved.

    But as Kent says, there is none of the advantage of the usual unequal temperaments in producing purer thirds which add to the sweetness of the overall sound and a sweetness of resonance. 

    This is why I have said that you've got to tune it on an acoustic piano, a good one, that a good and sensitive musician knows so that a relevant opinion can be found as to whether it improves the sound, or not. 

    There is the possibility that with such a temperament as this we lose what we admire in the sound of the conventional tuning and don't gain a lot besides. 

    This cannot be adjudicated by theory and only the experiment will confirm or otherwise the success of the intention. 

    Best wishes

    David P
    --
    - - - - - - - - - - - - - - - - - - - - - - - -
    David Pinnegar, B.Sc., A.R.C.S.
    - - - - - - - - - - - - - - - - - - - - - - - -
    +44 1342 850594





  • 25.  RE: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 8 days ago
    In your opinion, which temperament combines the best of both of the equal and unequal worlds?

    ------------------------------
    Roshan Kakiya
    ------------------------------