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Bruce Simonson
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temperaments available in myriad software?
« on: Jan 12th, 2023, 7:13am » |
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Greetings all!
I'm looking into alternative tunings and temperaments, and have found a couple of threads on this on the forum.
And the documentation on "microtonal adjustment" and "alternate tunings" is very helpful.
I'm curious, however, what, if any, work has already done on this. I'm especially interested in the Bach/Lehman temperament, defined as (cents from equal temperament):
C (+5.9), C# (+3.9), D (+2), Eb (+3.9), E (-2), F (+7.8),
F# (+2), G (+3.9), G# (+3.9), A (0), Bb (+3.9), B (0)
See this link for interesting details on this temperament:
http://www.larips.com/
For those with lots of free time, the following link provides a fine set of exercises for the reader:
http://www.instrument-tuner.com/temperaments.html
Has any of this work been done by over-achievers on the forum?
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Bruce Simonson
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Greetings,
I am working through alternate tunings, trying to set the Bach Lehmann temperament in my original post.
However, I think I am stymied at the point of defining the adjustment to the pitch for the tuning of a specified note (C, for this example).
I try to enter 5.9 cents on the micro-adjustment panel, and the decimal portion is dropped when I leave the data entry cell.
It appears that only integer cents are allowed in the "special" adjustment under the special affect.
Am I missing something? Temperaments are more finicky than to the nearest cent, they often go to thousandths, or at minimum, to hundredths of cents. At least tenths of cents from equal temperament are needed to begin to differentiate temperaments properly.
If the slider is limited to integers, I suggest that a different slider be used in a future release, as soon as practical.
I've attached a png of the work flow. As you can see, the 5.9 cents I enter in the slider is converted (truncated) to 5 when I leave the slider data entry cell. I would prefer to be able to enter +5.865 as the difference in cents from equal temperament C (all octaves), and have it stick.
Have I missed something?
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| « Last Edit: Jan 13th, 2023, 8:00am by Bruce Simonson » |
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Bruce Simonson
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For completeness, here are the actual offsets from ET that I require. I'll try to attach a copy of the draft rules file, as a zip. (This rules file is incorrect, as the offsets are truncated):
-- Bach Lehmann 1722 temperament ET offsets in cents
C 5.865
C# 3.910
D 2.000
D# 3.910
E -1.955
F 7.820
F# 2.000
G 3.910
G# 3.910
A 0.000
A# 3.910
B 0.000
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Sylvain Machefert
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Re: temperaments available in myriad software?
« Reply #3 on: Jan 13th, 2023, 9:23am » |
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Hi!
Just to say, you are right, the value is rounded or truncated to integer.
I'm very busy today to test, but you made great screen capture and you can contact info@myriad-online.com to suggest decimal cents values.
As I'm not Myriad team and dev, I can't tell you if it'll be very easy or very hard to change this data. File format should probably change, so it may wait for a "major" release (i.e. 9.7 instead of a subversion of 9.6).
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Tony Deff
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Quote:Some findings indicate that, for moderate loudness levels, humans can detect a frequency change of about 1 - 3 Hz for frequencies up to about 1000 Hz.
The plot shows the smallest frequency difference for which two tones can be discriminated for a number of reference tones.
You can see from the figure that up to about 1000 Hz, the threshold is between 1 and 3 Hz.
In fact, for frequencies between 500 - 2000 Hz, it is a constant fraction of the frequency to be discriminated, (ΔF/F)
Although this holds true for a wide range of intensities the intensity of the sound does affect the determination of the minimal discriminable change in frequency. As the intensity of the sound decreases, it is more difficult to detect it as being different from other sounds close to it in frequency. |
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https://www.open.edu/openlearn/science-maths-technology/biology/hearing/ content-section-11.2
Quote:The smallest frequency change that normal-hearing adults can detect is of the order of 0.2–0.3% for frequencies between 250–4000 Hz and increases rapidly with increasing frequency for frequencies above 4 kHz (Moore, 1973).
Frequency discrimination is related to pitch and music perception. |
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https://www.sciencedirect.com/topics/immunology-and-microbiology/frequen cy-discrimination.
Assuming our most hyper-sensitive listener can allegedly discriminate a change of 1Hz @ 1kHz, then a step of 1001/1000 is 1.7 cents.
To convert a frequency ratio (music interval) to cents, take its Natural Logarithm (Ln) and multiply by 1731.234
(A scientific calculator is available in Windows by typing calc in the search window, bottom-left.)
For an octave: 2 ln x 1731.234 = 1,199.99 (cents)
For a "Just" major third (5/4): 1.25 ln x 1731.234 = 386.3 (cents)
That is, we have become accustomed to an equal-temperament major third (400 cents) being 13.7 cents sharp.
Understandably, H.A rounds this to -14c in the "Barbershop" tuning.
Et quelle horreur, a "Just" minor seventh (7/4) = 968.8 cents (equal temperament @1000 cents is almost 1/3 semitone sharp.) 
Who is going to quibble about one-hundredth of a cent with that sort of discrepancy ?
Also to consider: accuracy in tuning a piano; long-term accuracy in holding any precise tuning; ability of other instruments to obtain such precision (e.g. a violinist); vibrato.
Quote:| The extent of vibrato for solo singers is usually less than a semitone (100 cents) either side of the note, while singers in a choir typically use narrower vibrato with an extent of less than a tenth of a semitone (10 cents) either side. Wind and bowed instruments generally use vibratos with an extent of less than half a semitone either side. |
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https://en.wikipedia.org/wiki/Vibrato
Edit by Sylvain: shortened the url that broke the width of the forum
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| « Last Edit: Jan 14th, 2023, 4:56pm by Sylvain Machefert » |
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Bruce Simonson
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Re: temperaments available in myriad software?
« Reply #5 on: Jan 14th, 2023, 4:23pm » |
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Greetings!
It's great to have folks with experience weigh in on this. Thanks!
"Precision vs Practicality" - a perfect subject line.1
First off, I find that most folks tune out, when temperaments are brought into conversation. Second, to really clear the room, talk about cents, and temperament recipes. Get to the level of precision of thousandths of a cent, and people flee, but, but ... voila ... in come the mathematicians!2
Anyway, Tony, I take your point, that resolutions to tenths of a cent in temperament recipes might be overkill, and that thousandths of a cent seems silly, for many of the reasons you identified (and more), including:
1) Most folks can't hear the difference between a cent, much less a hundredth, or a thousandth of a cent, at say, A=440 Hz.3
2) Almost no-one can tune, reliably, before, and during performance, to the level of fractional cents.4
3) Some performers try to play a trifle sharp, on purpose, in order to be heard and hear themselves, in the mayhem of ensemble playing.5
4) Some musicians, often older singers, have (shall we say) "artistically" wide vibrato, which renders temperament rather moot.6
5) ...
And yet, ...
I'd like to continue lobbying for micro-tuning in HA to at least the level of hundredths of a cent.
Here are a couple of quick observations7:
Regarding (1) above, sure, when played back to back, a 1 cent difference at A=440 is difficult to hear, but played simultaneously, a 1 cent difference in pitch (around A 440), beats about every 4 seconds.8
To add to this point, concurrent A=880 with a pitch 1 cent higher would beat every 2 seconds, and A=1760 with a pitch 1 cent higher would beat every second. And (please, if you haven't fled the room), consider two things: overtones and inharmonicity stretch.9
Regarding (2) above, a stable harpsichord10 would serve as an unassailable foundation to keep sensitive and temperamental players in tune, if playing in a temperament.
... I'll be back in a couple of days, with more thoughts on this. And perhaps a suggestion on how to (gently) accomplish 100ths of a cent in HA micro-tuning rules.
Cheers,
-Bruce
PS: I've tried to keep this conversation gentle and relaxed, as discussing temperaments (why? which?) can rise to the level of religion or politics. In the meantime, here's a friendly (perhaps too long) youtube I found on temperaments, that's worth a listen (sorry about lead-in advertisements, if they happen):
https://m.youtube.com/watch?v=nK2jYk37Rlg
and, starting here, the youtube drives the point home on using a good temperament, in the wrong key (about 28'06" in):
https://m.youtube.com/watch?v=nK2jYk37Rlg&t=28m6s
Peace.
1 Reminds me of my halcyon days in college (back in the day), when the gang would stay up all hours, arguing about "Platonic ideal vs Aristotlian reality".
2 If friends have a piano tuned, they are delighted, until I ask -- "um ... equal temperament?" And they frown when I say, "well, it can make a difference ...". And then they say, "well I had it tuned by the best tuner in town" (implying, like "what do you know?"). And if I bring up inharmonicity, and stretch, they quickly change the subject, or talk to someone else.
3 I have a touching story to tell about this, but this would probably best be done over beers, and not by being "chatty" in a great software forum.
4 Again, I remind myself of my third favorite musician joke -- Q: "how long does it take to tune a hurdy-gurdy?" -- A: "No-one knows." (I can say this, because I built a hurdy gurdy once, and I also play the viola.)
5 Usually violinists.
6 I can say this from experience, as I conduct an un-auditioned community chorus and orchestra. Besides, to be fair and honest, I have a hard time keeping the barbershop seventh in tune, if I don't really focus.
7 I have to dash off soon (delivering produce to a remote Alaskan village via a friend's landing craft this weekend).
8 If I have the math right ... figuring as I type: I believe it's about 8 cents between A=440 and A'=442, which would beat at two Hz. So one cent would be something like 440 vs 440.250, which beats at 0.25 Hz, or every four seconds ...?
9 Inharmonicity can be both "physical" (shorter strings (higher octoves) are typically "stiffer", and have to be tuned sharp in order to sound in tune), and "pyschological" (everyone's ears are different, and the "little hairs" in the cochlea respond differently at higher pitches for everyone, but typically perfect octaves sound flat. (sorry, I don't have references at hand, it's just something I read somewhere sometime ago, and haven't forgotten)). BTW, stretch goes the other way for lower octaves, and would better be called "relax", I guess.
10 Assuming such a thing, that is, a "stable harpsichord" exists. (But (horrors?), there are electronic versions of these things ...).
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Tony Deff
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Whereas equal temperament scales use steps of 1 or 2 semitones, the "just" (true) scale has 3 different ratiometric increments,
introducing a Minor Whole-tone (minor 2nd, 10/9), which differs from the Major Whole-tone by 21.5 cents ...
1 x 9/8 x 10/9 x 16/15 x 9/8 x 9/8 x 10/9 x 16/15
= 1 ... 9/8 .... 5/4 ..... 4/3 ..... 3/2 .... 5/3 .... 15/8 .... 2
do(h), re(ray), mi(me), fa(h), so(h), la(h), ti(te), do1
In Harmony Assistant, this scale is loaded by File > New > Alternate* tunings > Barbershop ... > Create
Then (once only) save this temperament for use with any voice by Staves > Edit rules > Save > {filename} > Save
(* An alternative word for alternative  , rare in the UK but popularised in the USA and stressed differently from the verb alternate.)
Copy this scale into each relevant staff (for example, violins) Staves > Edit rules > Load > {filename} > Open.
Major Chord Frequency-Ratios (I , IV, V)
{do : mi : so} = {1 : 5/4 : 3/2 } = {4 : 5 : 6}
{fa : la : do1} = {4/3 : 5/3 : 6/3} = {4 : 5 : 6}
{so : ti : re1} = {3/2 : 15/8 : 18/8} x 8/3 = {4 : 5 : 6}
Minor Chord Frequency-Ratios (III, VI)
{mi : so : ti} = {5/4 : 3/2 : 15/8} x 8 = {10 : 12 : 15} ÷60 = {1/6 : 1/5 : 1/4 }
{la : do1 : re1} = {5/3 : 6/3 : 10/4} x 6 = {10 : 12 : 15}
Discords" (II, VII)
The chord IIm has a "wolf fifth" of 680¢ (22¢ flat, the difference between a major and a minor tone) and hence needs resolution.
The chord VIIdim has a diminished fifth of 610¢ (a remote harmonic relationship of 64:45, but does also have a pure minor 3rd.
A diminished chord in equal temperament has two irrational intervals, 4√2 & 2√2).
(300¢ & 600¢ only seem like natural integer numbers because they are based on an artificial measurement.)
The above chords, as well as melodic major and minor seconds, can be compared in the above attachment.
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| « Last Edit: Feb 12th, 2023, 6:21pm by Tony Deff » |
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PaulL
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Re: temperaments available in myriad software?
« Reply #7 on: Jan 23rd, 2023, 5:06am » |
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I used to play a Bozeman organ that was tuned to Kirnberger III (which is just like Werckmeister III, only different  ).
I could never really tell the difference between that tuning and equal temperament, but perhaps part of the reason is that the Episcopal Church hymnal mostly uses only a few specific keys that happen to work well with that temperament (possibly one of the reasons the builder chose it; I wish I'd though to ask him).
The only thing I know about all this is that if you take care of the cents, the dollars will take care of themselves.
Okay, okay, I'll just show myself out, shall I?
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| « Last Edit: Jan 23rd, 2023, 5:07am by PaulL » |
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Tony Deff
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12-Tone Equal-Temperament (12-ET or 12 TET) is workable because 12 consecutive pure fifths (3/2)12 overshoots 7 octaves by only 1.4%.
12 fifths are "tempered" to share that error equally, each being flattened by just under 2 cents.
Side-effects are that 3rds. & 6ths. are 14 – 16 cents in error and the harmonic minor seventh (7/4).
The basis of 19-Tone Equal-Temperament is that 19 minor thirds (6/5)19 undershoots 5 octaves by less than 0.2%.
Each step represents a frequency ratio of 19√2 (63.16 cents).
Implemented as 16 steps of 63 cents interspersed with 3 steps of 64 cents, that still results in 19 near-perfect minor thirds
(Hmmm, I must try using many diminished chords.)
Major 3rds. are also closer to true than in 12-ET, at the expense of degraded 2nds., 4ths. & 5ths..
The good news is that you can use your existing note-input systems, and represent the music on existing 5-line staves, provided that the notes are "spelled properly" — that is, with no assumption that sharps and flats are enharmonic.
C♯ ¦ D♭ D♯ ¦ E♭ E♯=F♭ F♯ ¦ G♭ G♯ ¦ A♭ A♯ ¦ B♭ B♯=C♭
C ¦ ¦ D ¦ ¦ E ¦ ¦ F ¦ ¦ G ¦ ¦ A ¦ ¦ B ¦ ¦ C
0 63 126 189 253 316 379 442 505 568 632 695 758 821 884 947 1011 1074 1137 1200¢ (19-ET)
0 204 386 498 702 884 1088 1200¢ (Just)
0 200 400 500 700 900 1100 1200¢ (12-ET)
In the attached example, you might not notice any difference with diatonic (7-tone) melody, but will with chromatic chords.
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| « Last Edit: Feb 12th, 2023, 5:23pm by Tony Deff » |
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ANdre_B
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19-TET
« Reply #9 on: Feb 7th, 2023, 12:08pm » |
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This 19-TET is mathematically interesting. After all, it isn't more stupid to pile up minor thirds than fifths, though it's relatively far from handling harmonics.
However, other than bowed strings and voice, the instruments available are dodecaphonic. And are compatible with the medieval octophonic. Not sure if the latters fits into 19-TET if the sevenths are rough.
Notation, which is 21-based, is compatible, but keyboards would need, for instance, a 2-split of most black keys, as displayed on the Wikipedia page.
And, with third-tones instead of half-tones, we would create a totally different - "enneadecaphonic" - music, isn't it?
I don't know recorders very well, but I know that half tones are obtained by half-closing holes. Is it then possible (easy?) to obtain third tones by third closeing the holes?
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Tony Deff
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Instruments for 19-TET
« Reply #10 on: Feb 7th, 2023, 5:33pm » |
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Quote:| ... keyboards would need, for instance, a 2-split of most black keys, as displayed on the Wikipedia page. |
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Not if a pedal could control #/b mode.
This would be possible with an electronic keyboard but much more difficult for a mechanical (conventional) piano — not to mention the difficulty of tuning it.
19-ET guitars have been modified, but as for recorders, it would be more practical to construct a new instrument.
But with more holes to be drilled, accuracy is a problem and the playing technique becomes too complex.
(I can foresee problems drilling holes for 53-TET)
Such scales are the realm of electronics + software: realistically you can only compose and play it electronically.
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Tony Deff
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19-TET chords
« Reply #11 on: Feb 7th, 2023, 5:39pm » |
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Quote:... a major chord in root position consists of a root, the tone 6 steps higher, and then the note 5 steps higher.
Thus C-E-G and D#-Gb-A# (but not D#-F#-A#) are major chords.
Similarly, minor chords are defined as a root followed by the tone 5 steps higher and then the note 6 steps up.
Thus C-Eb-G and D#-F#-A# are minor chords.
Analogously, major and minor sevenths, dominant sevenths, major and minor sixths, ninth and diminished chords can all be defined... |
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https://sethares.engr.wisc.edu/tet19/guitarchords19.html
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Tony Deff
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on Feb 7th, 2023, 12:08pm, ANdre_B wrote:| This 19-TET is mathematically interesting... |
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It becomes yet more interesting when I demonstrate (accoustically, in the attachment) that -63 = 1137 
Ce forum permet de joindre un fichier .lex (lois) ; cependant, il est beaucoup plus facile de télécharger le fichier .myr (ci-joint) et de l'utiliser pour sauver ces lois.
Although this forum permits the attachment of a .lex (=rules) file, it is much easier to download the attached .myr file and use that to save the rules.
Note the enharmonics E#= Fb and B#= Cb — you may use either form for the same pitch.
It was great "fun" (?!) trying to implement these, leading to the discovery that -63¢ = +1137¢ (Deff's Theorem)
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ANdre_B
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Re: ET-19 : Rules & Regulations
« Reply #13 on: Feb 10th, 2023, 3:03pm » |
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on Feb 10th, 2023, 1:24pm, Tony Deff wrote:
-63¢ = +1137¢ (Deff's Theorem)
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But this is obvious, my dear! The arithmetic difference of 1200 cents is an octave, which as almost meaningless.
Harmony is not linear, but helicoidal.
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André Baeck, de retour en Belgique après 12 ans passés dans le Gard.
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Tony Deff
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Re: temperaments available in myriad software?
« Reply #14 on: Feb 10th, 2023, 3:44pm » |
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Intellectuals! That word is not even in my large dictionary!
Does it mean that as the pitch goes higher and higher, the circle of fifths gets smaller and smaller?
Does harmony end (way above our capability of hearing) in the theoretical singularity?
It dismays me to realise that my music is almost meaningless. 
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| « Last Edit: Feb 12th, 2023, 5:26pm by Tony Deff » |
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Bach-Lehmann_1722__draft__offsets_truncated__.zip