Just intonation in the Renaissance
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- เผยแพร่เมื่อ 15 มิ.ย. 2024
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As a cellist, it’s such a relief to know that “True just intonation is a forbidden fruit we are only allowed to dream of”.
It depends entirely on the nature of music one plays. Melody + drone can stay pure. Medieval Pythagorean polyphony doesn't drift by commas, although its thirds are not consonances. Limited harmonic mobility with consonant 5:4 thirds is possible. It is also possible tobplay a chord progression that ascends by comma and another that descends by comma to cancel it out.
@@taylordiclemente5163 Another approach would be to reset the comma drift whenever you get a break in the polyphony.
@@Lucius_Chiaraviglio clever. That doesn't solve the problem, but it compartmentalizes and distributes it. I think the question we should be asking ourselves is, how can we write music that aligns with nature instead of forcing nature to bend to it?
As a modern just intonation composer, don't keep your hopes up
Singers: "To solve the problem, we can use VIBRATO!"
To hide the problem maybe...
@@usernameonutube 9:44 The pig has spoken.
Good point and actually I suffer when singers as well as string players have too much vibrato. That is why I often love performances in historical style. If you want to hear a singer with a very beautiful clear style I recommend Maria Cristina Kiehr, I adore hers singing.
@@mardamlop a problem hidden is a problem solved!
I use a bit of vibrato to reach the microtones.
"True just intonation is a forbidden fruit we are only allowed to dream of."
*silent shot of blue plastic pig*
"So this was our show about just intonation..."
Just propaganda to keep people from the truth.
This isn't true. It works for purely modal music.
Bruh you can just fix the comas instead of avoiding it. Ur just being lazy after all lul
I do strongly not believe that.
@@branandubh It does not work for purely modal music, either. Even in a piece in which no accidentals are used at all, you quickly run into problems. Take the following simple sequence of triads: First you go from a C major triad (C-E-G) to a G major triad (G-B-D), while keeping the G at the same pitch, i.e. 3/2 the pitch of C. Then to a D minor triad (D-F-A), keeping the D at the same pitch, i.e. 3/2 the pitch of G, so 9/4 the pitch of our initial C, or 9/8 if we lower it by an octave. The F, a pure minor third from D, will be 6/5 the pitch of D, i.e. 27/20 the pitch of our initial C. Then to an F major triad (F-A-C), keeping the F and the A at the same pitch. The C in this triad is a perfect fifth from F, i.e. 3/2 the pitch of F. As this F was 27/20 the pitch of our initial C, the C in our F major triad turns out to be 81/40 the pitch of our initial C, or 81/80 if we bring it down an octave. It's a different pitch. Medieval and Renaissance composers did not take this into account when composing, so you will find many passages where melodies start sounding strange very quickly if absolute just intonation is attempted, as is discussed in the video. It is basically a flaw in the very fabric of European music as it has developed since the middle ages: different pitches are given the same name and treated as the same, both melodically and harmonically (C in my example).
a few years ago I was involved in a CD recording, performing a capella polyphony with one singer upon a part (well, I have done many of those). I took care of the cantus firmus. My colleages experienced great difficulties singing in tune during the always changing chords and melodies around my long note values. In the end I just flexed my own long notes a bit according to their functions in the chords (thirds vs sixths vs fifths). You hardly notice this when listening to the piece, but it solved the problem satisfactorily and all sounds nicely in tune. It is sometimes a question of being practical...
Cheers, Willem
I just loved the space given to the blue pig expressing himself on all this stuff at 13:40
I just discovered your channel tonight and I absolutely love your videos and content. I'm thrilled to see discussions about historic pitch and tuning discussed and demonstrated so beautifully and clearly. I'd like to toss some ideas your way for thought. Although I am professionally an opera singer, I spent most of my life as an instrumentalist, mostly on woodwinds and piano and my main instrument was bassoon. My teacher was very adamant about teaching his students about justified vs equal temperament and had us work diligently with drone tuners as well as in small ensembles to work out our intervals (perfect 5th are two cents sharp, major 3rd are fourteen cents flat, minor 3rds fourteen cents sharp, dominant 7ths roughly twenty-five cents flat!) so I loved seeing how you described and visualized this concept. I guess we had never discussed the notion that the pitch would change while using a justified system, but the fact is that even in modern orchestras that's how instrumentalists play "in tune." It's a constant and in-the-moment adjustment of intervals; the more trained the musician and their ear, the quicker they are capable of making that adjustments. Choirs "naturally" find themselves adjusting to these perfect intervals because singers can actually FEEL those consonant vibrations in our throats (and likewise can feel the "beats" of an out of tune interval), however so many are rarely specifically aware of all of this science and theory behind it. In fact, almost every solo instrument encounters a rather shocking experience the first time having to play with a piano...all of a sudden, they are constantly out of tune! This is because we are already (subconsciously, perhaps) trained to be able to adjust to other instruments that are capable of adjusting themselves--string players do it automatically by how they tilt their finger on a string. Thus, to suddenly have to play with an instrument that cannot make those adjustments, the intervals of the instrumentalist have to change more drastically than what they may have noticed in their solo practicing. It always seemed, however, that the general consensus to how best to apply a justified system (in chamber music without piano, for example) is that everyone generally tunes to the bass; to put it in the language you have used in the video, the bass instrument plays a melodic intonation that the upper voices tune down to in just intonation. Of course, all music is not necessarily written so simply as a continuo, but it's the basic understood practice that the lower voice is what determines the tuning. Thus, in your examples of how the tuning adjusts, let's say that moving from d minor to F major, the first F is going to be higher as the minor third, but rather than keeping the F as the tonic still so high, it would drop back down to "neutral" so that the A would then drop in pitch to accommodate the major triad, and so forth. This is, of course, given that today we play on instruments that are specifically built and designed according to one pitch (ie, 440) which is mutually agreed on as "neutral." Certainly they did not have this sort of system in the Renaissance and really nothing standardized until the mid-19th Century, but it would seem to me that it can help provide a clue as to the attitude and practice of aurally adjusting in the performance moment. It really is all endlessly fascinating...and, of course, because of my having studied so much in this manner, I do apply it in my singing, so much so that sometimes pianist coaches will accuse me of singing my major thirds and leading tones too low rather than defaulting to the melodic intonation expected!
I will add one further thought which is not addressed here: the presence of natural vibrato would be what would help avoid the changing of pitch as demonstrated in your video. By allowing a voice to vibrate naturally, there would be enough interference to avoid the attempt at pure intervals to destroy the vertical stability of a piece. On the subject of vibrato I will say this: I disagree with the idea that early music and musicians, especially singers, avoided natural vibrato in singing. The simple argument is this: if you go to a doctor and he tells you, "Your sodium levels are too high, you need to reduce your salt intake," the doctor is making that statement because you have OVERUSED salt in your diet. The wording of EVERY treatise that I have seen from the 17th-18th century as regards the "good taste" usage of vibrato is similar: there is always a caution against TOO MUCH vibrato, but never, NEVER an explicit statement indicating to avoid it completely. As such, it always seemed to me that these composers and theorists who "warned against" vibrato did so ONLY because the performers in their time were prone to overusing it!! If it were not so, there would be no need to issue such a warning. The default argument employed through so many musical generations, "in good taste," is akin to how we use salt and seasoning on food: you must salt your food, but please don't over salt your food; salt/season "to taste." Of course, there is always the additional dimension of this argument which is the terminology because there was also the presence of vibrato as a musical ornament, also in singing even in the early/mid 19th Century (Norma has a famous passage where Bellini wrote "vibrato" over the soprano's line, and this sure does not suggest that she had to tighten her vocal cords all night until that moment to finally produce a free and unrestricted sound!). All that said, I do believe that the natural vibrato of voices and the imitative vibrato (as we refer to it today) on mechanical instruments (strings, woodwinds) would have all helped cover up and ameliorate the tuning issues presented even in the Renaissance and beyond.
Of interest: My very same bassoon teacher, Mr. Christopher Weait, recognized as one of the preeminent bassoon pedagogues, had been curious to understand what causes vibrato and in the 70s had himself X-rayed while playing the bassoon and a video was made which he showed to us. The results were astonishing: when he played with vibrato, his vocal cords, completely and clearly visible in the X-ray, were moving WITH the vibrato that was sounding. When he played without vibrato, the vocal cords did not pulsate and were stiff. It was definitive proof that vibrato is a natural cause of the vocal folds when allowed to be unrestricted. By deliberately stiffening them, the audible vibrato of the sound was missing.
Anyway, I absolutely love your channel and videos...sorry that I've gone on so long, but it's a subject that I find endlessly fascinating and have a bit of a passion about! Keep up the great work! I hope you'll do some videos about Gesualdo and his crazy harmonies soon! Keep them coming!
I think this post really needs a reply linking to the following new video from the series, which covers the issue of vibrato: th-cam.com/video/7viQNYJ4sac/w-d-xo.html
I just recognized Elam, he and his group came to give a masterclass at my University! I think it was 2019, he came to McGill and gave demonstrations mostly of Italian Renaissance music and ornamentation. Very beautiful and intelligent singing. At one point, they handed out parts (in the original notation) and the whole audience sang from them.
Hi! 😊
1:31 "the pitch would change along the performance" -This is only a problem when it comes to comma pump progressions, e.g. the I vi ii V progression and it's many variants all temper out 81/80, and those kinds of progressions are quite common in music. Progressions like I IV vi V on the other hand are immune to this kind of drift. Also, the "forbidden fruit" isn't Just Intonation itself, it's the idea that you can have comma pump progressions while keeping everything pure. You will have to use a temperament instead for comma pump progressions.
You gotta like a video that explains the intricacies of pure vs. tempered tuning systems using characters from "The Simpsons" AND "Futurama"!
Rick and Morty
Theoretically, you could use pitch migration over and over and have a piece that slowly shifts keys over a long period of time, even fantastically long, like a massive year long Shepherd tone
look up Benedetti's comma pump! it's exactly what you're describing
Jacob Collier has a nice demonstration of that on TH-cam.
exactly. whilst practically this would be impossinle, with computers and synths we can do this now, right?
I'd love to hear a recording of that monstrous keyboard if there is one
th-cam.com/video/bhGwjgZ8zIY/w-d-xo.html
the channel Studio31 has a similar instrument
Perfect! Thank you.
For what it's worth, one might look at modern ensembles' tuning practices. The approach that I've found most often is that the roots of the harmonic progression strive to stay at the ^0 level, while the various parts of the chord will adjust up or down a sintonic comma as needed. This does result in some *slightly* skewed melodic intervals, but not the problems of being off by multiple sintonic commas.
Additionally, though it was (somewhat rightly) dismissed in the video as having no historical evidence, it is quite prevalent in highly skilled ensembles to adjust mid-note as a chord changes to have the new chord sound more fully in tune. However, we must remember that absence of evidence is not the same as evidence of absence. Just because there is no historical evidence for it does not mean it didn't happen either.
All this to say, just intonation is a goal that even modern ensembles strive for, but it's in an adjusted form.
I know that in a vocal group I performed with, we aimed at 7-limit JI, figured out where the comma pumps would result, and had the pitch of the segments connecting said comma pumps be on a very slightly slanted slope to even them out.
For example, if a four-chord progression caused a comma pump of about -13 cents across the span of four bars, then we would aim to slide our center _up_ by 13 cents over that same period of time, so we ended up back where we started. Sometimes, if two comma pumps canceled each other out in a relatively short span of time, we ignored both and just sang a section a comma high or low.
This required a lot of practice with a synthesized guide track, although I'd imagine in the Renaissance they'd use unfretted string instruments.
@@mal2ksc Thing is, in the Renaissance they frequently used fretted or fixed temperament instruments. The pitch of keyboards (virginals, harpsichord, spinet, clavichord and pipe organ), viols, lutes and recorders can't be shifted at all, or at least not mid-playing. And all of those instruments were very commonly used back then to accompany voice. This is why the video mentions acapella music specifically. As it is quoted in the video towards the end, in acapella (or variable pitch instruments like the violin family, for that matter) music it is possible to get close with superb efforts, but a completely just intonation will be always out of reach (unless you use only fifths).
We didn't use the guide track in performance. It was just there to make it possible to learn our parts, since they included microtonal shifts that wouldn't be possible on anything with fixed pitches. It is this purpose to which I assigned the unfretted strings -- rather than the piano accompaniment you usually see with a choir nowadays.
@@ido9988 While you are correct in saying that the pitch of keyboard instruments (excluding those with enharmonic keys) was fixed, that of recorders can be shifted at will, that of viols enough, and that of lutes at least a little, in the direction of pure intervals. Anyone who has played in a good recorder or viol consort knows this and uses it.
@@therealzilch That's right. Recorder's notes can be altered via blowing, or even alternative fingering.
Videos like this inspire me with awe at the depth and complexity of something we take so much for granted as singing in tune. Fascinating.
The same way that fretless instruments like the violin can follow tempered music yet fudge certain intervals to make them more pure, singers likely also do and did in the past.
This method of approximating some framework, but fudging the details is pretty natural to do intuitively when there aren't any fixed contraints to go by. (such as frets or keys)
These videos are TREMENDOUSLY HELPFUL and so well made. Thank you
Fantastic illustration! Love the subtle humor used in teaching this complex subject. Bravo!
Thank you so much for making these videos! They answer so many questions I had!
Wonderful video as always! I'll have to share this with some of the other members of my music department!
Marvellous work!
Thank you.
A difficult topic easily explained, shown and very funnily presented.
I've just discovered this channel and I find it wonderful. Thanks a lot
I just discovered your channel. Please continue making more videos, they are wonderful!
Absolutely fabulous that you are doing these excellent videos!
That was definitely an almost celestial explanation!
Thanks!
Many years ago I asked my violin teacher, Professor Eberhard Feltz in Berlin, how the equal temperament of the piano harmonizes with the "pure"/variable strings e.g. in a piano trio. He had me sing one note and harmonized it differently on the piano, and my voice had to adjust in order to sing in consonance with the piano string, even though it's the same string, just within different chords. It was quite spooky. It seems like our hearing "automatically" intones based on chord function, and even changes how we hear one and the same piano string.
Re: pitch drift, it's all about the 2nd step of the diatonic scale. The G in the F major or D minor. The G sometimes need to be sung at the '0' level (in the C major chord), and sometimes down one comma (in the G minor chord). This sometimes means that a tied or repeated G needs to adjust in pitch to stay consonant, but good singers do this sort of thing all the time. Moving the 2nd step is a very natural compromise.
Your work is great! Excellent explanations and illustrations, I'm a big fan of your videos. Thank you so much!
This is such a great video.
Incredibly well explained and cute at the same time! Love it!
really really really thankful for these video`s, things start to lighten up a little :)
i love this channel! This was really a worthy explanation of this theme! Thank you!
Idk why but morty at 5:23 had me dying
incredible quality of contents and presentation! admire your works and love your channel.
This is fascinating. When we sing (especially as non-perfect singers), we invariably end up at a different tonic pitch at the end of an a cappella song than we began it. I wonder if this video essentially explains why, as the perceived pitch centers seem to shift and we try to continuously find those perfect harmonic centers during our melodic journey, we imperceptibly arrive some distance from our tonic home. I wonder if that might be one of the reasons instruments began to be used to accompany voices in the first place, especially during very long pieces.
Excellent as always! Very well explained. Thanks a lot for sharing your knowledge with us!
I learn so much from your videos!!
Brilliant! It has parallels with the tedious debates that photographers have over image quality. They seem to forget that perfect image quality can never be achieved but what counts is the impression that the image makes.
The slowly approaching Morty is both disturbing and hilarious
Very Good Elam. Your videos are amazing. Thanks a lot. Cheers from Brazil.
The crafty TH-cam algorithm guessed right about my possible interest in this channel. I know a lot of chords, but still, I understand not terribly much of this. But I'm delighted the channel exists, and expect to be watching as long as you or I are around.
Thank you for this.
Great Video!!! Very informative, and great editing! I loved it.
Explaining music using memes, nerdy cartoons, and emojis - total win
Very interesting and well put video. Thank you
Barbershop 💈 singing is an art form that AIMS/for tempered intonation!
Strictly in
4 part chords 😅
the melodic line is called
“LEAD” -
and the harmonies
(Tenor - the line that largely sings above the melody
Bass will lock fifths and octaves with Lead,
And treat 4th interval and V7 intervals as sacred
Baritone sings the missing note of a 4 note chord 🤷🏻♂️
Generally the quartet
will temper Third intervals slightly flat -
and fifths are to
“LOCK 🔐”
As should Octaves (also ‘lock’… have NO ‘waves’)
I learn so much on all of your videos!
Magnífico canal. ¡Muchas gracias por compartir sus conocimientos!
A lot of fresh for me ideas. Will go now and practice intonation
I enjoyed this VERY much.
I really enjoy this
please make a video that explain why palestrina did good.. I will really appreciate it.
Thanks for the video! It was really well explained, and i'm only surprised i haven't found this channel earlier! :)))
Very good video, clear and well made, congrats + thanks for these very useful explanations, first time i clearly understand some very important hints on just intonation!
This is why, for many years now, I've always been advocating for the use of a 72-TET system!! The just intervals could be approximated better, while the proper pitch migrations could now be sufficiently approximated, via all the additional discrete pitches we'd be introducing to our current 12-TET standards.
Why not 31-TET or 53-TET which are simpler?
@@abramthiessen8749 What is "simpler" is a matter of perspective. In an ideal world, musical compositions would be written first, and its musical instruments would be suitably designed second. I don't see that happening anytime soon though, despite any current advances that might be making it more of a possibility than in the past.
The 31-TET and 53-TET systems could both be considered more complicated, from the perspective that they require obsoleting the 12-TET system we've already established as the universal standard we're used to. Unlike when advancing our EDO to being 72, we would no longer be able to persist with our use of it being 12. Why not choose a cake you can *have* AND can *eat* ?
Alas, 31 and 53 are prime numbers...verses 72 having 12 factors, which allows for plenty of other EDOs to be used (e.g., 8, 9, 18, 24, 36, etc.)!!
If you enjoyed this, please refer to the music of Ben Johnston (1926-21/7/2019) and his just intonation, especially his 10 string quartets. The world has yet to recognize his real contributions to music. And we are talking INSTRUMENTAL music, not electronic. Enjoy!
Excellent, loved the video, thanks!
I like this channel very much.
Very enlightening!
Excellent - thank you so much
5:20 those triads sound like cars honking
Great explanation!
Your channel is seriously epic.
6:09 YOYOYO IT'S THE THING! 1-6-2-5! My favorite progression
Do not forget the broader culture of the times. For a Renaissance Italian to say some performance reached almost to heaven is not a negative criticism. It probably implies the best any human could expect to hear this side of heaven and is based on necessary humility regarding statements about the divine. A true celestial harmony would be impossible for mere mortals.
Of course, if the author was concerned with theoretical concepts such as the divine music of the spheres, this could also come into play. However, this sounds to me like a supreme compliment rather than a backhanded one.
2:46. I'd like remark that, instead, by going eight perfect fifths(3/2) down from C you'll reach a Fb( 8192/6561) only a Schsima (32805/32768) down from a just 5/4 and a Pythagorean comma(531441/524288) down from the E(81/64) you find going up four perfect fifths. The same is true for the D#(19683/16384) you find going up nine perfect fifths from C. It is a shisma higher then a 6/5 just minor third and a comma higher then the Eb( 32/27) you get foing 3 fifths down from C. All this just to say that in a merely pythagorean setting you can get note that are almost pure thirds, a schisma measuring approximately 1.95 cents . A small differce far far smaller then the one we are used to in 12 et. Perhaps this just pyth close to just small integer ratios thirds are not there in every key(this depends on how long you want your fifths spiral to go) . But they are cool the same
Thanks for the video! It was very informative and easy to understand :)
i love this channel!!!
Note that Maqam players do NOT strive for just intonation, yet they are VERY particular about where the notes lie.
i love your channel; it doesn't give me all answers to questions i have been wondering about for years now but it gives me the optimistic dream that in the future, with the help of (rare) people like you we will have instruments that could aim for "real consonance" and thus really " surround" natural voices - I mean with real thirds and fifths etc., -that would be paradise for my old offended ears... Your research is not only about the past, it is also about the future, because, as far as I know, physics laws and resonances etc. won't change in the near future. We might as well go back to the infinite realm of colours that's inside any single note, instead of limiting our perceptions to the binary ( 2 exp 12) world of equal temperament. We (as a legacy of musicians) discovered the 12 tone temp to be able to travel round the keys, but then we lost the "world in between", a world so colourful that our feelings feel colourless without it... You research might be one of the past jumping into the future. I hope I'll live long enough to hear it without being (happily) stuck to subtle old music ensemble recordings. The final goal, after all, is to let us hear, enjoy and feel the sounds of the humankind of all times - and maybe more, but I'll be happy with "just" that ! Historical research and understanding is the key. And you're one of those lockpicking artists my middle-aged exiled ears love to learn from.
Well done.
Great video!!
fascinating, thank you!
A guy with such a t-shirt can't tell bullshit. Very-well put, and hexagons helped to see complex ideas!
Gold. All gold.
Great video.
8:01 I don't know why, but I laughed very hard at this little "conversation" between the painting people!
I wish this series was around when I was studying this stuff in college...
Fantastic video, so well done and informative, thank you! Love the meme- and emojji-punctuation as well :) :)
Love the pig. That’s my intonation exactly. Cool video.
Great!!
I had a four harp harpsichord. I tuned two to equal, one so flats were flat (would sound wrong as a sharp), and one as sharp. As long as one did not use them together, it was very nice to hear.
You got my like with that shirt
6:00 It's funny how you call this a problem, but when anyone talks about Jacob Collier's "In the Bleak Midwinter" arrangement, they say it's genius, because he takes in all the way to G half sharp.
This channel is awesome! and very schwifty after all!
Omg So clearly enlightened
great video!
Excellent and funny !
I'm not sure I agree with "True just intonation is a forbidden fruit we are only allowed to dream of." I've thought about it quite a bit, and thought of a solution that may work.
You can still justly tune each individual chord as a piece progresses. While the pitch may shift too high or too low, you can avoid that by utilizing equal temperament as the basis for the root of each chord (FYI, this would mostly apply to singers and wind instrument players, as other instruments likely have a fixed tuning).
For example, at 6:00 in the video, instead of shifting the bass notes of each chord down, you could simply use the equal tempered versions as the root of the chord (since all the chords are in root position) and adjust the rest of the chords as needed. So the A in the first chord would be lowered for the major third in F major, then shifts up to become the 5th of the D minor chord, and so on. This way, the root of each chord will always have a set tuning, even in inverted chords.
I guess you can argue utilizing this method may not be "true" just intonation, but I think this definitely solves the pitch shifting too high or too low. The musicians would just need to know which chord member they are playing and adjust each chord accordingly. As long as the root of the chord is always stable, the rest of the chord can easily be tuned.
Ever since learning about just intonation, this is how I've envisioned how every ensemble should sound like, combining the best of both equal temperament and just intonation.
Let me know your thoughts and critiques of this idea!
One of my choir leaders complained over me using to much of early intervals when singing :-) I have my base in renaissance music in my younger days. Thanks for interesting videos which I'm enjoying as an amateur musician and singer.
Change is constant.
Renaissance music was conceived and understood as the simultaneity of melodic lines, not as harmonies. Plus, they worked in modes, not harmonic keys as we think of them. This linear approach is vital and completely changes the questions around tuning. See Anne Smith, “The Performance of 16th Century Music,” in which she discusses the nature of each mode, the importance of a linear approach, and a great many other well-researched and well-practiced aspects of this wonderful period. Don’t apply a modern ear to post-medieval (so to speak) music. All of that said, it’s fascinating to see this notion pushed to its logical conclusion. Thanks!
wow you guys are adorable
Very nicely done and very useful. (I'll have to find out about levels and commas, though.)
If that's not 3 geeks who love music, I don't know what is!!! Thank you gentlemen and God bless!!
i miss read geeks to greaks and was about to 'correct' you. hahahahaa
Very interesting..
Very interesting. Long ago I tested myself playing the tenor saxophone and always on the same note I used a mid-tone second to remain 'pure' in the rest of the melody.
The concluding purpose of your highly informative and amusing video is probably best expressed by that crazy cat on piano!
7:38 i fall in the first category
I think, I just, now have an explanation why our Indian music (and other devotional music of the early centuries) - which is based on the tuning of pure 5ths (the drone of the tanpura) should not be attempted for harmonization rather than singing only in unison to bring out the "almost" celestial harmony. Thanks to EMS for such a detailed and enjoyable presentation of the situation that prevailed in my mind. I subscribed the channel long time ago, but viewed this only today. How can I help EMS.
Regards
Rupam J.B
This is the best description of the difference between different intervallic philosphies. I know almost nothing abouy music theory. And this is wonderful!
Ps I prefer melodies to harmonies/chords. So "just intonaton" (as impossible it may be) is much closer to my personal preference.
Actually if you look at some of the historical writings, like vallotti, you find a lot of mention of the major whole tone (9/8) an minor whole tone (10/9). This is how C-D-E sound like in just intonation. In fact mean tone intonation is called 'mean' tone because it averages out the major and minor whole tones and makes them equal.