(Originally posted in March, 2010)<\/p>\n
If you just thought this title didn\u2019t make sense, no, it does not make sense, at least to you and me. \u00a0Perhaps some of you can help me understand this stuff better \u2014 I don\u2019t have a mathematical mind and I can only go so far as to understand the temperament issues, but I don\u2019t think I\u2019m too stupid to understand the basics.\u00a0 Some of those claims out there doesn\u2019t quite make sense to me….<\/p>\n
Now, those of you who are thinking you\u2019ll be in over your head talking temperament, you\u2019ve got to stay with me. \u00a0 Let\u2019s clarify what the mean-tone temperament is.\u00a0 Well, if we are to polarize the temperament systems with their characters, we have melodic systems on one side and harmonic systems on the other.\u00a0 The Pythagorean system is basically a stack of pure (just intonation) perfect 5ths, and because of the sharp mediants and leading tones it has, it is a great melodic temperament for monophonic lines.\u00a0 if you play something unaccompanied on a violin-family instrument with perfect 5th tuning, it is likely that you are playing in a temperament that\u2019s pretty close to the Pythagorean.\u00a0 But because of pretty wide major 3rds that you find in this system, this isn\u2019t a great temperament harmonically \u2014 basic triads in this system would sound pretty ugly.\u00a0 If you can obtain pure (just intonation) major 3rds in this temperament, chords would sound much nicer!<\/p>\n
So, how are we going to incorporate narrower and pure 3rds into the stack of perfect 5ths?\u00a0 Let\u2019s figure out what the difference between pure perfect 5th stack from C to E and pure major 3rd interval C to E is.<\/p>\n
Just intonation is natural physics.\u00a0 The frequencies of the diatonic notes in just tuning are related by ratios of small whole numbers.<\/p>\n
Frequency ratio 2:1 = octave; 3:2 = perfect 5th; 4:3 = perfect 4th;<\/p>\n
5:4 = major 3rd; 6:5 = minor 3rd….\u00a0 and so on.<\/p>\n
If we are to get the interval of a pure perfect 5th in cents (1 octave = 1200 cents, in equal temperament one semitone is 100 cents; perfect 5th is 700 cents):<\/p>\n
log (3:2) x 1200 \u00f7 log (2) = 701.955 cents\u00a0 (don\u2019t ask me about logarithm!)<\/p>\n
In order for us to get to an E just stacking up perfect 5ths from the C in the cycle of 5ths, we need four 5ths.<\/p>\n
C \u2014 G, G \u2014 D, D \u2014 A, A \u2014 E<\/p>\n
701.955 + 701.955 + 701.955 + 701.955 = 2807.82 cents<\/p>\n
Pythagorean E is 2807.82 cents away from the bottom of the stack C.\u00a0 The interval between the bottom C and this E is a major 17th.\u00a0 Why don\u2019t we flatten this interval and make it a simple major 3rd?<\/p>\n
2807.82 \u2013 2400 (two octaves) = 407.82 cents<\/p>\n
Now, let\u2019s get the pure (just intonation) major 3rd in cents:<\/p>\n
log (5:4) x 1200 \u00f7 log (2) = 386.3137….<\/p>\n
To compare this with the Pythagorean C-E interval:<\/p>\n
Pythagorean major 3rd [407.82] \u2013 Just intonation major 3rd [386.31] = 21.51 cents<\/p>\n
Tada!\u00a0 This is the syntonic comma.\u00a0 As you remember, in the cycle of 5ths, we needed to stack up four perfect 5ths to get to the E.\u00a0 So, if you make each of those 5ths narrower by a 1\/4 syntonic comma, you get the harmonically happy major 3rd from C to E.\u00a0 And if you keep narrowing the rest of 5ths by 1\/4 commas, that\u2019s the mean-tone temperament.\u00a0 That\u2019s why, in the specific sense, mean-tone IS the 1\/4-comma mean-tone, and when people just say \u201cmean-tone,\u201d it means the 1\/4-comma mean-tone specifically.<\/p>\n
Now you understand why the title of this entry doesn\u2019t make sense.\u00a0 The mean-tone tuning is a syntonic-comma-based system, not a Pythagorean-comma-based system.\u00a0 But some piano workshop website has the instruction for tuning in 1\/6-Pythagorean-comma mean-tone IN ADDITION TO 1\/6-comma modified mean-tone and 1\/6-Pythagorean-comma (Vallotti & Young) temperaments…<\/p>\n
What is the broader definition of the term \u2018mean-tone?\u2019 here anyway…?<\/p>\n","protected":false},"excerpt":{"rendered":"
(Originally posted in March, 2010) If you just thought this title didn\u2019t make sense, no, it does not make sense, at least to you and me. \u00a0Perhaps some of you can help me understand this stuff better \u2014 I don\u2019t have a mathematical mind and I can only go so far as to understand the temperament issues, but I don\u2019t think I\u2019m too stupid to understand the basics.\u00a0 Some of those claims out there doesn\u2019t quite make sense to me…. Now, those of you who are thinking you\u2019ll be in over your head talking temperament, you\u2019ve got to stay with me. \u00a0 Let\u2019s clarify what the mean-tone temperament is.\u00a0 Well, […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"gallery","meta":{"nf_dc_page":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[15,5],"tags":[50,67,77,88,104,105],"class_list":["post-85","post","type-post","status-publish","format-gallery","hentry","category-early-music","category-essays","tag-frequency-ratio","tag-just-intonation","tag-meantone","tag-pythagorean-comma","tag-syntonic-comma","tag-temperament","post_format-post-format-gallery"],"_links":{"self":[{"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/posts\/85","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/comments?post=85"}],"version-history":[{"count":0,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/posts\/85\/revisions"}],"wp:attachment":[{"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/media?parent=85"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/categories?post=85"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/tags?post=85"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}