{"id":9021,"date":"2023-10-13T14:59:19","date_gmt":"2023-10-13T14:59:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=9021"},"modified":"2025-08-29T20:36:28","modified_gmt":"2025-08-29T20:36:28","slug":"math-in-music-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/math-in-music-learn-it-2\/","title":{"raw":"Math in Music: Learn It 2","rendered":"Math in Music: Learn It 2"},"content":{"raw":"<h2>Note Values, Half-Steps, Whole Steps, and Octaves<\/h2>\r\n<p>Moving our exploration to note values, the frequency of all notes is well defined by a specific and unique frequency for each note that is measurable. We will explore keys on a keyboard to discuss notes that have the same relationships with any instrument or musical piece.<\/p>\r\n<center>\r\n[caption id=\"attachment_9026\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-9026\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3-300x200.png\" alt=\"Piano keys and notes. The white keys are labeled A B C D E F G A. The black keys are labeled A flat and G sharp, B flat and A sharp, D flat and C sharp, E flat and D sharp, G flat and F sharp, A flat and G sharp, B flat and A sharp.\" width=\"500\" height=\"334\" \/> Figure 1. Labeled piano keys[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Let\u2019s look at the figure above. The white keys are labeled with the letters [latex]A\u2013G[\/latex] and the photo begins with middle [latex]C[\/latex], which can be found in the middle of a keyboard. This labeling of the keys repeats across an entire keyboard and keys to the right have a higher pitch and frequency than keys to the left. Each of the keys correlates to a musical note.<\/p>\r\n<p>Movement up or down between any two consecutive keys (black and white) or notes constitutes a <strong>half-step<\/strong>. Movement of one half-step sometimes involves a <strong>sharp <\/strong>([latex]\\#[\/latex]) or a <strong>flat <\/strong>([latex]\u266d[\/latex]) symbol. For example, [latex]D^\\#[\/latex] is one half-step above [latex]D[\/latex] and [latex]D^\u266d[\/latex] is one half-step below [latex]D[\/latex]. Note that this is not always true as one half-step above [latex]B[\/latex] is [latex]C[\/latex], and one half-step below [latex]F[\/latex] is [latex]E[\/latex]. In similar fashion, a <strong>whole step<\/strong> is movement up or down between any two half-steps on a keyboard.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>half-step, sharp, flat, and whole step<\/h3>\r\n<p><strong>Half-Step: <\/strong>A half-step is the smallest musical interval between two adjacent keys on a keyboard, often denoted by a sharp ([latex]\\#[\/latex]) or a flat ([latex]\u266d[\/latex]) symbol.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><strong>Sharp:<\/strong> A sharp ([latex]\\#[\/latex]) is a musical symbol that raises the pitch of a note by a half-step.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><strong>Flat:<\/strong> A flat ([latex]\u266d[\/latex]) is a musical symbol that lowers the pitch of a note by a half-step.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><strong>Whole Step:<\/strong> A whole step is a musical interval that consists of two half-steps, representing a larger distance between notes on a keyboard.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Name which keys are one half-step up and one half-step down from the following:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]D[\/latex]<\/li>\r\n\t<li>[latex]E[\/latex]<\/li>\r\n\t<li>[latex]G^\\#[\/latex]<\/li>\r\n<\/ol>\r\n\r\n[reveal-answer q=\"4331\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4331\"]\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>up [latex]D^\\#[\/latex], down [latex]D^\u266d[\/latex]<\/li>\r\n\t<li>up [latex]F[\/latex], down [latex]E^\u266d[\/latex]<\/li>\r\n\t<li>up [latex]A[\/latex], down [latex]G[\/latex]<\/li>\r\n<\/ol>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">\r\n<p>Name which keys are one whole step up and one whole step down from the following:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]F^\\#[\/latex]<\/li>\r\n\t<li>[latex]E[\/latex]<\/li>\r\n\t<li>[latex]A^\u266d[\/latex]<\/li>\r\n<\/ol>\r\n\r\n[reveal-answer q=\"4332\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4332\"]\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>up [latex]G^\\#[\/latex], down [latex]E[\/latex]<\/li>\r\n\t<li>up [latex]F^\\#[\/latex], down [latex]D[\/latex]<\/li>\r\n\t<li>up [latex]B^\u266d[\/latex], down [latex]G^\u266d[\/latex]<\/li>\r\n<\/ol>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13359[\/ohm2_question]<\/p>\r\n<\/section>\r\n<p>You may have noticed that there are eight letters of the alphabet used to label notes. Selecting any one note and counting up [latex]12[\/latex] half-steps you will find that the numbering for notes begins at the same value as you started from. This collection of [latex]12[\/latex] consecutive half-notes is called an <strong>octave <\/strong>and is a basic foundational component in music theory.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>octave<\/h3>\r\n<p>An <strong>octave <\/strong>serves as a foundational unit in music theory, encompassing 12 half-steps and marking the point where the note names repeat but at a higher pitch.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The octave is significant because it represents a doubling of frequency, creating a harmonious and consonant relationship between the starting and ending notes.<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>Let's look at an example.<\/p>\r\n<section class=\"textbox example\">\r\n<p>List the [latex]12[\/latex] notes forming an octave, beginning with the note [latex]C[\/latex].<\/p>\r\n<center>[latex]C, C^\\#, D, D^\\#, E, F, F^\\#, G, G^\\#, A, A^\\#, B[\/latex]<\/center><\/section>\r\n<p>Try it yourself.<\/p>\r\n<section class=\"textbox example\">\r\n<p>List the [latex]12[\/latex] notes forming an octave, beginning with the note [latex]G[\/latex].<\/p>\r\n\r\n[reveal-answer q=\"4333\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4333\"]<center>[latex]G, G^\\#, A, A^\\#, B, C, C^\\#, D, D^\\#, E, F, F^\\#[\/latex]<\/center>[\/hidden-answer]<\/section>","rendered":"<h2>Note Values, Half-Steps, Whole Steps, and Octaves<\/h2>\n<p>Moving our exploration to note values, the frequency of all notes is well defined by a specific and unique frequency for each note that is measurable. We will explore keys on a keyboard to discuss notes that have the same relationships with any instrument or musical piece.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_9026\" aria-describedby=\"caption-attachment-9026\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-9026\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3-300x200.png\" alt=\"Piano keys and notes. The white keys are labeled A B C D E F G A. The black keys are labeled A flat and G sharp, B flat and A sharp, D flat and C sharp, E flat and D sharp, G flat and F sharp, A flat and G sharp, B flat and A sharp.\" width=\"500\" height=\"334\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3-300x200.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3-1024x684.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3-768x513.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3-1200x802.png 1200w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3-65x43.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3-225x150.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3-350x234.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/13151333\/ed57794ae8171aaab4ad4a4a0c319d65acfe68e3.png 1251w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-9026\" class=\"wp-caption-text\">Figure 1. Labeled piano keys<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Let\u2019s look at the figure above. The white keys are labeled with the letters [latex]A\u2013G[\/latex] and the photo begins with middle [latex]C[\/latex], which can be found in the middle of a keyboard. This labeling of the keys repeats across an entire keyboard and keys to the right have a higher pitch and frequency than keys to the left. Each of the keys correlates to a musical note.<\/p>\n<p>Movement up or down between any two consecutive keys (black and white) or notes constitutes a <strong>half-step<\/strong>. Movement of one half-step sometimes involves a <strong>sharp <\/strong>([latex]\\#[\/latex]) or a <strong>flat <\/strong>([latex]\u266d[\/latex]) symbol. For example, [latex]D^\\#[\/latex] is one half-step above [latex]D[\/latex] and [latex]D^\u266d[\/latex] is one half-step below [latex]D[\/latex]. Note that this is not always true as one half-step above [latex]B[\/latex] is [latex]C[\/latex], and one half-step below [latex]F[\/latex] is [latex]E[\/latex]. In similar fashion, a <strong>whole step<\/strong> is movement up or down between any two half-steps on a keyboard.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>half-step, sharp, flat, and whole step<\/h3>\n<p><strong>Half-Step: <\/strong>A half-step is the smallest musical interval between two adjacent keys on a keyboard, often denoted by a sharp ([latex]\\#[\/latex]) or a flat ([latex]\u266d[\/latex]) symbol.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Sharp:<\/strong> A sharp ([latex]\\#[\/latex]) is a musical symbol that raises the pitch of a note by a half-step.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Flat:<\/strong> A flat ([latex]\u266d[\/latex]) is a musical symbol that lowers the pitch of a note by a half-step.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Whole Step:<\/strong> A whole step is a musical interval that consists of two half-steps, representing a larger distance between notes on a keyboard.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Name which keys are one half-step up and one half-step down from the following:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]D[\/latex]<\/li>\n<li>[latex]E[\/latex]<\/li>\n<li>[latex]G^\\#[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4331\">Show Solution<\/button><\/p>\n<div id=\"q4331\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>up [latex]D^\\#[\/latex], down [latex]D^\u266d[\/latex]<\/li>\n<li>up [latex]F[\/latex], down [latex]E^\u266d[\/latex]<\/li>\n<li>up [latex]A[\/latex], down [latex]G[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Name which keys are one whole step up and one whole step down from the following:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]F^\\#[\/latex]<\/li>\n<li>[latex]E[\/latex]<\/li>\n<li>[latex]A^\u266d[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4332\">Show Solution<\/button><\/p>\n<div id=\"q4332\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>up [latex]G^\\#[\/latex], down [latex]E[\/latex]<\/li>\n<li>up [latex]F^\\#[\/latex], down [latex]D[\/latex]<\/li>\n<li>up [latex]B^\u266d[\/latex], down [latex]G^\u266d[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13359\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13359&theme=lumen&iframe_resize_id=ohm13359&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>You may have noticed that there are eight letters of the alphabet used to label notes. Selecting any one note and counting up [latex]12[\/latex] half-steps you will find that the numbering for notes begins at the same value as you started from. This collection of [latex]12[\/latex] consecutive half-notes is called an <strong>octave <\/strong>and is a basic foundational component in music theory.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>octave<\/h3>\n<p>An <strong>octave <\/strong>serves as a foundational unit in music theory, encompassing 12 half-steps and marking the point where the note names repeat but at a higher pitch.<\/p>\n<p>&nbsp;<\/p>\n<p>The octave is significant because it represents a doubling of frequency, creating a harmonious and consonant relationship between the starting and ending notes.<\/p>\n<\/div>\n<\/section>\n<p>Let&#8217;s look at an example.<\/p>\n<section class=\"textbox example\">\n<p>List the [latex]12[\/latex] notes forming an octave, beginning with the note [latex]C[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]C, C^\\#, D, D^\\#, E, F, F^\\#, G, G^\\#, A, A^\\#, B[\/latex]<\/div>\n<\/section>\n<p>Try it yourself.<\/p>\n<section class=\"textbox example\">\n<p>List the [latex]12[\/latex] notes forming an octave, beginning with the note [latex]G[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4333\">Show Solution<\/button><\/p>\n<div id=\"q4333\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]G, G^\\#, A, A^\\#, B, C, C^\\#, D, D^\\#, E, F, F^\\#[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Contemporary Mathematics\",\"author\":\"Donna Kirk\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/13-4-math-and-music\",\"project\":\"13.4 Math and Music\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/1-introduction\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":8095,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Contemporary Mathematics","author":"Donna Kirk","organization":"OpenStax","url":"https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/13-4-math-and-music","project":"13.4 Math and Music","license":"cc-by","license_terms":"Access for free at https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/1-introduction"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/9021"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":16,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/9021\/revisions"}],"predecessor-version":[{"id":15939,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/9021\/revisions\/15939"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/8095"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/9021\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=9021"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=9021"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=9021"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=9021"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}